Communication and Behavior in Centralized and Decentralized
Communication and Behavior in Centralized and Decentralized Coordination Games Piotr Evdokimov∗ and Umberto Garfagnini†
November 23, 2015
Abstract Using novel experimental methods, we study how communication is used to coordinate in centralized and decentralized games where players’ incentives are private and misaligned. We find that subjects respond to changes in incentives to coordinate strategically. In particular, the quality of vertical communication with the principal is significantly higher than the quality of horizontal communication between agents if and only if the importance of coordination is low. Surprisingly, decisions in centralized games underweigh and decisions in decentralized games overweigh the importance of coordination. These distortions in decision rules account for 94% of subjects’ welfare losses, with the remaining 6% due to miscommunication. The distortions can be explained by attitudes toward ambiguity and risk and disappear in an additional experiment with complete information. JEL Classification: C70, D03, C92. Keywords: communication, coordination, decentralization, experiment
∗
ITAM. University of Surrey. This paper was previously circulated as “Incentives to Coordinate in Organizations” and “Mend Your Speech a Little: Authority, Communication, and Incentives to Coordinate.” We are grateful to Wouter Dessein, Daniel Friedman, Felipe Meza, Ryan Oprea, Joel Sobel, Alistair Wilson, and seminar participants at the ITAM Theory Workshop, the University of Minnesota Microeconomic Theory Workshop, the London Experimental Workshop 2015, M-BEES 2015, and the Econometric Society World Congress 2015 for helpful comments. We also thank Luis Aguirre and Alfredo Rubio for excellent research assistance. Financial support from the Asociaci´ on Mexicana de Cultura A.C. is acknowledged. †
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1
Introduction
Firms coordinate production decisions across divisions; districts in federal systems coordinate policies; international organizations coordinate decisions across countries. Coordination problems play an important role in economic activity, and economists’ interest in them has a long history.1 Decision makers in these problems often have privately known motives affecting their incentives to coordinate.2 In the presence of incomplete information, it is well known that coordination can be facilitated by delegation of authority to the best informed parties3 and by information sharing through communication.4 While much theoretical work has addressed the question of how communication is used in centralized and decentralized coordination problems with incomplete information, the question has not been explored in any existing empirical or experimental study. Our work unifies the experimental literatures on coordination, delegation of authority, and strategic communication to achieve three basic goals. First, we quantify how the quality of communication responds to changes in incentives to coordinate under centralization and decentralization. Second, we explore how subjects make decisions conditional on the communicated information. Third, we assess quantitative deviations from optimal behavior, provide explanations for the observed coordination failures, and show that they are reduced in an otherwise similar setting with complete information. In doing so, we make several methodological contributions to the experimental cheap talk literature. First, we elicit subjects’ beliefs about their matched subjects’ states and use the elicited beliefs to construct an empirical counterpart to the residual variance of communication, a measure commonly used in theoretical work.5 This allows us to formulate predictions about how well subjects communicate without relying on assumptions about how they do it. Second, we use the elicited beliefs together with the equilibrium decision 1
See, for example, Hayek (1945); Chandler (1977); Aoki (1986). Several literatures build on this insight. For example, Carlsson and Van Damme (1993) apply this idea in the context of global games; Baliga and Sj¨ostr¨om (2004) in the context of games of conflict; Dessein and Santos (2006) in the context of organizational economics. 3 See, for example, Milgrom and Roberts (1992) for a discussion of the “delegation principle.” 4 See Alonso et al. (2008), Rantakari (2008), Dessein et al. (2010a), Hagenbach and Koessler (2010), Alonso et al. (2015). 5 We also perform robustness checks of our results that do not rely on belief elicitation. Theoretically, communication quality is measured as the residual variance of the message receiver’s posterior around the sender’s privately known state. I.e., letting θi denote player i’s state and mi theh player’s message i 2 about the state to the receiver, the residual variance of communication is defined as E (θi − E[θi |mi ]) . In contrast, other studies of cheap talk games, such as Cai and Wang (2006), have used the correlation between messages and states to measure the quality of communication. 2
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rules to study how subjects decide conditional on the communicated information.6 This allows us to test theoretical predictions about subjects’ decision rules directly. Third, we use the elicited beliefs to perform a detailed welfare analysis that decomposes subjects’ welfare losses into a component due to miscommunication and a component due to deviations from equilibrium behavior. Our experiment uses a 2x2 design that independently manipulates the structure of a coordination game (centralized vs. decentralized) and the importance of coordination for individual payoffs (high vs. low). A decentralized game is played between two agents, with each agent having to make a decision.7 Each agent has private information about his local conditions, which affect the payoff the agent receives from his own decision. The agent incurs an adaptation loss if his decision fails to adapt to his local conditions, and a coordination loss if his decision is not perfectly aligned with the decision of the other agent. He therefore faces a trade-off between adaptation and coordination. The agents can communicate with each other before making their decisions. In a centralized coordination game, decision rights are delegated to an unbiased coordinator, referred to as the principal, who maximizes joint profits and is uninformed about both local conditions. Agents can communicate their private information to the principal before the decisions are made. In both types of games, communication is informal and occurs through cheap talk. The first of our main findings is that the quality of (vertical) communication with the principal is significantly higher than the quality of (horizontal) communication between agents if and only if the importance of coordination is low. This result is remarkably consistent with the theoretical predictions of Alonso et al. (2008) and Rantakari (2008). In theory, the fact that the principal maximizes joint payoffs implies that the agents’ incentives are more aligned with the principal than they are with each other. Thus, the quality of vertical communication should always be higher than that of horizontal communication.8 As the importance of coordination grows, the agents’ incentives become more aligned with each other but less aligned with the incentives of the principal, and the difference in quality of vertical and horizontal communication should shrink to zero.9 This is what we observe 6
As discussed in Section 6.2.1, we also use other proxies for the message receivers’ posteriors. In applications of the model, an agent could be a manager in charge of a division or a function within a firm, a local district, a state government, etc. 8 It is well-known that cheap talk games admit a multiplicity of equilibria, including an equilibrium which is completely uninformative. The theoretical literature usually focuses on the most informative equilibrium from an ex-ante perspective. We follow the approach used in the literature because it allows to formulate a clear set of equilibrium predictions about communication quality. 9 An increase in the importance of coordination requires the agents to coordinate their decisions more to avoid large losses from miscoordination. This implies that agents have lower incentives to strategically 7
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in the data. Predictions about communication quality underlie predictions about coordination. The second of our main findings is that behavior responds to changes in incentives to coordinate roughly as theory predicts. Taking subjects’ (elicited) posterior beliefs as given, we estimate the coefficients on states and posteriors in subjects’ decision rules under the null hypothesis of equilibrium behavior.10 Both under centralization and decentralization, we find that the decision rules show significantly more coordination and significantly less adaptation when the incentives to coordinate are greater. The third of our main findings is that subjects’ decision rules show significant quantitative deviations from equilibrium. The agent systematically underweighs the importance of adaptation to himself, while the principal systematically underweighs the importance of coordination for the maximization of joint payoffs. As a result, decisions under decentralization tend to be more coordinated (and thus less adapted to local conditions) than decisions under centralization, despite the fact that the principal could improve performance by coordinating the agents’ decisions more. Using subjects’ recovered beliefs together with the equilibrium decision rules, we perform a structural estimation exercise to quantify the extent to which subjects’ decisions are distorted away from the theoretical optima.11 We also estimate the effect of these distortions on welfare. We find that subjects’ observed welfare losses amount to approximately 3.75 times the losses predicted by the payoff-maximizing benchmark. Moreover, 94% of the difference between optimal and observed losses can be explained by distortions in decision rules, with the remaining 6% due to miscommunication. Thus, the welfare cost of these distortions is large. Under decentralization, adaptation involves no uncertainty while coordination involves the other agent’s unknown decision. Under centralization, both decisions are made by the principal, and coordination involves no uncertainty, while adaptation involves the two agents’ unknown states. Thus, the decision maker overweighs the uncertain part of his manipulate their information. On the other hand, the principal cares less about the agents’ local conditions when coordination becomes more important. Therefore, the agents have higher incentives to misreport. 10 To our knowledge, no previous study of cheap talk games has applied belief elicitation in this manner. 11 Our design parameterizes the importance of coordination through a parameter γ which varies on a scale from 0 (only adaptation matters) to 1 (only coordination matters). We find that the principal in a centralized coordination game behaves as if coordination was irrelevant (ˆ γ ≈ 0) when the importance of coordination is low (γ = 0.25), and less than half as important as it should be (ˆ γ = 0.356) when the need to coordinate is high (γ = 0.75). Behavior of the agent in a decentralized coordination game exhibits the opposite pattern. Thus, the agent behaves as if coordination was twice as important as it should be when γ is low, and as if adaptation is almost irrelevant (ˆ γ = 0.9396) when γ is high.
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payoff function in both centralized and decentralized coordination games. In Section 7, we provide some possible underlying reasons for this overweighing. In particular, we show that risk-aversion can generate distortions of decision rules in the observed direction, and that the magnitudes of these distortions can be amplified by strategic uncertainty and ambiguity-aversion. To test the hypothesis that the distortions are generated by uncertainty, we run an additional experiment with complete information. In line with the hypothesis, we find no systematic distortions in decision weights in this second experiment. Some of our results stand in contrast to others reported in the cheap talk literature. Thus, unlike Cai and Wang (2006), we find no evidence of overcommunication in the experiment. Indeed, relative to the Most Informative Equilibrium (MIE) benchmark, we find evidence of undercommunication in one of our treatments. Importantly, the same treatment is also the only treatment in which we document significant losses in welfare as a result of subjects not communicating optimally. Overall, however, the welfare costs of undercommunication in the experiment are small. Our findings have important implications for the study and design of institutions. In particular, we highlight a novel hidden cost that is systematically related to uncertainty in coordination problems. While we focused attentions on coordination games in which decision rights are either completely centralized or completely decentralized, the results can be used to predict behavior in related settings. For example, in partially centralized organizations where the principal retains authority over some but not all of the decisions, we predict that the agents entitled with authority will still under-adapt their decisions to their local conditions. Our findings are also relevant to the design of political institutions. If policy coordination requires expertise on different dimensions, our results suggest that an incumbent politician may underestimate the importance of coordination and put excessive weight on the information provided by experts. Therefore, the value of expertise might have to be weighed against uncertainty-induced distortions in decision making. The remainder of the paper is structured as follows. Section 2 summarizes the related literature; Section 3 introduces the model; Section 4 discusses the theoretical predictions; Section 5 explains our experimental design; Section 6 describes the results; Section 7 provides a discussion of these results; Section 8 describes the results of the second experiment with complete information; Section 9 concludes.
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2 2.1
Literature Review Theoretical Literature
Since the seminal contributions of Green and Stokey (1980) and Crawford and Sobel (1982), the theoretical literature on strategic communication has grown significantly. The studies most related to ours are those that applied strategic communication to the study of organizations, such as Alonso et al. (2008), Rantakari (2008), and Dessein et al. (2010b). A line of literature dating back to Holmstr¨om (1977) has studied delegation of authority in organizations. Dessein (2002), Harris and Raviv (2005), and Alonso and Matouschek (2007) do this in a setting with cheap talk communication. A common thread in this literature is the “delegation principle,” which is the notion that decision power in a setting with private information should be granted or delegated to the most informed agent. While testing the delegation principle is not the purpose of our study, our design allows us to study how subjects behave in settings where the question of delegation is important. Another strand of the literature that relates to ours has studied communication with multiple senders. Battaglini (2002) shows that full revelation of information can be an equilibrium even when players’ preferences are arbitrarily misaligned. Ambrus and Takahashi (2008) shows that full revelation may not be achieved in equilibrium if the state space is restricted. These studies, however, focus on communication of private information about the same random variable. In our setting, the players communicate private information about their own states.
2.2
Experimental and Empirical Literature
Experimental economists have long been interested in coordination in the laboratory. Van Huyck et al. (1990) study a minimum-effort coordination game and show that large groups tend to choose the lowest effort level even after several repeated interactions. Using a similar game, Brandts and Cooper (2006) show that strengthening financial incentives can overcome coordination failures. More related to ours are studies exploring the role of communication as coordination device (see, e.g., Cooper et al., 1992; Blume and Ortmann, 2007). While in these studies subjects communicate information about their strategies, in our setting subjects communicate information about their privately known states.
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The experimental literature on cheap talk games is reviewed in Crawford (1998). This literature is large, and the strand most relevant to our study is that which focused on communication of private information in settings similar to Crawford and Sobel (1982), such as Dickhaut et al. (1995), Blume et al. (2001), Cai and Wang (2006), S´anchez-Pag´es and Vorsatz (2007), and Wang et al. (2010).12 Dickhaut et al. (1995) study communication in sender-receiver games, manipulating the degree to which players’ preferences diverge. They find that as preferences diverge less, more information is transmitted, results broadly in line with the Crawford and Sobel (1982) prediction. Cai and Wang (2006) find a similar result in a setting which allows for a more direct test of the theory. Their study also provides some evidence that participants can be broadly classified in two groups: those with a preference for truth-telling, and those driven by material incentives. S´anchez-Pag´es and Vorsatz (2007) provide additional evidence for this observation. Unlike previous studies of cheap talk games, we investigate the role of communication in coordinating subjects’ decisions. Two related papers are Palfrey and Rosenthal (1991) and Palfrey et al. (2015). Both papers study the effect of cheap talk communication on the efficient provision of a public good with heterogenous and privately known costs. However, our framework differs substantially from theirs. First, we focus on the impact of a higher need to coordinate and of the allocation of decision rights on agents’ behavior. These effects cannot be jointly tested in the context of a public good game. Second, incentives for information transmission in our experiment arise from a private adaptation motive, while those authors focus on free-riding. Finally, public good games endow each agent with an action, no contribution, which effectively insulates that agent from the actions chosen by other agents. By contrast, in our framework each agent is always affected by the action chosen by his partner, thus affecting both the incentives to communicate and coordinate. Therefore, the environment and the forces at work in our study are different from those in Palfrey and Rosenthal (1991) and Palfrey et al.(2015), albeit complementary. Heinemann et al. (2004) study coordination under incomplete information in the context of a global game where each subject receives a conditionally independent signal about the common state of the world. However, subjects face no private adaptation motive and no communication takes place. Qu (2013) experimentally investigates communication in a global game environment with private signals, but still no incentive misalignment, 12
While most experimental studies of cheap talk games consider correlations between messages, decisions, and states as measures of the informativeness of communication, in Vespa and Wilson (2015), the players’ actions should at least in theory be equal to their posteriors, making their measure of communication quality closer to ours. Their study, however, tests the predictions of Battaglini (2002) and is therefore only loosely related.
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similar to Heinemann et al., and finds that communication increases the likelihood of attack/investment. These experiments differ from ours because coordination failure always affects an agent’s payoff in our experiments, while in global games environments (as in the case of public goods) agents can decide to take an outside option.13 We also investigate both centralized and decentralized coordination games. Delegation of authority has been investigated experimentally by Lai and Lim (2012), Fehr et al.(2013), and Dominguez-Martinez et al. (2014). Of these, the former paper is most related to ours since its environment allows for cheap talk communication. Unlike all previous delegation studies, we investigate the role of communication in coordination decisions. We are also not interested in subjects’ decisions to delegate authority. We manipulate decision authority experimentally and instead ask the question of how communication quality and behavior depend on the degree of centralization of decision rights. A concurrent related study is Brandts and Cooper (2015). While they also compare centralized and decentralized coordination games, they do not test predictions about communication quality, and the games they use are substantially different from ours. For example, agents are symmetrically informed about each others’ local conditions as well as a global state of the world, which affects the payoffs of each player, while the principal is uninformed about the global state but informed about the agents’ local conditions. They focus on testing a payoff prediction similar to the one in Alonso et al. (2008) regarding the dominance of decentralization over centralization (under some conditions), and instead find that performance is generally higher under centralization. Unlike Brandts and Cooper (2015), we focus on communication quality and its effect on coordination. While we formulate predictions about adaptation and coordination losses, for reasons discussed in Section 4 we do not focus on predictions about performance of different types of organizations. Finally, coordination with incomplete information has been explored in recent empirical work (McElheran, 2014; Thomas, 2011), which has focused on the interaction between incentives to coordinate and the allocation of decision rights. While data on communication (e.g., through e-mail) can in principle be collected and analyzed, the laboratory provides the ideal environment to measure communication quality and its effect on decision making. 13
McDaniel (2011) studies cheap talk communication in an environment with one-sided incomplete information. Only one of two players knows which game is being played, a prisoner’s dilemma game or a stag-hunt game. The informed player can send a binary message to the uninformed player before actions are chosen. Therefore, subjects face uncertainty about whether the incentives of the informed party are misaligned. We instead study environments with two-sided incomplete information in which incentives are misaligned with probability one.
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3
Model
Our experimental design is based on the theoretical models of Alonso et al. (2008) and Rantakari (2008). We consider two different types of coordination games which we refer to as centralized and decentralized. Both types of games share the following features: (1) there are two decisions (d1 and d2 ) to be made; (2) there are two agents: Player 1, and Player 2; (3) the agents face a trade-off between coordinating those decisions and adapting each decision to some local conditions. In both games, the payoff of Player i, i = 1, 2, is given by πi = −(1 − γ)(di − θi )2 − γ(di − dj )2 ,
i = 1, 2,
i 6= j,
(3.1)
where θi is Player i’s state, or local conditions. The payoff of Player i has two components. The first component captures the adaptation loss which arises when the decision di is far from Player i’s local conditions, θi . The second component represents the possible coordination loss which increases in the distance between the two decisions. The parameter γ ∈ [0, 1] measures the importance of coordination for the players. The information structure is the following. Player 1 knows his own state, θ1 , but not Player 2’s state, θ2 . Similarly, Player 2 knows his own state, θ2 , but not Player 1’s state, θ1 . It is common knowledge that θ1 and θ2 are uniformly distributed on the interval [−1, 1].14 The states are drawn independently. Under decentralization, Player 1 makes decision d1 ∈ R and Player 2 simultaneously makes decision d2 ∈ R. Before the decisions are made, Player 1 sends a cheap talk message m1 ∈ M1 to Player 2, and Player 2 simultaneously sends a cheap talk message m2 ∈ M2 to Player 1. This is referred to as horizontal communication. Under centralization, there is an additional Player 3 (the principal), who chooses both d1 ∈ R and d2 ∈ R. Player 3 observes neither θ1 nor θ2 , and her payoff from the decisions is given by Π = π1 + π2 . Before Player 3 makes her two decisions, Player 1 and Player 2 simultaneously send cheap talk messages m1 ∈ M1 and m2 ∈ M2 to Player 3. This is referred to as vertical communication. 14
Alonso et al. (2008) and Rantakari (2008) consider a more general framework in which θi is drawn uniformly from [−si , si ], with si > 0.
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4
Predictions
4.1
Communication
A communication equilibrium is formally defined by communication rules for Player 1 and Player 2, decision rules for the decision makers, and belief functions for the message receivers such that the communication rules are optimal given the decision rules, the decision rules are optimal given beliefs, and the beliefs are derived from the communication rules using Bayes’ Rule whenever possible. It is well-known that games of strategic information transmission admit a multiplicity of such equilibria.15 We formulate our predictions about equilibrium communication around the Most Informative Equilibrium (MIE).16 This selection principle is appealing for two reasons: it maximizes ex ante payoffs and generates the elegant quantitative predictions described below. We take the MIE predictions as a useful guide for how incentives to coordinate affect communication and behavior in our coordination games, recognizing that similar incentives affect behavior in equilibria close to the most informative one, and interpret quantitative deviations as evidence against MIE. Alonso et al. (2008) show that any finite communication equilibrium is economically equivalent to a partitional one. Partitional equilibria share the intuitive property that the incentives to misrepresent information are increasing in |θi | under both centralization and decentralization.17 While studying the form of equilibrium communication is not the focus of our paper, this prediction is easily testable, and we explore it in Section 6.1. Following the theoretical literature, we measure the quality of communication through 15
In a setting similar to Crawford and Sobel (1982), any equilibrium is also economically equivalent to one in which the communication rules take a partitional form: a sender partitions the state space and only communicates which element of the partition the realized state belongs to. It is also well-known that an uninformative equilibrium always exists in such games. 16 In Crawford and Sobel (1982), only a finite number of messages can be sent in the Most Informative Equilibrium (MIE). In the framework of Alonso et al. (2008) and Rantakari (2008), however, a countably infinite number of messages can be sent because there exists a state where the preferences of the sender and receiver coincide. Recently, Chen et al. (2008) provide conditions which uniquely select MIE in cheap talk games. 17 Alonso et al. (2008) and Rantakari (2008) show that an arbitrary partitional equilibrium with N partition elements solves the following difference equation: ai,n+1 − ai,n = ai,n − ai,n−1 + 4bai,n .
(4.1)
This shows that moving away from θi = 0 in either direction leads to progressively larger partition elements.
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the residual variance of the posterior around the true value of the state, defined as E [(θi − E[θi |mi ])2 ]. The advantage of such a measure in experimental settings is that the residual variance of communication is defined independently of the partitional structure of equilibrium. Therefore, it can be used to measure communication quality whether or not players are conforming to any particular equilibrium or even exhibiting non-equilibrium behavior. Residual Variance Most Informative Equilibrium
1 12
Horizontal
1 21
Vertical Truth-telling equilibrium 0
Importance of coordination, γ
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Figure 1: Predicted communication quality as a function of γ. If γ ∈ (0, 1), it is shown in Alonso et al. (2008) that the residual variance of communication in MIE under decentralization is given by 1 E (θi − E[θi |mi ])2 = if i = 1, 2. (4.2) 12 + 9γ Under centralization, the residual variance of communication is given by γ E (θi − E[θi |mi ])2 = if i = 1, 2. 9 + 12γ
(4.3)
Figure 1 plots the residual variance of communication in MIE.18 18
Note that when coordination is irrelevant (γ = 0), it is an equilibrium to tell the truth about one’s state and set d1 equal to θ1 and d2 equal to θ2 . This is true in both the centralized and the decentralized game. Because the residual variance of communication in the truth-telling equilibrium is equal to zero, the residual variance of horizontal communication exhibits a discontinuity at γ = 0. Both residual variances also exhibit a discontinuity at γ = 1 in MIE, because truth-telling can be sustained in equilibrium when coordination is the only relevant task, given that private information has no value. In principle, these discontinuities may be behaviorally relevant. For example, it could be that when γ is low the players decide to play the game ignoring coordination, in which case full revelation is an equilibrium. This, however, is not observed in our data.
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Because the incentives of Player 1 and Player 3 are more aligned than the incentives of Player 1 and Player 2 (and the incentives of Player 2 and Player 3 are more aligned than the incentives of Player 2 and Player 1), the quality of vertical communication is greater than that of horizontal communication for any value of γ. However, the comparative statics in the two regimes are different. Under horizontal communication, a higher γ makes players’ preferences more aligned. As a result, the players misrepresent their states less and the quality of communication improves. Under centralization, a higher γ makes Player 3 care less about meeting the states of Player 1 and Player 2, who in response misrepresent their states more. Thus, a higher need for coordination worsens communication quality under centralization. This implies the following prediction: Prediction 1. As the importance of coordination, γ, increases, the quality of horizontal communication improves, while the quality of vertical communication declines. Notice that when γ is close to zero, the difference between the quality of vertical and horizontal communication is nearly maximal. As γ grows, this difference tends to zero, although a discontinuity is present at γ = 1 as noted above. This suggests the following: Prediction 2. When γ is small, the quality of vertical communication is significantly higher than that of horizontal communication. When γ is large, the difference in quality of horizontal and vertical communication is small. We note that Prediction 1 and Prediction 2 are robust to Player 1 and Player 2 having social preferences.19 Alonso et al. (2008) consider a variation of the model described above in which Player 1 maximizes λπ1 + (1 − λ)π2 and Player 2 maximizes (1 − λ)π1 + λπ2 , where λ ∈ 12 , 1 . Although the payoff functions used in the experiment set λ = 1, it is in principle plausible that Player 1 and Player 2 assign strictly positive weights to each other’s payoffs. However, for a fixed γ, it can be shown that the difference in quality of vertical and horizontal communication, while shrinking as λ approaches 12 , remains strictly positive. Similarly, for any fixed λ, the arguments regarding the effect of γ on the quality of communication remain valid. 19
Player 3 is already maximizing joint payoffs.
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4.2
Decisions
In any 20 communication equilibrium, players’ decisions depend on their beliefs about other players’ local conditions, which in turn depend on messages sent by the other players.21 Suppose that Player i sent a message mi . We denote by νi = E[θi |mi ] the posterior expectation about i’s state held by the receiver of the message. Under decentralization, Player i makes the following decision in equilibrium after receiving the message mj : dD i = (1 − γ)θi +
γ2 γ νi + νj , 1+γ 1+γ
i = 1, 2,
i 6= j.
(4.4)
Thus, each player’s decision rule is a linear function of his own state θi , his own posterior νj , and the other player’s posterior νi . In the experiment, we consider two parameters of γ: 14 and 34 . Let us consider the decision rule of Player i. When γ = 14 , the importance of coordination is low, and the weight that Player i assigns to his own state is high (and equal to 34 ). There is a small weight of 0.05 on the other player’s posterior about Player i’s state, νi , and a small weight of 0.2 on Player i’s own posterior about the other player’s state, νj . When the importance of coordination is high, Player i’s own state matters less (weight 41 ). The other player’s posterior about Player i’s state, νi , becomes more important (with a weight of ≈ 0.32), and there is a substantially larger weight on Player i’s posterior about the other player’s state, νj (≈ 0.43). This leads to the following prediction: Prediction 3. As the importance of coordination increases, the decision rule under decentralization puts a smaller weight on the state and larger weights on posterior beliefs. Under centralization, Player 3 makes the following decisions in equilibrium after receiving the message m = (m1 , m2 ) dC i =
1+γ 2γ νi + νj , 1 + 3γ 1 + 3γ
i = 1, 2,
i 6= j.
(4.5)
Notice that now the decision rules are only functions of the decision maker’s posteriors about the states of Player 1 and Player 2. When the importance of coordination is low, the posterior about Player i’s state has a much larger weight in determining di (≈ 0.71) than the posterior about the state of Player j (≈ 0.29). When the importance of coordination is high, the weights on the two posteriors are closer to each other (with a weight of ≈ 0.54 on own posterior). The following prediction is implied: 20
I.e., not only the most informative one. See Alonso et al. (2008) and Rantakari (2008) for derivations of the decision rules described in this section. 21
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Prediction 4. As the importance of coordination increases, the decision rule under centralization puts a smaller weight on the division’s own posterior and a larger weight on the posterior about the other division’s state.
4.3
Adaptation and Coordination
We can use the decision rules above to formulate predictions about average degrees of adaptation and coordination in the experiment. The advantage of centralization lies in the principal’s ability to perfectly control the degree to which the decisions are coordinated with each other. The principal, however, lacks complete knowledge of the agents’ local conditions, which makes adaptation difficult. On the other hand, the agents under decentralization can perfectly control the degree of adaptation of their decisions to their own local conditions. Coordination, however, is more difficult in this case because the agents only control their own decisions. Let CLk = E[(d1 − d2 )2 ], k ∈ {C, D}, denote the expected (normalized) coordination loss.22 In MIE, the principal’s comparative advantage at coordination generates a smaller coordination loss under centralization than under decentralization. Moreover, as coordination becomes more important, decisions become more correlated in both centralized and decentralized games, which leads to lower coordination losses.23 This implies the following predictions, proved in Proposition 1 of the appendix: Prediction 5. For both centralized and decentralized coordination games, an increase in the importance of coordination reduces the average coordination loss. Prediction 6. The average coordination loss is larger under decentralization than under centralization. Similarly, we can compute the expected (normalized) adaptation loss in MIE for an arbitrary agent i, denoted by ALik = E[(dki − θik )2 ], k ∈ {C, D}. The principal’s full internalization of the need to coordinate leads to larger adaptation losses under centralization than under decentralization. As coordination becomes more important, adaptation 22
This quantity is normalized by γ and hence does not represent actual utility or point losses. The model is sufficiently tractable to also allow the computation of correlations between decisions, and between decisions and states, in closed form. However, we base our predictions on coordination and adaptation losses (see below) because they are easier to test compared to predictions based on correlation coefficients. 23
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losses increase both under centralization and decentralization.24 This implies the following predictions, proved in Proposition 2 in the appendix: Prediction 7. For both centralized and decentralized coordination games, an increase in the importance of coordination increases the average adaptation loss of each agent. Prediction 8. The average adaptation loss is larger under centralization than under decentralization. We note that Prediction 5 through Prediction 8 are robust to social preferences, as proved in Lemma 2 of Alonso et al. (2008).
4.4
Payoffs
When γ = 0, the best that subjects can do in equilibrium under centralization is to communicate truthfully, set di = θi , and incur zero losses. Likewise, when γ = 1, the best the principal can do is to set the decisions equal to each other. This shows that the relationship between γ and output under centralization is non-monotonic. A similar argument holds for decentralization: the smallest losses are zero both when γ = 0 and when γ = 1. Nevertheless, it can be shown theoretically that centralization dominates decentralization if and only if γ > γ¯ ≡ 0.161.25 We do not emphasize this prediction, however, because it is much weaker than Predictions 1-8. In particular, the prediction is valid only if players are sufficiently self-interested. Indeed, it is the message of Alonso et al. (2008) that if the social preferences parameter λ is close to 12 , expected joint payoffs under decentralization can be higher that under centralization for any value of γ. Moreover, the difference in payoffs under centralization and decentralization is maximized at γ ≈ 0.45. Our experimental design is not optimized to capture differences in payoffs, and we leave this task to a future study. 24
This has no immediate implication for welfare because an increase in γ also implies that adaptation losses have a lower impact on welfare. 25 When γ ∈ [0, 1) and players only care about their own earnings, dividing the payoff functions (3.1) γ by (1 − γ) and setting 1−γ = δ, our payoff functions coincide with those of Alonso et al. (2008) for the case in which Player 1 and Player 2 only care about their own payoff. Alonso et al. (2008) show in the proof of their Proposition 5 that ΠC − ΠD > 0 if, and only if, 33δ 3 + 32δ 2 + 3δ − 2 > 0. This inequality is equivalent to 6γ 3 + 20γ 2 + 9γ − 2 > 0. γ¯ is the only positive root of the equation 6γ 3 + 20γ 2 + 9γ − 2 = 0. Intuitively, when γ is large, the players mostly care about coordination, and it is much easier for Player 3 to coordinate d1 and d2 (he or she can always set the two decisions equal to each other) than it is for Player 1 and Player 2, who need to rely on each other’s messages.
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5
Experimental Design
The experiment was conducted at Instituto Tecnol´ogico Aut´onomo de M´exico in Mexico City between October and November 2014 using the software z -Tree (Fischbacher, 2007). After entering the laboratory, sitting down at their computer terminals, and signing the consent forms, the subjects were distributed their treatment’s instructions.26 At the same time that the subjects were reading the instructions, a quiz was displayed on their computer screens. The subjects were informed that they have 20 minutes to read the instructions and complete the quiz. The quiz had 8 questions that were identical for all treatments, and 4 that differed across the decentralized and centralized treatments. It tested the subjects’ understanding of statistical independence, how they were to be matched in the experiment, the conversion of points to pesos, and the game’s basic structure. The answers to all of the quiz questions were incentivized: each subject gained one Mexican peso for each quiz question correctly answered. The questions are included in the online appendix. Each session consisted of two practice rounds followed by fifteen rounds which counted towards each subject’s earnings. Subjects were randomly and anonymously matched at the beginning of each round. The subjects’ earnings were determined as follows. Every subject was guaranteed a 30 Mexican pesos (≈ US$2) show up fee in addition to the earnings from the quiz. These earnings were called the subject’s “guaranteed earnings.” In addition, each subject was given 210 Mexican pesos (≈ US$15). In each (non-practice) round of the game, the decisions of each subject and his matched partners led the subject to lose a number of points. The subject’s “additional earnings” were determined as follows: Additional earnings = 210 − 3 × T otal points lost during the experiment. In addition to losing points from the game, the subjects lost points from their conjectures of other players’ states in accordance to a quadratic scoring rule. This “guessing game” is described in more detail below. Each subject’s total earnings were given by the sum of his/her guaranteed and additional earnings.27 It was explained to the subjects that any subject losing more than 50 cumulative points (150 Mexican pesos) would be excluded from further matches, and that in the event this happens, the remaining subjects will be rematched with each other, with some randomly chosen subjects sitting out in each subsequent match. In practice, this never happened during any of our sessions, although 26
See the online appendix at http://piotr-evdokimov.com/Appendix-Instructions.pdf. While the sample instructions are in English, the actual instructions were administered in Spanish. 27 The instructions provided subjects with several examples of final earnings as a function of points lost, and the quiz tested their understanding of the payment rules with yet another example.
16
the program we used allowed for the contingency.
5.1
Treatments
Our first experiment included four treatments: Decentralized-High, Centralized-High, Decentralized-Low, Centralized-Low.28 In the two Decentralized treatments, decision d1 was made by Player 1, decision d2 was made by Player 2, and each match consisted of two players. In the two Centralized treatments, each match had three players, and the decisions d1 and d2 were made by Player 3, whose points lost in every round were given by the average of the points lost by Player 1 and Player 2: π3 =
π1 + π 2 . 2
In the High treatments, the points lost by Player 1 and Player 2 in each round of the game were determined by the following formula: πi = −(di − θi )2 − 3 · (di − dj )2
i = 1, 2, i 6= j.
(5.1)
Thus, the High treatments placed a higher weight on coordinating d1 and d2 than on adapting to each state θi . The Low treatments placed a high weight on adaptation to θi : πi = −3 · (di − θi )2 − (di − dj )2
i = 1, 2, i 6= j.
(5.2)
In all of our treatments, the state, message, and decision spaces were restricted to be equal to each other. The θi , mi , and di variables were all selected in increments of 0.01 from the set {−1, −0.99, −0.98, ..., 0.98, 0.99, 1}. The decision space was restricted because allowing the decisions to be elements of R would have made it possible for a player who behaves randomly to sustain enormous losses, making the experiment infeasible. As several previous experimental studies, such as Cai and Wang (2006), we restricted the message space to be equal to the state space. This restriction can be motivated from the observation that subjects tend to interpret messages in cheap talk games using a natural language (see, e.g., Blume et al., 2001). Note that this restriction leaves the predictions described above unchanged. The timing in the Decentralized treatments proceeded as follows. First, the players i = 1, 2 were asked to simultaneously send their messages. Second, they were asked to 28
The second experiment is described below in Section 8.
17
simultaneously make their decisions. After the decisions were made, the players simultaneously made incentivized conjectures of each other’s states. At the end of each match, the players received feedback about the other player’s state, the other player’s decision, own points lost due to the decisions made, own points lost from the conjecture about the other player’s state, points lost in the round, points lost so far, and pesos lost so far. In the Centralized treatments, Player 1 and Player 2 started each match by sending their messages to Player 3. The screens they saw at this stage were identical to those displayed in the decentralized treatments. While these players decided what messages to send, Player 3 waited. After the messages were sent, Player 3 made his/her decisions. After the decisions were made, all players made conjectures about the states: Player 1 guessed θ2 ; Player 2 guessed θ1 , and Player 3 guessed both θ1 and θ2 . As in the decentralized treatments, these conjectures were incentivized. Finally, all players received feedback about the unknown state(s), Player 3’s decisions, own points lost from the decisions made, own points lost from the conjecture(s) about the other state(s), points lost in the round, points lost so far, and pesos lost so far. Subjects’ conjectures of each other’s states were obtained with quadratic scoring rules (Nyarko and Schotter, 2002). In both the centralized and the decentralized treatments, Player 1 guessed the state of Player 2, and Player 2 guessed the state of Player 1. For these players, the points lost for the guesses were equal to the square of the distance between the conjecture and the true value of the state being guessed. Formally, denote Player i’s conjecture about θj , conditional on having received message mj , by p(θj |mj ). The points lost for the conjecture are given by (p(θj |mj ) − θj )2 . In the centralized treatments, Player 3 also guessed the states of both Player 1 and Player 2. For this player, the points lost were equal to the average of the two squared distances to ensure that his losses for the guesses do not differ in magnitude from those of Player 1 and Player 2. Quadratic scoring rules incentivize risk-neutral subjects to report their mean beliefs truthfully. While there have been attempts at incentive compatible belief elicitation under risk-aversion, such scoring rules are more complicated and the evidence regarding their performance is mixed (Schotter and Trevino, 2014). In our analysis, we take the conjectures made by the subjects about their partners’ states to be proxies of the posteriors, E[θ1 |m1 ] and E[θ2 |m2 ], described in Section 3. We refer to Section 6.2.1 for a discussion of robustness checks supporting the validity of our belief elicitation procedure. Recall that subjects’ decision rules are in theory functions of their posterior beliefs. This allows us to obtain implied (implicit) equilibrium beliefs by inverting the decision rules. 18
In Section 6.1.2, we test the predictions about communication quality using both explicit and implicit beliefs and show that the results are qualitatively similar. Belief elicitation, however, is a necessary component of our experimental design because it allows to analyze how subjects’ decision rules deviate from equilibrium conditional on communication. Moreover, we argue in Section 6.1.2 that the implicit beliefs–which assume equilibrium–are much less likely to reflect subjects’ true beliefs in the experiment than the elicited ones.
6
Results
For the first experiment, we collected data from 238 undergraduate students recruited from introductory level classes. A total of 14 experimental sessions were conducted with a minimum of 11 students and a maximum of 21 students in each session. An average session lasted for 75 minutes with an individual average payment of 163.5 Mexican pesos (≈ US$11), excluding the show-up fee and the payment from the quiz. The distribution of these subjects among treatments is shown in Table 1. Note that more subjects participated in the centralized treatments. This was to ensure that the amount of observations (e.g., for d1 and d2 ) is not too unbalanced.29
Decentralized Centralized
Low γ 3 sessions N = 48 4 sessions N = 66
High γ 3 sessions N = 56 4 sessions N = 68
Table 1: Subjects per treatment.
Our results are summarized in Table 2. We find that the main prediction of Alonso et al. (2008) and Rantakari (2008) regarding the response of communication quality to a need to coordinate cannot be rejected: the quality of horizontal communication is significantly lower than that of vertical communication if and only if the need to coordinate is low. 29
The number of participants in the Centralized-High treatment is not divisible by three. This is because one of the subjects experienced a health issue while the instructions were being administered and had to leave the room. We re-calibrated the program in this session to accommodate 11 subjects rather than 12. This was accomplished by matching 9 people in every round of the game, with two remaining participants sitting out randomly. We also informed the participants in this session about the new rematching procedure. Our results do not significantly change if this session is excluded from the analysis.
19
Moreover, subjects’ decision rules respond to changes in incentives for coordination largely as theory predicts. We also find, however, that subjects in centralized (decentralized) treatments coordinate (adapt) less than predicted. In Section 7, we hypothesize that this is caused by uncertainty and provide a theoretical discussion of possible underlying channels for the effect of uncertainty on subjects’ decisions. In Section 8, we present the results of an additional experiment without incomplete information in which subjects’ behavior is significantly more consistent with equilibrium. The remainder of this section describes the results of the first experiment in detail.
Communication Quality Decision rules
Adaptation and Coordination
Prediction Prediction Prediction Prediction Prediction Prediction Prediction Prediction Prediction
1 2 3 4 5 6 7 8
Equilibrium Concept MIE MIE PBE PBE MIE MIE MIE MIE
Evidence?
Where
Yes
Result 2
Partial
Result 3
Partial
Result 5
Table 2: Predictions and results.
6.1 6.1.1
Communication Summary of the Data
While 63.78% of the messages in the experiment are truthful, only 44.48% of messages are believed. Counting each decision as an observation, we find that 32.67% of observations are associated with both a truthful sender and a trusting receiver. 22.27% of subjects are always truthful in their messages; the rest misrepresent the true value of the state at least once.30 These results suggest that the subjects are neither completely honest nor completely trusting. In theory, all of our treatments provide participants with incentives to exaggerate their states. The standard argument is that other players care less about θi than Player i does. This leads Player i’s partners to make decisions that are not sufficiently extreme from i’s perspective, and, to induce a more extreme decision, i reports mi > θi if θi > 0 and mi < θi 30
Every subject in the dataset had at least one opportunity to be a message sender.
20
if θi < 0.31 We test the hypothesis of exaggeration by regressing mi against θi (Table 3, column 1). Contrary to exaggeration, we find that subjects’ messages are significantly biased toward zero, i.e., the coefficient on θi in the regression is significantly smaller than one (P < 0.001 with H0 : β = 1). Note that while this result is inconsistent with the exaggeration hypothesis, it does not provide evidence against the theory because theory is agnostic about what language is used by the players. Importantly, the coefficient on θi is highly significant and close to one in magnitude. When we regress subjects’ guesses against their received messages (Table 3, column 2), we also find a highly significant coefficient close to one. This suggests that subjects’ guesses are highly correlated with the unobserved states. In a regression of subjects’ guesses against the states being guessed, the R2 is approximately equal to 0.7. This suggests that variation in states explains about 70% of the variation in the elicited beliefs. We interpret this as strong evidence of the efficacy of our belief elicitation procedure. In Table 18 in the appendix, we modify the regressions in Table 3 to include treatment dummies and all relevant interactions. As reported there, subjects in Centralized-Low send more truthful messages (P < 0.05) than subjects in Decentralized-Low, while subjects in DecentralizedHigh believe their messages more than subjects in Decentralized-Low (P < 0.001). Both of these results are consistent with our predictions about communication quality and the results in Section 6.1.2.
State
Message 0.890**** (0.0235)
Message
Constant Observations
Guess
Guess 0.805**** (0.0358)
0.902**** (0.0213) 0.0149* (0.00758) 2880
0.0169** (0.00566) 2880
0.0303*** (0.00997) 2880
Session-clustered standard errors in parentheses * p < 0.10, ** p < 0.05, *** p < 0.01, **** p < 0.001
Table 3: Subjects’ guesses and messages are biased toward zero. 31
Note that this is not an equilibrium argument. E.g., a noninformative equilibrium is a possibility in every treatment.
21
Recall that communication in a partitional equilibrium is noisier when the state is further away from zero. We test this prediction by regressing the absolute deviation between the message and the state, |mi − θi |, against the absolute value of the state, with treatment dummies used as explanatory variables. The results are reported in Table 4. We do not find evidence that less information is transmitted the larger |θi | in any of our treatments. These findings cast doubt on the ability of partitional equilibria to rationalize communication in our data. The coefficients involving the treatment dummies are significant with signs consistent with Prediction 1 and Prediction 2. Thus, when the incentives to coordinate are low, communication is less noisy under centralization than under decentralization (P < 0.01). When these incentives are high, |θi | is not significantly different under centralization and decentralization (P = 0.863). We test predictions about communication quality more rigorously in Section 6.1.2 below.
6.1.2
Communication Quality
The quality of communication is theoretically measured as the residual variance of the message receiver’s posterior, νi = E[θi |mi ], around the sender’s privately known state, that is, E [(θi − νi )2 ]. Our experimental data permits the construction of the empirical analogue of the residual variance of communication by exploiting subjects’ elicited posterior beliefs ν¯i as follows: N 1 X [θn − ν¯n ]2 . N n=1 Here, n denotes an observation and θn the true value of the state being guessed.32 Figure 2 shows how the quality of communication differs across treatments. Note that it is not significantly higher than predicted in any of the treatments. Moreover, in one of the treatments (Decentralized-Low), it is significantly lower than the MIE prediction (P < 0.01).33 This suggests an important counterpoint to Cai and Wang (2006), who find 32
Note that the way in which the centralized treatments were designed allowed us a choice of which observations to include when computing our measure of residual variance. Since all players made guesses about the state(s) unknown to them, we could use (a) only guesses of Player 3, (b) only guesses of Player 1 and Player 2, or (c) everybody’s guesses. To be closest to theory, we take the first approach and use Player 3’s guesses in our analysis of the centralized treatments. The results, however, are qualitatively the same regardless of which of these approaches is used. 33 In Section 6.4, we show that Decentralized-Low is the only treatment in our data where miscommunication led to statistically significant losses in welfare.
22
|mi − θi | 0.00894 (0.0378)
|θi |
High (dummy = 1 if γ = 34 )
-0.0622** (0.0241)
Vertical (dummy = 1 if centralized)
-0.0808*** (0.0221)
High × Vertical
0.0692** (0.0278)
High × |θi |
0.0631 (0.0556)
Vertical × |θi |
0.0214 (0.0482)
High × Vertical × |θi |
0.0195 (0.0830)
Constant
0.103**** (0.0220) 2880
Observations
Session-clustered standard errors in parentheses * p < 0.10, ** p < 0.05, *** p < 0.01, **** p < 0.001
Table 4: Noisiness in communication. The Decentralized-Low treatment serves as a baseline.
23
Centralized
.3
Decentralized
.2
**
0
.1
**
Low gamma
High gamma Low gamma
Data
High gamma
Theory
Figure 2: Residual variance of communication and MIE predictions. The double stars denote a significant difference at P < 0.05. overcommunication in cheap talk games relative to the MIE benchmark.34 Result 1. Communication quality is not higher than MIE predicts in any of our treatments, and significantly lower than predicted in Decentralized-Low. As the importance of coordination increases, the quality of communication improves in decentralized games and worsens in centralized ones, although the high standard error in the Centralized-High treatment suggests that the latter effect may not be statistically significant.35 The figure also suggests that the quality of communication in decentralized treatments is lower than that in centralized ones when the importance of coordination is low. To test these hypotheses more rigorously, we compare mean residual variances across treatments with the residual variance of horizontal communication in the DecentralizedLow treatment as a baseline. The results are reported in Table 5, column 1. The regression results are consistent with several of our theoretical predictions. First, 34
Another study that finds undercommunication relevant to MIE is Wilson (2014). That the standard error in Centralized-High is higher than in all other treatments provides some evidence that subjects find incentives in Centralized-High more difficult to understand. We also find that their answers to the quiz are significantly worse in this treatment than in any other. 35
24
(1) All periods Decentralized-High -0.0703** (0.0325)
(2) Last 8 periods -0.0508 (0.0446)
(3) Implicit Beliefs -0.0696 (0.0508)
(4) Implicit Beliefs (all) 1.055 (1.109)
Centralized-Low
-0.0807** (0.0351)
-0.0684** (0.0288)
-0.302**** (0.0478)
-4.381**** (0.728)
Centralized-High
0.0132 (0.0797)
0.00732 (0.0789)
-0.123 (0.0734)
5.503**** (1.207)
0.142**** (0.0191) 2880
0.121**** (0.0107) 1536
0.446**** (0.0470) 1636
4.694**** (0.727) 2880
Constant Observations
Session-clustered standard errors in parentheses * p < 0.10, ** p < 0.05, *** p < 0.01, **** p < 0.001
Table 5: Treatment effects on residual variance of communication. The Decentralized-Low treatment serves as a baseline. the coefficient on Decentralized-High is negative (P < 0.05), suggesting that the quality of horizontal communication increases with the importance of coordination. The coefficient on Centralized-Low is also negative (P < 0.05), suggesting that the quality of communication in Decentralized-Low is significantly lower than that in Centralized-Low. Third, there is no significant difference between coefficients on Decentralized-High and Centralized-High (P = 0.326), although the residual variance in the latter case is greater. This is also consistent with the theory, since it predicts that residual variances of horizontal and vertical communication are close when the importance of coordination is high. We highlight these results below: Result 2. The quality of vertical communication is higher than that of horizontal communication if and only if the incentive to coordinate is low. Inconsistent with the predictions, we find no significant difference between the coefficients on Centralized-Low and Centralized-High (P = 0.277). We interpret this to mean that it is difficult for subjects in centralized coordination games to see incentives become misaligned as the importance of coordination grows. This observation is consistent with
25
the finding that the average distance between messages and states is insignificantly different in the centralized treatments. We also find that the effect of incentives to coordinate on the quality of horizontal communication is less robust to learning than its effect on the quality of vertical communication. The second column of Table 5 re-estimates the same regression using only observations from the last eight periods of the experiment. Note that the coefficient on Decentralized-High is not significant in this specification (P = 0.276).36 One interpretation is that the difference between Decentralized-Low and DecentralizedHigh is more difficult for subjects to grasp than the difference between Decentralized-Low and Centralized-Low. Note, however, that our other results remain significant even if the observations are restricted to the last eight periods. Our findings in regard to communication quality can be summarized as follows. First, contrary to many other studies in the experimental literature, we find that subjects do not overcommunicate information in any of our treatments. To the extent that we find undercommunication in Decentralized-Low, we expect this to lead to a loss in welfare. In Section 6.4, we explore the welfare consequences of miscommunication in detail. Second, Alonso et al. (2008) and Rantakari (2008) provide a good if imperfect description of how communication quality responds to incentives to coordinate. In particular, we find that as the importance of coordination increases, the difference in quality of vertical and horizontal communication tends to zero. Thus, the main theoretical predictions of Alonso et al.(2008) and Rantakari (2008) about communication quality are consistent with the data.
6.1.3
Robustness to Implicit Beliefs
While our focus so far has been on subjects’ elicited beliefs, our experiment allows us to derive subjects’ implicit beliefs under the assumption of equilibrium behavior. Treating the posterior beliefs νi and νj as unknowns, this can be done using Equation 4.4 for the decentralized and using Equation 4.5 for the centralized treatments. When we do this, we find that 43.19% of implicit beliefs lie outside the interval [−1, 1]. While the correlation between subjects’ explicit guesses and the true values of the states being guessed is 0.83, the correlation between implicit beliefs and the true states is 0.40. This suggests that elicited beliefs are a much better proxy for participants’ true beliefs than the implicit equilibrium 36
It also loses significance in late periods of the interaction if we estimate a regression which uses all observations but interacts the treatment dummies with the period variable.
26
ones.37 Nevertheless, our main results regarding communication quality are robust to using implicit beliefs in place of elicited ones. The third column of Table 5 shows the results of a regression in which the residual variance of communication, formed using implicit beliefs, is regressed against the treatment dummies. Because subjects knew that the true value of the state was contained in the interval [−1, 1], we only include observations with beliefs in this range. The results, same as those in the first two columns of the table, show a negative and highly significant coefficient on Centralized-Low (P < 0.001). This shows that the residual variance of communication is significantly higher under decentralization when the importance of coordination is low. When the importance of coordination is high, we find no significant difference in communication quality under decentralization and centralization (P = 0.3853). This is consistent with Prediction 1, Prediction 2, and Result 2. Results of the same regression that includes all observations are reported in the fourth column of Table 5. As in all other regressions, the coefficient on Centralized-Low is negative and significant. While the results show that the residual variance of communication is higher under decentralization when the importance of coordination is low (P < 0.001), it is instead higher under centralization when the importance of coordination is high (P < 0.01). Note, however, that the coefficients are extremely large in magnitude. It is unlikely that subjects’ beliefs differed from the states’ true values by such large amounts. We interpret implicit beliefs outside of the [-1,1] range as not reflecting subjects’ true beliefs and therefore inappropriate for an analysis of communication quality. One possibility is that such beliefs are miscalculated by the inversion procedure, e.g. because of the failure of the equilibrium assumption. As we argue in Section 6.2, our data is also consistent with equilibrium behavior and systematic distortions of decision weights.
6.2
Decisions
We now explore how well the predicted decision rules (Equation 4.4 and Equation 4.5) correlate with those observed in the data. To this end, we calculate the equilibrium predicD 38 tions for dC i and di conditional on the realized states as well as subjects’ reported beliefs. 37
In Section 6.2, we provide a possible justification for the observed dispersion in implicit beliefs using subjects’ behavior in the decision making stage. 38 Note that this exercise would have been impossible without belief elicitation. In particular, it is impossible if we restrict our analysis to implicit beliefs, since these take equilibrium decision weights as given.
27
Centralized, Low gamma (P=0.8685)
Decentralized, High gamma (P=0.6913)
Centralized, High gamma (P=0.6829)
−1
−.5
0
.5
1
Data
−1
−.5
0
.5
1
Decentralized, Low gamma (P=0.7068)
−1
−.5
0
.5
1
−1
−.5
0
.5
1
Theory
Figure 3: Decision rules: theory and data. Figure 3 provides a scatter plot of what theory predicts the decisions should be and the decisions observed in the data. The figure suggests that the correlation in every treatment is high. Nevertheless, subjects’ decisions also show substantial deviations from the theoretical optima. As we show below, these deviations are in large part systematic. Using the decision rule described by Equation 4.4, we estimate the decision weights by regressing di against θi , ν¯i , and ν¯j .39 To accommodate the effect of γ on subjects’ decisions, the variables θi , ν¯i , and ν¯j are interacted with a dummy that takes on the value of one for treatments with γ = 43 . We also place the restriction that the weights add up to 1 both when γ = 41 and when γ = 34 .40 Note that the results of this regression allow us to discuss deviation from equilibrium decision rules conditional on the communication rules used by the subjects. The results are reported in the first column of Table 6. Qualitatively, subjects respond to incentives to coordinate and adapt roughly as theory predicts. As the importance of coordination increases, the weight on θi decreases, and the weight on ν¯j increases (P < 0.05 and P < 0.001, respectively). While the change in the weight on ν¯i is not statistically 39 40
Recall that νi = E[θi |mi ]. The results are qualitatively similar if this restriction is removed.
28
significant, the sign is in the right direction. Thus, subjects’ decisions put a smaller weight on the state and a larger weight on posterior beliefs when the incentive to coordinate is greater. We estimate an analogous regression for the centralized treatments. Using Equation 4.5, we regress the principals’ decisions against their elicited posterior beliefs with the restriction that the weights sum up to one. The results are reported in the second column of Table 6. These results also suggest that subjects’ decision rules respond as predicted to incentives to coordinate. Thus, the weight on ν¯i is significantly lower and the weight on ν¯j significantly higher when the importance of coordination is high (P < 0.001 in both cases). I.e., as predicted, the principal weighs the belief about i’s state less, and the belief about the state of i’s partner more, when the importance of coordination is high. We summarize these findings below: Result 3. Qualitatively, subjects’ decision rules respond to incentives to coordinate as theory predicts. Importantly, the estimated decision weights show significant and systematic deviations from those implied by the theory. Table 7 compares the estimated and predicted decision weights under decentralization. In Decentralized-Low, subjects underweigh their own states (P < 0.001), overweigh their partners’ posteriors (P < 0.001), and overweigh their own posteriors about their partners’ states (P < 0.001). In Decentralized-High, subjects overweigh their own posteriors (P < 0.001). Thus, subjects in the decentralized treatments underweigh their own states and overweigh their own and their partners’ beliefs. Table 8 compares the estimated and predicted decision weights under centralization. In both Centralized-Low and Centralized-High, the principal puts too much weight on the belief about θi and too little weight on the belief about θj when making the decision di (all P < 0.001). The deviations reported above are consistent with the agent overweighing the importance of coordination under decentralization and the principal overweighing the importance of adaptation under centralization. To quantify the degree to which coordination is over- or underweighed, we structurally estimate the γ implied by the subjects’ decisions under the null hypothesis of equilibrium. We do this separately for each of the experimental treatments using non-linear least squares. The estimation results are shown in Table 9. The estimated γ’s are significantly higher than what they should be (i.e., those specified in the instructions) in both Decentralized-Low and Decentralized-High (P < 0.05 in P < 0.01, respectively). For example, when γ = 43 , the agent acts as if adaptation is almost irrelevant. 29
Decentralized
Centralized
-0.00735 (0.0131)
0.0149 (0.0149)
High (dummy=1 if γ = 34 )
θi
0.493**** (0.0775)
ν¯i
0.162**** (0.0224)
0.946**** (0.0196)
ν¯j
0.345**** (0.0572)
0.0544*** (0.0196)
θi ×High
-0.270** (0.115)
ν¯i ×High
0.0576 (0.0939)
-0.290**** (0.0323)
ν¯j ×High
0.213**** (0.0598)
0.290**** (0.0323)
Constant
0.0197 (0.0123) 1560
0.00615 (0.0118) 1320
Observations
Session-clustered standard errors in parentheses * p < 0.10, ** p < 0.05, *** p < 0.01, **** p < 0.001
Table 6: Estimated decision weights. As the importance of coordination increases (High=1), agents in decentralized games put a smaller weight on their own state and a larger weight on their posterior belief about the state of the other agent. This is consistent with Prediction 4. In centralized treatments, when the importance of coordination is high, the weight on the posterior belief about the local conditions of the agent for which the decision is made is smaller, and the weight on the posterior belief about the other agent is larger. This is consistent with Prediction 3.
30
Decentralized Treatments Low (Predicted) Low (Actual) High (Predicted) High (Actual)
θi 0.75 0.49**** 0.25 0.22
νi 0.05 0.16**** 0.32 0.22
νj 0.2 0.34** 0.43 0.56****
* p < 0.10, ** p < 0.05, *** p < 0.01, **** p < 0.001
Table 7: Predicted equilibrium weights and actual weights in the decision rules of decentralized treatments. The significance levels refer to the difference between actual and predicted weights. Centralized Treatments Low (Predicted) Low (Actual) High (Predicted) High (Actual)
νi 0.71 0.95**** 0.54 0.66****
νj 0.29 0.05**** 0.46 0.34****
* p < 0.10, ** p < 0.05, *** p < 0.01, **** p < 0.001
Table 8: Predicted equilibrium weights and actual weights in the decision rules of centralized treatments. The significance levels refer to the difference between actual and predicted weights. In Centralized-Low and Centralized-High, the weights are significantly lower than what they should be (P < 0.001 in both treatments). For example, when γ = 41 , the principal acts as if the weight on coordination were almost zero. We summarize these results below and postpone a discussion of how the distortions in decision weights relate to uncertainty to Section 7. Result 4. Agents in decentralized coordination games overweigh, and principals in centralized coordination games underweigh, the importance of coordination.
6.2.1
Robustness Checks
In principle, risk-aversion might have affected belief elicitation. If this were the case, the documented distortions in decision rules might simply be a result of estimation based 31
γˆ when γ = 0.25
γˆ when γ = 0.75 Observations
Decentralized 0.517 >∗∗ 0.25 (0.103)
Centralized 0.0296 ∗∗∗ 0.75 (0.0375) 1560
0.356 0.1 on the test of the constant plus any of the indicator variables being equal to zero). This is consistent with the results on communication quality reported in Section 6.1.2, where we find that the quality of communication is significantly different from MIE only in Decentralized-Low.
7 7.1
Discussion Social Preferences
Under centralization, social preferences cannot explain our observed distortions in the principal’s decision rules. Conditional on the messages received, the principal is already maximizing the agents’ joint payoffs. Social preferences therefore leave the theoretical predictions about the principal’s decisions unaltered. Under decentralization, it is in theory possible that social preferences lead the agents to over-coordinate relative to the selfinterested benchmark. If social preferences caused the distortions in Decentralized-Low and 40
Decentralized-High, similar distortions should be observed in otherwise identical treatments where both agents observe each other’s states. We highlight this below as a prediction: Prediction 9. If social preferences caused the distortions in Decentralized-Low and Decentralized-High, we should observe similar distortions in analogous treatments with complete information. We test this prediction in an additional experiment whose results are reported in Section 8.
7.2
Risk Aversion
Unlike social preferences, risk aversion can generate distortions in the directions we observe both under centralization and decentralization. Thus, under centralization, a risk-averse principal will under-coordinate relative to the risk-neutral baseline. Under decentralization, a risk-averse agent will under-adapt. To get some intuition for these results, consider the case of centralization. Note that the principal’s payoff can be decomposed into two adaptation losses (one for each agent) and a coordination loss. The coordination loss is non-stochastic, while the adaptation losses depend on unknown states. If the agents report truthfully and the principal believes the messages received, the adaptation losses are also non-stochastic–conditional on the information received–and risk preferences have no bite. Noise in messages introduces uncertainty about the principal’s adaptation losses. While a risk-neutral decision maker only cares about the posterior expectations of the states, a risk-averse principal also cares about the variance of her conditional expectation, and she will under-coordinate to decrease this variance.46 Introducing risk preferences into the model described in Section 3 and deriving the associated predictions is a difficult task because the decision rules have no closed-form solutions.47 Nevertheless, we can get a sense for the effect of curvature on optimal decisions by considering a simple reduced-form version of the baseline model. We describe such a model in Section A.4 for the centralized and Section A.5 for the decentralized case and perform simulations to calculate average degrees of under-coordination and underadaptation for different values of the risk aversion parameter. Both under centralization 46
In the simulations discussed below, we are agnostic about how subjects’ beliefs are related to the realized states. We therefore abstract away from the issue of over-adaptation under centralization. 47 We know of no extensions of Alonso et al. (2008), Rantakari (2008), or similar models to risk-averse agents.
41
and decentralization, we find that risk aversion generates distortions that go in the right direction, although the model performs somewhat better under decentralization. Thus, even if we allow risk aversion to be unreasonably high, the simulations can only account for 13%-34% of the under-coordination observed in the centralized treatments. For reasonable degrees of risk aversion, we can explain only 4%-17% of the observed under-coordination. In contrast, under decentralization, reasonable degrees of risk aversion can explain 28%-50% of the under-adaptation observed in the data. Consider now the following modification of our original experiment. Under decentralization, both agents are informed of each other’s states, and the game is otherwise identical. Under centralization, both agents and the principal are informed of the agents’ states. As we discuss in Section 8, the centralized and decentralized games with complete information have unique equilibria and hence no uncertainty under the null hypothesis of equilibrium behavior. This suggests the following prediction: Prediction 10. If risk aversion caused the distortions in Centralized-Low, CentralizedHigh, Decentralized-Low and Decentralized-High, we should observe no distortions in analogous treatments with complete information.
7.3
Risk Seeking
In our experiment, subjects only incur losses, and a large literature starting from Kahneman and Tversky (1979) found subjects to be risk-seeking in the loss domain.48 For example, with the utility function used in our simulations, U (x) = −(−x)α , x ≤ 0, Tversky and Kahneman (1992) have estimated the curvature of subjects’ utility function to be around 0.88, which implies risk seeking over the loss domain and risk aversion otherwise. Our simulation results in Section A.4 and Section A.5, however, show that risk seeking generates distortions in the opposite direction to those observed in the data both under centralization and decentralization. I.e., a risk-seeking principal over-coordinates under centralization, while a risk-seeking agent over-adapts under decentralization. There are several possible explanations for why our results differ from those of Tversky and Kahneman (1992). First, our experiment differs substantially from the choice-theoretic experiments in the prospect theory literature or related experiments such as Myagkov and Plott (1997). The noise in our game is not objective but derived from other subjects’ communication rules. Second, it’s possible that subjects’ reference points are not zero (Kah48
See also Myagkov and Plott (1997), which documents risk-seeking with losses in a market experiment.
42
neman and Tversky, 1979). Indeed, incurring no loss is difficult in our environment and subjects might have formed expectations accordingly (K˝oszegi and Rabin, 2006). Third, as we show below, ambiguity-aversion can generate distortions in the right direction even with moderate risk-seeking preferences.
7.4
Strategic Uncertainty and Ambiguity
If the players fail to coordinate on the same equilibrium, they face strategic uncertainty. To study the effect of strategic uncertainty on behavior, consider first the case of centralization with incomplete information. Here, the principal potentially faces uncertainty about the communication rules used by the agents. She therefore needs to form subjective beliefs about how states are communicated through subjects’ messages. Assuming ambiguity-neutral preferences, the principal can easily form posterior beliefs E(θi |mi ), and the relevant theoretical (resp., numerical) predictions are in Section 4.2 for risk-neutral, Section 7.2 for risk-seeking, and Section 7.3 for risk-averse preferences. This shows that strategic uncertainty about communication rules has no bite under ambiguity-neutral preferences. In the presence of ambiguity aversion, however, the predictions change. In Section A.6, we perform simulations to solve the problem of an ambiguity-averse principal under centralization. The simulation assumes that the principal has preferences of the max min sort, where the min is taken over different beliefs about the agents’ states. We find that ambiguity aversion substantially amplifies the distortions due to risk aversion. Thus, with α = 2 and ambiguity-averse preferences, the model accounts for 17% to 55% of the undercoordination observed in the data. Moreover, if the principal is ambiguity-averse, the model can generate distortions in the right direction even if her utility function is moderately risk-seeking. If, following Van Huyck et al. (1990), we interpret strategic uncertainty as arising from multiplicity of equilibria, then strategic uncertainty is not a concern under complete information regardless of whether the game is centralized. Under centralization, this is because–conditional on her beliefs–the principal faces an optimization problem with a unique solution. Under decentralization, the equilibrium is unique if both agents know each other’s states. This suggests the following prediction: Prediction 11. If ambiguity aversion caused the distortions in Centralized-Low, CentralizedHigh, Decentralized-Low and Decentralized-High, we should observe no distortions in analogous treatments with complete information.
43
8
The Second Experiment
To test Prediction 9, Prediction 10, and Prediction 11, we ran an additional experiment in September 2015. We collected data from 60 subjects for two treatments: DecentralizedComplete Info and Centralized-Complete Info (see Table 14). For both of these treatments, we chose γ = 34 because the observed distortions in this case were most robust to learning. The treatments were identical to Decentralized-High and Centralized-High in all respects but the following. First, every player observed every state θi (i = 1, 2) before making any decision. Second, the players did not make any guesses about the states of other players. The instructions and quizzes were modified accordingly.49
Decentralized-Complete Info Centralized-Complete Info
High γ 2 sessions N = 30 2 sessions N = 30
Table 14: Subjects per treatment in the second experiment.
The predictions for these treatments are simple extensions of the results in Section 4. In particular, the complete information decision rules are the same as those in Equation 4.4 and Equation 4.5, with the modification that each posterior belief νi is replaced by the true value of the corresponding state θi . While the decision rule under decentralization is the unique equilibrium solution, the decision rule under centralization is the unique solution to the principal’s decision problem:
dD i =
dC i =
1 γ θi + θj , 1+γ 1+γ
1+γ 2γ θi + θj , 1 + 3γ 1 + 3γ
i = 1, 2,
i = 1, 2,
i 6= j.
i 6= j.
(8.1)
(8.2)
In principle, given complete information, subjects’ messages can be interpreted as suggestions.50 I.e., under decentralization, the agents who understood the equilibrium solution 49
See the online appendix. While we could have removed communication from the complete information treatments, we avoided doing this so that uncertainty and communication are not manipulated at the same time. Thus, the only 50
44
can send messages to other agents to inform them what this solution is. Under centralization, the agents can send messages to the principal to inform them of the unique solution to the principal’s problem. From a theoretical point of view, however, these messages are irrelevant. We therefore make no predictions about communication quality for CentralizedComplete Info and Decentralized-Complete Info. The predictions about coordination and adaptation losses are simple extensions of Prediction 5, Prediction 6, Prediction 7, and Prediction 8. Thus, both under centralization and decentralization the decisions are better coordinated and less adapted to the states when γ is higher. The adaptation loss remains greater under centralization, and the coordination loss remains greater under decentralization. Intuitively, it can be seen from Equation 8.1 and Equation 8.2 that both decision rules decrease the weight on θi when γ is greater. It can also be seen that the decision rule under centralization puts a greater weight on coordination both when γ = 41 and γ = 34 . We re-estimated the models discussed in Section 6 using data from treatments with both complete and incomplete information. Our first set of results deals with adaptation and coordination losses. These are reported in Table 15. First, we find that coordination losses are greater under decentralization than under centralization in the complete information treatments (P < 0.001). This stands in stark contrast to our results with incomplete information, where the opposite was true. Second, we find that adaptation losses are not significantly different under decentralization and centralization with complete information (P = 0.801). This also stands in contrast to the incomplete information results, where we found greater adaptation losses under decentralization than under centralization. We highlight these results below: Result 8. With complete information, centralized treatments show more coordination than decentralized treatments and as much adaptation. Result 8 suggests that subjects in centralized treatments overweighed adaptation less and subjects in decentralized treatments overweighed coordination less with complete information. To quantify the magnitudes of these distortions, we estimated subjects’ decision weights structurally taking the same approach as in Section 6.2. In particular, we augmented Table 9 to include estimation results for treatments with complete information.51 The results are reported in Table 16. The table shows that the estimated γˆ ’s for the comdifference between the complete and incomplete information treatments is uncertainty about θi . 51 To compare the distortions to their analogues with incomplete information, we carry out the estimation for all decentralized treatments together. Likewise, a second non-linear regression is estimated for all centralized treatments.
45
(1) (d1 − d2 )2 -0.0935* (0.0528)
(2) (di − θi )2 0.117** (0.0424)
Centralized-Low
0.204*** (0.0695)
-0.145*** (0.0399)
Centralized-High
-0.0596 (0.0465)
0.0590 (0.0735)
Decentralized-Complete Info
-0.122** (0.0472)
0.00881 (0.0453)
Centralized-Complete Info
-0.220**** (0.0422)
0.00105 (0.0394)
Constant
0.319**** (0.0420) 1815
0.220**** (0.0366) 3630
Decentralized-High
Observations
Session-clustered standard errors in parentheses * p < 0.10, ** p < 0.05, *** p < 0.01, **** p < 0.001
Table 15: Degrees of adaptation and coordination in different experimental treatments. The Decentralized-Low treatment serves as a baseline.
46
plete information treatments do not significantly differ from the optimal γ = 0.75. We highlight this result below: Result 9. With complete information, the hypothesis that subjects use the correct γ cannot be rejected.
γˆ when γ = 0.25
γˆ when γ = 0.75
γˆ when γ = 0.75 (Complete Information) Observations
Decentralized 0.517 >∗∗ 0.25 (0.1)
Centralized 0.0296 ∗∗∗ 0.75 (0.037)
0.356 ALiD . Also,
dALiC dγ
> 0 and
dALiD dγ
> 0.
Proof. From the proof of Proposition 2 in Alonso et al. (2008, pag. 174), it follows again that 1 (1 + γ)(1 + 6γ + γ 2 ) ALiC = − , (A.5) 3 (1 + 3γ)2 (3 + 4γ) 7γ 2 + γ 3 . 3(1 + γ)(4 + 3γ) Differentiating (A.5) and (A.6) with respect to γ gives ALiD =
(A.6)
dALiC 1 + 57γ + 131γ 2 + 67γ 3 = > 0, dγ (1 + 3γ)3 (3 + 4γ)2
(A.7)
dALiD γ 56 + 61γ + 14γ 2 + 3γ 3 > 0. = dγ 3 (1 + γ)2 (4 + 3γ)2
(A.8)
Finally, ALiC > ALiD , for any γ ∈ (0, 1), follows from Lemma 2 in Alonso et al. (2008). 49
A.2
Additional Analysis of Adaptation and Coordination Losses
Table 17 breaks point losses of teams in different treatments of the experiment into miscomunication and miscoordination components. Thus, for example, the miscommunication component of the relative coordination loss in the treatments with γ = 34 is calculated as:53 ) ( 2 X belief s belief s 2 IE 2 IE − dM 3 ∗ (dreported − dreported ) − (dM −i ) i i −i i=1
Note that this table can be used to recover the overall relative losses due to distortions or communication reported in Table 13. For example, to compute the relative losses due to distortions in Decentralized-Low (Table 13, constant term in the second column), add the relative coordination losses due to distortions in Decentralized-Low (Table 17, first column, second row) to the relative adaptation losses due to distortions in Decentralized-Low (Table 17, first column, fifth row). Recall from the second column of Table 13 that the relative welfare losses due to distortions were smaller in Centralized-Low than in any of the other treatments. Table 17 provides evidence for our conjecture that this was driven by coordination losses being smaller in Centralized-Low (where the over-weighting of adaptation was less costly) than in Centralized-High. Thus, while the coordination loss due to distortions was greater in Centralized-High than in Centralized-Low, distortions in decision rules did not lead to adaptation losses under centralization (all P > 0.1). The table also provides additional evidence for Result 7: very little of the significant loss in welfare is due to miscommunication. As discussed above, the only treatment showing significant welfare loss due to miscommunication is Decentralized-Low.
53
The sum is necessary in the expression because the analysis of the decompositions is carried out in terms of units of welfare and not individual payoffs.
50
D-L 0.125 (0.081)
D-H C-L C-H 1.236**** 0.810**** 1.533**** (0.191) (0.111) (0.120)
Relative coordination loss (Distortions)
0.111 (0.070)
1.233**** 0.826**** 1.533**** (0.200) (0.097) (0.120)
Relative coordination loss (Miscommunication)
0.014 (0.014)
0.003 (0.012)
-0.017 (0.016)
0.0001 (0.001)
Relative adaptation loss
1.173**** (0.220)
0.397**** 0.028 (0.038) (0.082)
0.219 (0.134)
Relative adaptation loss (Distortions)
1.157**** (0.221)
0.381**** -0.158 (0.042) (0.106)
0.088* (0.050)
Relative adaptation loss (Miscommunication)
0.016**** 0.016 (0.002) (0.012) 360 420
Relative coordination loss
Observations
0.187 (0.134) 330
0.131 (0.088) 330
Session-clustered standard errors in parentheses * p < 0.10, ** p < 0.05, *** p < 0.01, **** p < 0.001
Table 17: Decompositions of adaptation and coordination losses into a component due to distortions of decision weights and a component due to miscommunication. The standard errors are obtained by regressing each of the variables (e.g., relative coordination loss) against the treatment dummies.
51
A.3
Omitted Tables
State
Message 0.867**** (0.0183)
Message
Guess
Guess 0.794**** (0.0247)
0.921**** (0.00291)
Decentralized-High
-0.00211 (0.0153)
-0.00300 (0.0138)
-0.00483 (0.0157)
Centralized-Low
-0.0118 (0.0141)
-0.00503 (0.0168)
-0.0160 (0.0111)
Centralized-High
0.0214 (0.0259)
0.00792 (0.0173)
0.0245 (0.0297)
Decentralized-High × State
0.0215 (0.0477)
0.0402 (0.0506)
Centralized-Low × State
0.0762** (0.0341)
0.0721 (0.0558)
Centralized-High × State
-0.00111 (0.0683)
-0.0711 (0.112)
Decentralized-High × Message
0.0214**** (0.00465)
Centralized-Low × Message
-0.0139 (0.0321)
Centralized-High × Message
-0.0916 (0.0641)
Constant Observations
0.0133 (0.0136) 2880
0.0173 (0.0108) 2880
0.0296**** (0.00352) 2880
Session-clustered standard errors in parentheses * p < 0.10, ** p < 0.05, *** p < 0.01, **** p < 0.001
Table 18: Subjects’ guesses, messages, and interactions with the treatment dummies.
52
A.4
Simulations for Risk Preferences (Centralization)
To accommodate risk-seeking as well as risk-averse preferences, we assume that the decision maker in the experiment has a utility function of the form U (x) = −(−x)α over her point losses x < 0 (Tversky and Kahneman, 1992). The parameter α > 0 determines the risk attitudes of the decision maker, with α > 1 leading to risk-averse and α ∈ (0, 1) to risk-seeking behavior. Our goal is to explore how changes in this parameter affect decision making. Suppose that authority is centralized. Let νi be the principal’s posterior expectation about θi after having received a message about θi . Suppose that ν˜i ∈ {νi − , νi + }, with > 0, i = 1, 2. Let p = Prob(˜ νi = νi + ). Then, E[˜ νi ] = νi + (2p − 1) and 1 2 V ar(˜ νi ) = 4p(1 − p) . When p is close to 2 , the distribution of νi is a proxy for a uniform posterior distribution around the posterior mean νi , as it would be in communication equilibria with risk-neutrality.54 The parameter can be interpreted as a measure of uncertainty about the posterior expectation νi . The problem of the principal can therefore be written as: max −E
d1 ,d2 ∈R
(1 − γ)(d1 − ν˜1 )2 + (1 − γ)(d2 − ν˜2 )2 + 2γ(d1 − d2 )2
α
.
(A.9)
If the principal were risk-neutral (α = 1), she would choose di =
2γ 1+γ E[˜ νi ] + E[˜ νj ], 1 + 3γ 1 + 3γ
i = 1, 2, i 6= j.
(A.10)
Note that the decision rules are exactly those used by the principal in our baseline model. We perform simulations to calculate the average distance between the principal’s decisions, |d1 − d2 |, for different values of ν1 , ν2 , α, and .55 In the simulations, we assume that p = 21 56 and consider νi ∈ [−0.6, 0.6].57 Figure 6 shows the simulated average distances RS RN − dRN for different values of α and , with D(, α) ≡ M ean(ν1 ,ν2 ) |dRS 2 | 1 − d2 | − |d1 α ∈ [0, 1].58 The figure shows that the simulated average distance is negative, which means that the decisions are on average more coordinated under risk-seeking than risk-neutrality. If we average over ∈ [0, 0.4], and α ∈ [0, 1], we obtain that the average distance between 54
Although communication equilibria could have different features under risk aversion, we make this distributional assumption for tractability. 55 We set the grid sizes to 0.05 for νi , i = 1, 2, 0.02 for . 56 Robustness checks suggest that the magnitude of distortions is little affected by relaxing it. 57 The values of ν1 and ν2 are chosen in such a way that max {|νi − |, |νi + |} ≤ 1, i = 1, 2, given the simulated values of ∈ [0, 0.4]. Varying over a smaller interval leads to smaller distortions. 58 More precisely, we calculate the distance D for each vector (α, , ν1 , ν2 ) and, holding α and fixed, average the distances obtained for different values of (ν1 , ν2 ). The grid size for α is set at 0.05.
53
decisions under risk-seeking is -0.04 for γ = 1/4 and -0.01 for γ = 3/4.59 In the experiment, the average absolute distance between the observed and risk-neutral equilibrium decisions, Eq |dObserved − dObserved | − |dEq 1 − d2 |, is approximately 0.29 for Centralized-Low and 0.22 for 1 2 Centralized-High. Based on these simulation results, we conclude that risk-seeking cannot explain the over-coordination observed in the centralized treatments in the data. Simulation results for risk-averse preferences are shown in Figure 7. The figure gives us a rough idea of how much risk aversion is necessary to generate distortions of the order observed in the experiment. With α ∈ [1, 5],60 we obtain that the average difference in the distances is 0.05 for γ = 1/4 and 0.01 for γ = 3/4. Although the simulated distortions go in the same direction as what we observe in our data, the magnitudes are of a different order even with highly unreasonable degrees of risk aversion. For example, averaging over α ∈ [10, 20] only raises the average between distances to 0.098 for γ = 1/4 and 0.029 γ = 3/4. We performed simulations with alternative, standard, utility functions such as the log and CRRA and obtained similar results. This shows that risk aversion can partly explain the distortions observed in the centralized treatments with incomplete information but cannot fully accommodate them.
We also performed simulations with a larger number of states, namely, νi − , νi − 3 , νi + 3 , νi + , i = 1, 2. We found similar qualitative and quantitative results. 60 The grid size for α was increased to 0.1 due to the larger parameter interval. 59
54
(a) Comparison of average distances between optimal risk-seeking and risk-neutral decisions, RS RN − dRN | for γ = 1 . Each point corresponds to the average over (ν , ν ) ∈ |dRS 1 2 1 − d2 | − |d1 2 4 2 [−0.6, 0.6] with a grid of size 0.05. The grid size for α is 0.05.
(b) Comparison of average distances between optimal risk-seeking and risk-neutral decisions, RS RN − dRN | for γ = 3 . Each point corresponds to the average over (ν , ν ) ∈ |dRS 1 2 1 − d2 | − |d1 2 4 2 [−0.6, 0.6] with a grid of size 0.05. The grid size for α is 0.05.
Figure 6: The effect of risk-seeking on coordination behavior in centralized coordination games. 55
(a) Comparison of average distances between optimal risk-averse and risk-neutral decisions, |dRA 1 − 1 RA RN RN d2 | − |d1 − d2 | for γ = 4 . Each point corresponds to the average over (ν1 , ν2 ) ∈ [−0.6, 0.6]2 with a grid of size 0.05.
(b) Comparison of average distances between optimal risk-averse and risk-neutral decisions, |dRA 1 − 3 RA RN RN d2 | − |d1 − d2 | for γ = 4 . Each point corresponds to the average over (ν1 , ν2 ) ∈ [−0.6, 0.6]2 with a grid of size 0.05.
Figure 7: The effect of risk aversion on coordination behavior in centralized coordination games.
56
A.5
Simulations for Risk Preferences (Decentralization)
Under decentralization, we can without loss of generality consider the decision problem of Player 1. Player 1 observes her own local conditions θ1 and needs to make a single decision without knowing the decision made by Player 2. Let us reformulate the problem assuming that the decision of Player 2, d˜2 , is random from Player 1’s perspective and could take on the value d2 + with probability p, or d2 − otherwise, where d2 ∈ (−1, 1) and ∈ (0, 1 − |d2 |). We interpret d2 as the expected decision of Player 2 from Player 1’s perspective. Given a risk aversion coefficient α, Player 1’s decision problem can be written as h α i max −E (1 − γ)(d1 − θ1 )2 + γ(d1 − d˜2 )2 . (A.11) d1 ∈R
If Player 1 were risk-neutral (α = 1), she would choose d1 = (1 − γ)θ1 + γE[d˜2 ].
(A.12)
Note that this decision rule is the same as the one used by Player 1 in the baseline model, given our interpretation of d2 . We perform simulations to calculate the degree of adaptation, |d1 − θ1 |, for different values of θ1 , d2 , α, and . In the simulations, we assume that p = 21 and consider values of θ1 ∈ [−1, 1], d2 ∈ [−0.6, 0.6], and ∈ [0, 0.4].61 Figure 8 shows the simulated average RN distances D(, α) ≡ M ean(θ1 ,d2 ) |dR − θ | − |d − θ | for different values of α and . 1 1 1 1 The figure shows that the decisions are on average more adapted under risk-seeking than risk neutrality, and more adapted under risk neutrality than under risk aversion. More precisely, averaging over ∈ [0, 0.4] and α ∈ {0.2, 0.4, 0.6, 0.8}, for a risk seeking decision maker, we obtain that the average distances are approximately -0.02 for γ = 1/4, and -0.05 for γ = 3/4. For degrees of risk aversion in the set {2, 3, 4, 5}, the same average leads to 0.036 for γ = 1/4, and 0.05 for γ = 3/4. For comparison, the average distance between decisions and states in the data is approximately 0.128 for γ = 1/4, and 0.10 for γ = 3/4. Thus, risk aversion explains the direction of the observed distortions under decentralization. It is also provides quantitative benchmarks that are closer to the data than their counterparts in the centralized case.
61
The grids for θ1 , d2 , and are 0.01, 0.01, and 0.02, respectively.
57
(a) Comparison of average distances between optimal risky and risk neutral level of adaptation, RN − θ | for γ = 1 , different attitudes toward risk, α ∈ {0.2, 0.4, 0.6, 0.8, 1, 2, 3, 4, 5}, |dR 1 1 − θ1 | − |d1 4 and different values of ∈ [0, 0.4] with grid of size 0.02. Each point corresponds to the average over (θ1 , d2 ) ∈ [−1, 1] × [−0.6, 0.6] with a grid of size 0.01.
(b) Comparison of average distances between optimal risky and risk neutral level of adaptation, RN − θ | for γ = 3 , different attitudes toward risk, α ∈ {0.2, 0.4, 0.6, 0.8, 1, 2, 3, 4, 5}, |dR 1 1 − θ1 | − |d1 4 and different values of ∈ [0, 0.4] with grid of size 0.02. Each point corresponds to the average over (θ1 , d2 ) ∈ [−1, 1] × [−0.6, 0.6] with a grid of size 0.01.
Figure 8: The effect of attitudes toward risk on the degree of adaptation in decentralized coordination games. 58
A.6
Simulations for Ambiguity Preferences (Centralization)
We now use simulations similar to those described in Sections A.4 and A.5 to argue that strategic uncertainty about communication rules combined with ambiguity-aversion can generate distortions of larger magnitudes than those generated by risk-aversion alone. Moreover, ambiguity-aversion can generate distortions in the right direction even with risk-seeking preferences. To see this, assume that the principal solves the following optimization problem: max
min
d1 ,d2 ∈R µ∈{(p,1−p),(1−p,p)}
−Eµ
(1 − γ)(d1 − ν˜1 )2 + (1 − γ)(d2 − ν˜2 )2 + 2γ(d1 − d2 )2
α
.
Here, µ indexes the principal’s belief system, which specifies beliefs both about ν1 and about ν2 .62 The belief system can be either (p, 1 − p) or (1 − p, p). If µ = (p, 1 − p), p is the probability that ν1 is high as well as the probability that ν2 is low.63 If µ = (1 − p, p), then p is the probability that ν1 is low as well as the probability that ν2 is high. Thus, for any p 6= 1/2, the principal considers two belief systems: one in which the probability that ν1 is high is greater than the probability that ν2 is high, and another in which the probability that ν2 is high is greater than the probability that ν1 is high. Intuitively, for any (ν1 , ν2 ), the principal posterior beliefs can take on one of four values: (ν1 − , ν2 − ), (ν1 − , ν2 + ), (ν1 + , ν2 − ), or (ν1 + , ν2 + ). The principal will use one of two belief systems (p, 1 − p) and (1 − p, p) to compute her expected utility. Ambiguity-aversion will make the principal select the belief system under which “bad” posteriors–posteriors where beliefs about ν1 and ν2 are further apart–are more likely. In the simulation, we consider three possible values for p ∈ {0.1, 0.3, 0.6}. To complete the description of the simulation, we assume that both belief systems are equally likely, so that an ambiguity neutral decision maker will have a posterior belief equal to 1/2 for any of our possible values of p. Our simulation results with different values of p and α are reported in Figure 9 in the appendix. These results show that introducing ambiguity-aversion amplifies the distortions caused by risk aversion considerably. Thus, even with risk neutrality, that is, α = 1, we obtain that the average difference in the distances, over our simulated values of p, is 0.1369 for γ = 1/4 and 0.031 for γ = 3/4. Increasing the risk aversion coefficient to α = 2 increases the average difference in distance to 0.1547 for γ = 1/4 and 0.039 for γ = 3/4, thus more than tripling the average distances compared to an ambiguity neutral but risk averse agent 62 63
Recall that we assume ν1 and ν2 are independent. Formally, p = Prob(˜ ν1 = ν1 + ) and p = Prob(˜ ν1 = ν1 − ).
59
with the same attitudes toward risk. We conclude that reasonable degrees of risk aversion (i.e., α = 2), coupled with extreme aversion to ambiguity, can account for 17% to 55% of the distortions observed in the data. Moreover, note that when p is either sufficiently low or sufficiently high, the simulated distortions are quantitatively close to those for an ambiguity neutral decision maker for values of α in the upper part of the interval [0, 1]. This suggests that ambiguity-aversion can generate a reasonable fit to the data even with moderate risk-seeking preferences.
60
(a) Comparison of average distances between optimal maxmin and risk/ambiguity-neutral deciM − dM M | − |dRN − dRN | for γ = 1 , different degrees of risk aversion α, and different sions, |dM 1 2 1 2 4 distribution parameters p. Each point corresponds to the average over (ν1 , ν2 ) ∈ [−0.6, 0.6]2 with a grid of size 0.05. The grid size for the decisions is 0.05, and the value of = 0.4.
(b) Comparison of average distances between optimal maxmin and risk/ambiguity-neutral deciM − dM M | − |dRN − dRN | for γ = 3 , different degrees of risk aversion α, and different sions, |dM 1 2 1 2 4 distribution parameters p. Each point corresponds to the average over (ν1 , ν2 ) ∈ [−0.6, 0.6]2 with a grid of size 0.05. The grid size for the decisions is 0.05, and the value of = 0.4.
Figure 9: The effect of ambiguity-aversion on coordination behavior in centralized coordination games. 61
References Alonso, R., W. Dessein, and N. Matouschek (2008): “When Does Coordination Require Centralization?” American Economic Review, 98, 145–179. 2, 3, 6, 8, 9, 10, 11, 12, 13, 15, 19, 26, 39, 41, 48, 49 ——— (2015): “Organizing to adapt and compete,” American Economic Journal: Microeconomics, 7, 158–187. 2 Alonso, R. and N. Matouschek (2007): “Relational delegation,” The RAND Journal of Economics, 38, 1070–1089. 6 Ambrus, A. and S. Takahashi (2008): “Multi-sender cheap talk with restricted state spaces,” Theoretical Economics, 3, 1–27. 6 Aoki, M. (1986): “Horizontal vs. Vertical Information Structure of the Firm,” American Economic Review, 76, 971–983. 2 Baliga, S. and T. Sj¨ostr¨om (2004): “Arms races and negotiations,” The Review of Economic Studies, 71, 351–369. 2 Battaglini, M. (2002): “Multiple referrals and multidimensional cheap talk,” Econometrica, 70, 1379–1401. 6, 7 Blume, A., D. V. DeJong, Y.-G. Kim, and G. B. Sprinkle (2001): “Evolution of communication with partial common interest,” Games and Economic Behavior, 37, 79–120. 7, 17 Blume, A. and A. Ortmann (2007): “The effects of costless pre-play communication: Experimental evidence from games with Pareto-ranked equilibria,” Journal of Economic theory, 132, 274–290. 6 Brandts, J. and D. J. Cooper (2006): “A change would do you good....An experimental study on how to overcome coordination failure in organizations,” American Economic Review, 96, 669–693. 6 ——— (2015): “Centralized vs. Decentralized Management: An Experimental Study,” mimeo. 8 Cai, H. and J. T.-Y. Wang (2006): “Overcommunication in Strategic Information Transmission Games,” Games and Economic Behavior, 56, 7–36. 2, 5, 7, 17, 22
Carlsson, H. and E. Van Damme (1993): “Global games and equilibrium selection,” Econometrica: Journal of the Econometric Society, 989–1018. 2 Chandler, A. D. (1977): The visible hand: the managerial revolution in American business, Cambridge, MA: Belknap Press. 2 Chen, Y., N. Kartik, and J. Sobel (2008): “Selecting Cheap-Talk Equilibria,” Econometrica, 76, 117–136. 10 Cooper, R., D. V. DeJong, R. Forsythe, and T. W. Ross (1992): “Communication in coordination games,” The Quarterly Journal of Economics, 107, 739–771. 6 Crawford, V. (1998): “A survey of experiments on communication via cheap talk,” Journal of Economic theory, 78, 286–298. 7 Crawford, V. and J. Sobel (1982): “Strategic Information Transmission,” Econometrica, 50, 1431–1451. 6, 7, 10 Dessein, W. (2002): “Authority and communication in organizations,” The Review of Economic Studies, 69, 811–838. 6 Dessein, W., L. Garicano, and R. Gertner (2010a): “Organizing for Synergies,” American Economic Journal: Microeconomics, 2, 77–114. 2 ——— (2010b): “Organizing for synergies,” American Economic Journal: Microeconomics, 2, 77–114. 6 Dessein, W. and T. Santos (2006): “Adaptive Organizations,” Journal of Political Economy, 114, 956–995. 2 Dickhaut, J. W., K. A. McCabe, and A. Mukherji (1995): “An Experimental Study of Strategic Information Transmission,” Economic Theory, 6, 389–403. 7 Dominguez-Martinez, S., R. Sloof, and F. A. von Siemens (2014): “Monitored by your friends, not your foes: Strategic ignorance and the delegation of real authority,” Games and Economic Behavior, 85, 289–305. 8 Fehr, E., H. Herz, and T. Wilkening (2013): “The lure of authority: Motivation and incentive effects of power,” American Economic Review, 103, 1325–1359. 8 Fischbacher, U. (2007): “z-Tree: Zurich Toolbox for Ready-Made Economic Experiments,” Experimental Economics, 10, 171–178. 16
Green, J. R. and N. L. Stokey (1980): “A two-person game of information transmission,” mimeo. 6 Hagenbach, J. and F. Koessler (2010): “Strategic Communication Networks,” Review of Economic Studies, 77, 1072–1099. 2 Harris, M. and A. Raviv (2005): “Allocation of decision-making authority,” Review of Finance, 9, 353–383. 6 Hayek, F. A. (1945): “The use of knowledge in society,” American economic review, 35, 519–530. 2 Heinemann, F., R. Nagel, and P. Ockenfels (2004): “The theory of global games on test: Experimental analysis of coordination games with public and private information,” Econometrica, 72, 1583–1599. 7, 8 Holmstr¨om, B. R. (1977): “On incentives and control in organizations,” PhD Dissertation. 6 Kahneman, D. and A. Tversky (1979): “Prospect theory: An analysis of decision under risk,” Econometrica, 47, 263–291. 42 K˝oszegi, B. and M. Rabin (2006): “A model of reference-dependent preferences,” The Quarterly Journal of Economics, 1133–1165. 43 Lai, E. K. and W. Lim (2012): “Authority and Communication in the Laboratory,” Games and Economic Behavior, 74, 541–560. 8 McDaniel, T. M. (2011): “Coordination in games with incomplete information: experimental results,” International Game Theory Review, 13, 461–474. 8 McElheran, K. (2014): “Delegation in Multi-Establishment Firms: Evidence from I.T. Purchasing,” Journal of Economics & Management Strategy, 23, 225–258. 8 Milgrom, P. and J. Roberts (1992): Economics, Organization and Management, Englewood Cliffs, NJ: Prentice Hall. 2 Myagkov, M. and C. R. Plott (1997): “Exchange economies and loss exposure: Experiments exploring prospect theory and competitive equilibria in market environments,” The American Economic Review, 801–828. 42 Nyarko, Y. and A. Schotter (2002): “An experimental study of belief learning using elicited beliefs,” Econometrica, 70, 971–1005. 18
Palfrey, T., H. Rosenthal, and N. Roy (2015): “How Cheap Talk Enhances Efficiency in Threshold Public Goods Games,” Games and Economic Behavior, forthcoming. 7 Palfrey, T. R. and H. Rosenthal (1991): “Testing for Effects of Cheap Talk in a Public Goods GHame with Private Information,” Games and economic behavior, 3, 183–220. 7 Qu, H. (2013): “How do market prices and cheap talk affect coordination?” Journal of Accounting Research, 51, 1221–1260. 7 Rantakari, H. (2008): “Governing adaptation,” The Review of Economic Studies, 75, 1257– 1285. 2, 3, 6, 9, 10, 13, 19, 26, 41, 48 S´anchez-Pag´es, S. and M. Vorsatz (2007): “An experimental study of truth-telling in a sender–receiver game,” Games and Economic Behavior, 61, 86–112. 7 Schotter, A. and I. Trevino (2014): “Belief Elicitation in the Laboratory,” Annual Review of Economics, 6, 103–128. 18 Thomas, C. (2011): “Too Many Products: Decentralized Decision Making in Multinational Firms,” America Economic Journal: Microeconomics, 3, 280–306. 8 Tversky, A. and D. Kahneman (1992): “Advances in prospect theory: Cumulative representation of uncertainty,” Journal of Risk and uncertainty, 5, 297–323. 42, 53 Van Huyck, J. B., R. C. Battalio, and R. O. Beil (1990): “Tacit coordination games, strategic uncertainty, and coordination failure,” American Economic Review, 80, 234– 248. 6, 43 Vespa, E. and A. J. Wilson (2015): “Communication with multiple senders: An experiment,” Quantitative Economics, forthcoming. 7 Wang, J. T.-y., M. Spezio, and C. F. Camerer (2010): “Pinocchio’s pupil: Using eyetracking and pupil dilation to understand truth telling and deception in sender-receiver games,” American Economic Review, 100, 984–1007. 7 Wilson, A. J. (2014): “Costly Communication in Groups: Theory and an Experiment,” mimeo. 24 Yang, H. (2012): “Firm Size, Strategic Communication, and Organization,” mimeo. 48
November 23, 2015
Abstract Using novel experimental methods, we study how communication is used to coordinate in centralized and decentralized games where players’ incentives are private and misaligned. We find that subjects respond to changes in incentives to coordinate strategically. In particular, the quality of vertical communication with the principal is significantly higher than the quality of horizontal communication between agents if and only if the importance of coordination is low. Surprisingly, decisions in centralized games underweigh and decisions in decentralized games overweigh the importance of coordination. These distortions in decision rules account for 94% of subjects’ welfare losses, with the remaining 6% due to miscommunication. The distortions can be explained by attitudes toward ambiguity and risk and disappear in an additional experiment with complete information. JEL Classification: C70, D03, C92. Keywords: communication, coordination, decentralization, experiment
∗
ITAM. University of Surrey. This paper was previously circulated as “Incentives to Coordinate in Organizations” and “Mend Your Speech a Little: Authority, Communication, and Incentives to Coordinate.” We are grateful to Wouter Dessein, Daniel Friedman, Felipe Meza, Ryan Oprea, Joel Sobel, Alistair Wilson, and seminar participants at the ITAM Theory Workshop, the University of Minnesota Microeconomic Theory Workshop, the London Experimental Workshop 2015, M-BEES 2015, and the Econometric Society World Congress 2015 for helpful comments. We also thank Luis Aguirre and Alfredo Rubio for excellent research assistance. Financial support from the Asociaci´ on Mexicana de Cultura A.C. is acknowledged. †
1
1
Introduction
Firms coordinate production decisions across divisions; districts in federal systems coordinate policies; international organizations coordinate decisions across countries. Coordination problems play an important role in economic activity, and economists’ interest in them has a long history.1 Decision makers in these problems often have privately known motives affecting their incentives to coordinate.2 In the presence of incomplete information, it is well known that coordination can be facilitated by delegation of authority to the best informed parties3 and by information sharing through communication.4 While much theoretical work has addressed the question of how communication is used in centralized and decentralized coordination problems with incomplete information, the question has not been explored in any existing empirical or experimental study. Our work unifies the experimental literatures on coordination, delegation of authority, and strategic communication to achieve three basic goals. First, we quantify how the quality of communication responds to changes in incentives to coordinate under centralization and decentralization. Second, we explore how subjects make decisions conditional on the communicated information. Third, we assess quantitative deviations from optimal behavior, provide explanations for the observed coordination failures, and show that they are reduced in an otherwise similar setting with complete information. In doing so, we make several methodological contributions to the experimental cheap talk literature. First, we elicit subjects’ beliefs about their matched subjects’ states and use the elicited beliefs to construct an empirical counterpart to the residual variance of communication, a measure commonly used in theoretical work.5 This allows us to formulate predictions about how well subjects communicate without relying on assumptions about how they do it. Second, we use the elicited beliefs together with the equilibrium decision 1
See, for example, Hayek (1945); Chandler (1977); Aoki (1986). Several literatures build on this insight. For example, Carlsson and Van Damme (1993) apply this idea in the context of global games; Baliga and Sj¨ostr¨om (2004) in the context of games of conflict; Dessein and Santos (2006) in the context of organizational economics. 3 See, for example, Milgrom and Roberts (1992) for a discussion of the “delegation principle.” 4 See Alonso et al. (2008), Rantakari (2008), Dessein et al. (2010a), Hagenbach and Koessler (2010), Alonso et al. (2015). 5 We also perform robustness checks of our results that do not rely on belief elicitation. Theoretically, communication quality is measured as the residual variance of the message receiver’s posterior around the sender’s privately known state. I.e., letting θi denote player i’s state and mi theh player’s message i 2 about the state to the receiver, the residual variance of communication is defined as E (θi − E[θi |mi ]) . In contrast, other studies of cheap talk games, such as Cai and Wang (2006), have used the correlation between messages and states to measure the quality of communication. 2
2
rules to study how subjects decide conditional on the communicated information.6 This allows us to test theoretical predictions about subjects’ decision rules directly. Third, we use the elicited beliefs to perform a detailed welfare analysis that decomposes subjects’ welfare losses into a component due to miscommunication and a component due to deviations from equilibrium behavior. Our experiment uses a 2x2 design that independently manipulates the structure of a coordination game (centralized vs. decentralized) and the importance of coordination for individual payoffs (high vs. low). A decentralized game is played between two agents, with each agent having to make a decision.7 Each agent has private information about his local conditions, which affect the payoff the agent receives from his own decision. The agent incurs an adaptation loss if his decision fails to adapt to his local conditions, and a coordination loss if his decision is not perfectly aligned with the decision of the other agent. He therefore faces a trade-off between adaptation and coordination. The agents can communicate with each other before making their decisions. In a centralized coordination game, decision rights are delegated to an unbiased coordinator, referred to as the principal, who maximizes joint profits and is uninformed about both local conditions. Agents can communicate their private information to the principal before the decisions are made. In both types of games, communication is informal and occurs through cheap talk. The first of our main findings is that the quality of (vertical) communication with the principal is significantly higher than the quality of (horizontal) communication between agents if and only if the importance of coordination is low. This result is remarkably consistent with the theoretical predictions of Alonso et al. (2008) and Rantakari (2008). In theory, the fact that the principal maximizes joint payoffs implies that the agents’ incentives are more aligned with the principal than they are with each other. Thus, the quality of vertical communication should always be higher than that of horizontal communication.8 As the importance of coordination grows, the agents’ incentives become more aligned with each other but less aligned with the incentives of the principal, and the difference in quality of vertical and horizontal communication should shrink to zero.9 This is what we observe 6
As discussed in Section 6.2.1, we also use other proxies for the message receivers’ posteriors. In applications of the model, an agent could be a manager in charge of a division or a function within a firm, a local district, a state government, etc. 8 It is well-known that cheap talk games admit a multiplicity of equilibria, including an equilibrium which is completely uninformative. The theoretical literature usually focuses on the most informative equilibrium from an ex-ante perspective. We follow the approach used in the literature because it allows to formulate a clear set of equilibrium predictions about communication quality. 9 An increase in the importance of coordination requires the agents to coordinate their decisions more to avoid large losses from miscoordination. This implies that agents have lower incentives to strategically 7
3
in the data. Predictions about communication quality underlie predictions about coordination. The second of our main findings is that behavior responds to changes in incentives to coordinate roughly as theory predicts. Taking subjects’ (elicited) posterior beliefs as given, we estimate the coefficients on states and posteriors in subjects’ decision rules under the null hypothesis of equilibrium behavior.10 Both under centralization and decentralization, we find that the decision rules show significantly more coordination and significantly less adaptation when the incentives to coordinate are greater. The third of our main findings is that subjects’ decision rules show significant quantitative deviations from equilibrium. The agent systematically underweighs the importance of adaptation to himself, while the principal systematically underweighs the importance of coordination for the maximization of joint payoffs. As a result, decisions under decentralization tend to be more coordinated (and thus less adapted to local conditions) than decisions under centralization, despite the fact that the principal could improve performance by coordinating the agents’ decisions more. Using subjects’ recovered beliefs together with the equilibrium decision rules, we perform a structural estimation exercise to quantify the extent to which subjects’ decisions are distorted away from the theoretical optima.11 We also estimate the effect of these distortions on welfare. We find that subjects’ observed welfare losses amount to approximately 3.75 times the losses predicted by the payoff-maximizing benchmark. Moreover, 94% of the difference between optimal and observed losses can be explained by distortions in decision rules, with the remaining 6% due to miscommunication. Thus, the welfare cost of these distortions is large. Under decentralization, adaptation involves no uncertainty while coordination involves the other agent’s unknown decision. Under centralization, both decisions are made by the principal, and coordination involves no uncertainty, while adaptation involves the two agents’ unknown states. Thus, the decision maker overweighs the uncertain part of his manipulate their information. On the other hand, the principal cares less about the agents’ local conditions when coordination becomes more important. Therefore, the agents have higher incentives to misreport. 10 To our knowledge, no previous study of cheap talk games has applied belief elicitation in this manner. 11 Our design parameterizes the importance of coordination through a parameter γ which varies on a scale from 0 (only adaptation matters) to 1 (only coordination matters). We find that the principal in a centralized coordination game behaves as if coordination was irrelevant (ˆ γ ≈ 0) when the importance of coordination is low (γ = 0.25), and less than half as important as it should be (ˆ γ = 0.356) when the need to coordinate is high (γ = 0.75). Behavior of the agent in a decentralized coordination game exhibits the opposite pattern. Thus, the agent behaves as if coordination was twice as important as it should be when γ is low, and as if adaptation is almost irrelevant (ˆ γ = 0.9396) when γ is high.
4
payoff function in both centralized and decentralized coordination games. In Section 7, we provide some possible underlying reasons for this overweighing. In particular, we show that risk-aversion can generate distortions of decision rules in the observed direction, and that the magnitudes of these distortions can be amplified by strategic uncertainty and ambiguity-aversion. To test the hypothesis that the distortions are generated by uncertainty, we run an additional experiment with complete information. In line with the hypothesis, we find no systematic distortions in decision weights in this second experiment. Some of our results stand in contrast to others reported in the cheap talk literature. Thus, unlike Cai and Wang (2006), we find no evidence of overcommunication in the experiment. Indeed, relative to the Most Informative Equilibrium (MIE) benchmark, we find evidence of undercommunication in one of our treatments. Importantly, the same treatment is also the only treatment in which we document significant losses in welfare as a result of subjects not communicating optimally. Overall, however, the welfare costs of undercommunication in the experiment are small. Our findings have important implications for the study and design of institutions. In particular, we highlight a novel hidden cost that is systematically related to uncertainty in coordination problems. While we focused attentions on coordination games in which decision rights are either completely centralized or completely decentralized, the results can be used to predict behavior in related settings. For example, in partially centralized organizations where the principal retains authority over some but not all of the decisions, we predict that the agents entitled with authority will still under-adapt their decisions to their local conditions. Our findings are also relevant to the design of political institutions. If policy coordination requires expertise on different dimensions, our results suggest that an incumbent politician may underestimate the importance of coordination and put excessive weight on the information provided by experts. Therefore, the value of expertise might have to be weighed against uncertainty-induced distortions in decision making. The remainder of the paper is structured as follows. Section 2 summarizes the related literature; Section 3 introduces the model; Section 4 discusses the theoretical predictions; Section 5 explains our experimental design; Section 6 describes the results; Section 7 provides a discussion of these results; Section 8 describes the results of the second experiment with complete information; Section 9 concludes.
5
2 2.1
Literature Review Theoretical Literature
Since the seminal contributions of Green and Stokey (1980) and Crawford and Sobel (1982), the theoretical literature on strategic communication has grown significantly. The studies most related to ours are those that applied strategic communication to the study of organizations, such as Alonso et al. (2008), Rantakari (2008), and Dessein et al. (2010b). A line of literature dating back to Holmstr¨om (1977) has studied delegation of authority in organizations. Dessein (2002), Harris and Raviv (2005), and Alonso and Matouschek (2007) do this in a setting with cheap talk communication. A common thread in this literature is the “delegation principle,” which is the notion that decision power in a setting with private information should be granted or delegated to the most informed agent. While testing the delegation principle is not the purpose of our study, our design allows us to study how subjects behave in settings where the question of delegation is important. Another strand of the literature that relates to ours has studied communication with multiple senders. Battaglini (2002) shows that full revelation of information can be an equilibrium even when players’ preferences are arbitrarily misaligned. Ambrus and Takahashi (2008) shows that full revelation may not be achieved in equilibrium if the state space is restricted. These studies, however, focus on communication of private information about the same random variable. In our setting, the players communicate private information about their own states.
2.2
Experimental and Empirical Literature
Experimental economists have long been interested in coordination in the laboratory. Van Huyck et al. (1990) study a minimum-effort coordination game and show that large groups tend to choose the lowest effort level even after several repeated interactions. Using a similar game, Brandts and Cooper (2006) show that strengthening financial incentives can overcome coordination failures. More related to ours are studies exploring the role of communication as coordination device (see, e.g., Cooper et al., 1992; Blume and Ortmann, 2007). While in these studies subjects communicate information about their strategies, in our setting subjects communicate information about their privately known states.
6
The experimental literature on cheap talk games is reviewed in Crawford (1998). This literature is large, and the strand most relevant to our study is that which focused on communication of private information in settings similar to Crawford and Sobel (1982), such as Dickhaut et al. (1995), Blume et al. (2001), Cai and Wang (2006), S´anchez-Pag´es and Vorsatz (2007), and Wang et al. (2010).12 Dickhaut et al. (1995) study communication in sender-receiver games, manipulating the degree to which players’ preferences diverge. They find that as preferences diverge less, more information is transmitted, results broadly in line with the Crawford and Sobel (1982) prediction. Cai and Wang (2006) find a similar result in a setting which allows for a more direct test of the theory. Their study also provides some evidence that participants can be broadly classified in two groups: those with a preference for truth-telling, and those driven by material incentives. S´anchez-Pag´es and Vorsatz (2007) provide additional evidence for this observation. Unlike previous studies of cheap talk games, we investigate the role of communication in coordinating subjects’ decisions. Two related papers are Palfrey and Rosenthal (1991) and Palfrey et al. (2015). Both papers study the effect of cheap talk communication on the efficient provision of a public good with heterogenous and privately known costs. However, our framework differs substantially from theirs. First, we focus on the impact of a higher need to coordinate and of the allocation of decision rights on agents’ behavior. These effects cannot be jointly tested in the context of a public good game. Second, incentives for information transmission in our experiment arise from a private adaptation motive, while those authors focus on free-riding. Finally, public good games endow each agent with an action, no contribution, which effectively insulates that agent from the actions chosen by other agents. By contrast, in our framework each agent is always affected by the action chosen by his partner, thus affecting both the incentives to communicate and coordinate. Therefore, the environment and the forces at work in our study are different from those in Palfrey and Rosenthal (1991) and Palfrey et al.(2015), albeit complementary. Heinemann et al. (2004) study coordination under incomplete information in the context of a global game where each subject receives a conditionally independent signal about the common state of the world. However, subjects face no private adaptation motive and no communication takes place. Qu (2013) experimentally investigates communication in a global game environment with private signals, but still no incentive misalignment, 12
While most experimental studies of cheap talk games consider correlations between messages, decisions, and states as measures of the informativeness of communication, in Vespa and Wilson (2015), the players’ actions should at least in theory be equal to their posteriors, making their measure of communication quality closer to ours. Their study, however, tests the predictions of Battaglini (2002) and is therefore only loosely related.
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similar to Heinemann et al., and finds that communication increases the likelihood of attack/investment. These experiments differ from ours because coordination failure always affects an agent’s payoff in our experiments, while in global games environments (as in the case of public goods) agents can decide to take an outside option.13 We also investigate both centralized and decentralized coordination games. Delegation of authority has been investigated experimentally by Lai and Lim (2012), Fehr et al.(2013), and Dominguez-Martinez et al. (2014). Of these, the former paper is most related to ours since its environment allows for cheap talk communication. Unlike all previous delegation studies, we investigate the role of communication in coordination decisions. We are also not interested in subjects’ decisions to delegate authority. We manipulate decision authority experimentally and instead ask the question of how communication quality and behavior depend on the degree of centralization of decision rights. A concurrent related study is Brandts and Cooper (2015). While they also compare centralized and decentralized coordination games, they do not test predictions about communication quality, and the games they use are substantially different from ours. For example, agents are symmetrically informed about each others’ local conditions as well as a global state of the world, which affects the payoffs of each player, while the principal is uninformed about the global state but informed about the agents’ local conditions. They focus on testing a payoff prediction similar to the one in Alonso et al. (2008) regarding the dominance of decentralization over centralization (under some conditions), and instead find that performance is generally higher under centralization. Unlike Brandts and Cooper (2015), we focus on communication quality and its effect on coordination. While we formulate predictions about adaptation and coordination losses, for reasons discussed in Section 4 we do not focus on predictions about performance of different types of organizations. Finally, coordination with incomplete information has been explored in recent empirical work (McElheran, 2014; Thomas, 2011), which has focused on the interaction between incentives to coordinate and the allocation of decision rights. While data on communication (e.g., through e-mail) can in principle be collected and analyzed, the laboratory provides the ideal environment to measure communication quality and its effect on decision making. 13
McDaniel (2011) studies cheap talk communication in an environment with one-sided incomplete information. Only one of two players knows which game is being played, a prisoner’s dilemma game or a stag-hunt game. The informed player can send a binary message to the uninformed player before actions are chosen. Therefore, subjects face uncertainty about whether the incentives of the informed party are misaligned. We instead study environments with two-sided incomplete information in which incentives are misaligned with probability one.
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3
Model
Our experimental design is based on the theoretical models of Alonso et al. (2008) and Rantakari (2008). We consider two different types of coordination games which we refer to as centralized and decentralized. Both types of games share the following features: (1) there are two decisions (d1 and d2 ) to be made; (2) there are two agents: Player 1, and Player 2; (3) the agents face a trade-off between coordinating those decisions and adapting each decision to some local conditions. In both games, the payoff of Player i, i = 1, 2, is given by πi = −(1 − γ)(di − θi )2 − γ(di − dj )2 ,
i = 1, 2,
i 6= j,
(3.1)
where θi is Player i’s state, or local conditions. The payoff of Player i has two components. The first component captures the adaptation loss which arises when the decision di is far from Player i’s local conditions, θi . The second component represents the possible coordination loss which increases in the distance between the two decisions. The parameter γ ∈ [0, 1] measures the importance of coordination for the players. The information structure is the following. Player 1 knows his own state, θ1 , but not Player 2’s state, θ2 . Similarly, Player 2 knows his own state, θ2 , but not Player 1’s state, θ1 . It is common knowledge that θ1 and θ2 are uniformly distributed on the interval [−1, 1].14 The states are drawn independently. Under decentralization, Player 1 makes decision d1 ∈ R and Player 2 simultaneously makes decision d2 ∈ R. Before the decisions are made, Player 1 sends a cheap talk message m1 ∈ M1 to Player 2, and Player 2 simultaneously sends a cheap talk message m2 ∈ M2 to Player 1. This is referred to as horizontal communication. Under centralization, there is an additional Player 3 (the principal), who chooses both d1 ∈ R and d2 ∈ R. Player 3 observes neither θ1 nor θ2 , and her payoff from the decisions is given by Π = π1 + π2 . Before Player 3 makes her two decisions, Player 1 and Player 2 simultaneously send cheap talk messages m1 ∈ M1 and m2 ∈ M2 to Player 3. This is referred to as vertical communication. 14
Alonso et al. (2008) and Rantakari (2008) consider a more general framework in which θi is drawn uniformly from [−si , si ], with si > 0.
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4
Predictions
4.1
Communication
A communication equilibrium is formally defined by communication rules for Player 1 and Player 2, decision rules for the decision makers, and belief functions for the message receivers such that the communication rules are optimal given the decision rules, the decision rules are optimal given beliefs, and the beliefs are derived from the communication rules using Bayes’ Rule whenever possible. It is well-known that games of strategic information transmission admit a multiplicity of such equilibria.15 We formulate our predictions about equilibrium communication around the Most Informative Equilibrium (MIE).16 This selection principle is appealing for two reasons: it maximizes ex ante payoffs and generates the elegant quantitative predictions described below. We take the MIE predictions as a useful guide for how incentives to coordinate affect communication and behavior in our coordination games, recognizing that similar incentives affect behavior in equilibria close to the most informative one, and interpret quantitative deviations as evidence against MIE. Alonso et al. (2008) show that any finite communication equilibrium is economically equivalent to a partitional one. Partitional equilibria share the intuitive property that the incentives to misrepresent information are increasing in |θi | under both centralization and decentralization.17 While studying the form of equilibrium communication is not the focus of our paper, this prediction is easily testable, and we explore it in Section 6.1. Following the theoretical literature, we measure the quality of communication through 15
In a setting similar to Crawford and Sobel (1982), any equilibrium is also economically equivalent to one in which the communication rules take a partitional form: a sender partitions the state space and only communicates which element of the partition the realized state belongs to. It is also well-known that an uninformative equilibrium always exists in such games. 16 In Crawford and Sobel (1982), only a finite number of messages can be sent in the Most Informative Equilibrium (MIE). In the framework of Alonso et al. (2008) and Rantakari (2008), however, a countably infinite number of messages can be sent because there exists a state where the preferences of the sender and receiver coincide. Recently, Chen et al. (2008) provide conditions which uniquely select MIE in cheap talk games. 17 Alonso et al. (2008) and Rantakari (2008) show that an arbitrary partitional equilibrium with N partition elements solves the following difference equation: ai,n+1 − ai,n = ai,n − ai,n−1 + 4bai,n .
(4.1)
This shows that moving away from θi = 0 in either direction leads to progressively larger partition elements.
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the residual variance of the posterior around the true value of the state, defined as E [(θi − E[θi |mi ])2 ]. The advantage of such a measure in experimental settings is that the residual variance of communication is defined independently of the partitional structure of equilibrium. Therefore, it can be used to measure communication quality whether or not players are conforming to any particular equilibrium or even exhibiting non-equilibrium behavior. Residual Variance Most Informative Equilibrium
1 12
Horizontal
1 21
Vertical Truth-telling equilibrium 0
Importance of coordination, γ
1
Figure 1: Predicted communication quality as a function of γ. If γ ∈ (0, 1), it is shown in Alonso et al. (2008) that the residual variance of communication in MIE under decentralization is given by 1 E (θi − E[θi |mi ])2 = if i = 1, 2. (4.2) 12 + 9γ Under centralization, the residual variance of communication is given by γ E (θi − E[θi |mi ])2 = if i = 1, 2. 9 + 12γ
(4.3)
Figure 1 plots the residual variance of communication in MIE.18 18
Note that when coordination is irrelevant (γ = 0), it is an equilibrium to tell the truth about one’s state and set d1 equal to θ1 and d2 equal to θ2 . This is true in both the centralized and the decentralized game. Because the residual variance of communication in the truth-telling equilibrium is equal to zero, the residual variance of horizontal communication exhibits a discontinuity at γ = 0. Both residual variances also exhibit a discontinuity at γ = 1 in MIE, because truth-telling can be sustained in equilibrium when coordination is the only relevant task, given that private information has no value. In principle, these discontinuities may be behaviorally relevant. For example, it could be that when γ is low the players decide to play the game ignoring coordination, in which case full revelation is an equilibrium. This, however, is not observed in our data.
11
Because the incentives of Player 1 and Player 3 are more aligned than the incentives of Player 1 and Player 2 (and the incentives of Player 2 and Player 3 are more aligned than the incentives of Player 2 and Player 1), the quality of vertical communication is greater than that of horizontal communication for any value of γ. However, the comparative statics in the two regimes are different. Under horizontal communication, a higher γ makes players’ preferences more aligned. As a result, the players misrepresent their states less and the quality of communication improves. Under centralization, a higher γ makes Player 3 care less about meeting the states of Player 1 and Player 2, who in response misrepresent their states more. Thus, a higher need for coordination worsens communication quality under centralization. This implies the following prediction: Prediction 1. As the importance of coordination, γ, increases, the quality of horizontal communication improves, while the quality of vertical communication declines. Notice that when γ is close to zero, the difference between the quality of vertical and horizontal communication is nearly maximal. As γ grows, this difference tends to zero, although a discontinuity is present at γ = 1 as noted above. This suggests the following: Prediction 2. When γ is small, the quality of vertical communication is significantly higher than that of horizontal communication. When γ is large, the difference in quality of horizontal and vertical communication is small. We note that Prediction 1 and Prediction 2 are robust to Player 1 and Player 2 having social preferences.19 Alonso et al. (2008) consider a variation of the model described above in which Player 1 maximizes λπ1 + (1 − λ)π2 and Player 2 maximizes (1 − λ)π1 + λπ2 , where λ ∈ 12 , 1 . Although the payoff functions used in the experiment set λ = 1, it is in principle plausible that Player 1 and Player 2 assign strictly positive weights to each other’s payoffs. However, for a fixed γ, it can be shown that the difference in quality of vertical and horizontal communication, while shrinking as λ approaches 12 , remains strictly positive. Similarly, for any fixed λ, the arguments regarding the effect of γ on the quality of communication remain valid. 19
Player 3 is already maximizing joint payoffs.
12
4.2
Decisions
In any 20 communication equilibrium, players’ decisions depend on their beliefs about other players’ local conditions, which in turn depend on messages sent by the other players.21 Suppose that Player i sent a message mi . We denote by νi = E[θi |mi ] the posterior expectation about i’s state held by the receiver of the message. Under decentralization, Player i makes the following decision in equilibrium after receiving the message mj : dD i = (1 − γ)θi +
γ2 γ νi + νj , 1+γ 1+γ
i = 1, 2,
i 6= j.
(4.4)
Thus, each player’s decision rule is a linear function of his own state θi , his own posterior νj , and the other player’s posterior νi . In the experiment, we consider two parameters of γ: 14 and 34 . Let us consider the decision rule of Player i. When γ = 14 , the importance of coordination is low, and the weight that Player i assigns to his own state is high (and equal to 34 ). There is a small weight of 0.05 on the other player’s posterior about Player i’s state, νi , and a small weight of 0.2 on Player i’s own posterior about the other player’s state, νj . When the importance of coordination is high, Player i’s own state matters less (weight 41 ). The other player’s posterior about Player i’s state, νi , becomes more important (with a weight of ≈ 0.32), and there is a substantially larger weight on Player i’s posterior about the other player’s state, νj (≈ 0.43). This leads to the following prediction: Prediction 3. As the importance of coordination increases, the decision rule under decentralization puts a smaller weight on the state and larger weights on posterior beliefs. Under centralization, Player 3 makes the following decisions in equilibrium after receiving the message m = (m1 , m2 ) dC i =
1+γ 2γ νi + νj , 1 + 3γ 1 + 3γ
i = 1, 2,
i 6= j.
(4.5)
Notice that now the decision rules are only functions of the decision maker’s posteriors about the states of Player 1 and Player 2. When the importance of coordination is low, the posterior about Player i’s state has a much larger weight in determining di (≈ 0.71) than the posterior about the state of Player j (≈ 0.29). When the importance of coordination is high, the weights on the two posteriors are closer to each other (with a weight of ≈ 0.54 on own posterior). The following prediction is implied: 20
I.e., not only the most informative one. See Alonso et al. (2008) and Rantakari (2008) for derivations of the decision rules described in this section. 21
13
Prediction 4. As the importance of coordination increases, the decision rule under centralization puts a smaller weight on the division’s own posterior and a larger weight on the posterior about the other division’s state.
4.3
Adaptation and Coordination
We can use the decision rules above to formulate predictions about average degrees of adaptation and coordination in the experiment. The advantage of centralization lies in the principal’s ability to perfectly control the degree to which the decisions are coordinated with each other. The principal, however, lacks complete knowledge of the agents’ local conditions, which makes adaptation difficult. On the other hand, the agents under decentralization can perfectly control the degree of adaptation of their decisions to their own local conditions. Coordination, however, is more difficult in this case because the agents only control their own decisions. Let CLk = E[(d1 − d2 )2 ], k ∈ {C, D}, denote the expected (normalized) coordination loss.22 In MIE, the principal’s comparative advantage at coordination generates a smaller coordination loss under centralization than under decentralization. Moreover, as coordination becomes more important, decisions become more correlated in both centralized and decentralized games, which leads to lower coordination losses.23 This implies the following predictions, proved in Proposition 1 of the appendix: Prediction 5. For both centralized and decentralized coordination games, an increase in the importance of coordination reduces the average coordination loss. Prediction 6. The average coordination loss is larger under decentralization than under centralization. Similarly, we can compute the expected (normalized) adaptation loss in MIE for an arbitrary agent i, denoted by ALik = E[(dki − θik )2 ], k ∈ {C, D}. The principal’s full internalization of the need to coordinate leads to larger adaptation losses under centralization than under decentralization. As coordination becomes more important, adaptation 22
This quantity is normalized by γ and hence does not represent actual utility or point losses. The model is sufficiently tractable to also allow the computation of correlations between decisions, and between decisions and states, in closed form. However, we base our predictions on coordination and adaptation losses (see below) because they are easier to test compared to predictions based on correlation coefficients. 23
14
losses increase both under centralization and decentralization.24 This implies the following predictions, proved in Proposition 2 in the appendix: Prediction 7. For both centralized and decentralized coordination games, an increase in the importance of coordination increases the average adaptation loss of each agent. Prediction 8. The average adaptation loss is larger under centralization than under decentralization. We note that Prediction 5 through Prediction 8 are robust to social preferences, as proved in Lemma 2 of Alonso et al. (2008).
4.4
Payoffs
When γ = 0, the best that subjects can do in equilibrium under centralization is to communicate truthfully, set di = θi , and incur zero losses. Likewise, when γ = 1, the best the principal can do is to set the decisions equal to each other. This shows that the relationship between γ and output under centralization is non-monotonic. A similar argument holds for decentralization: the smallest losses are zero both when γ = 0 and when γ = 1. Nevertheless, it can be shown theoretically that centralization dominates decentralization if and only if γ > γ¯ ≡ 0.161.25 We do not emphasize this prediction, however, because it is much weaker than Predictions 1-8. In particular, the prediction is valid only if players are sufficiently self-interested. Indeed, it is the message of Alonso et al. (2008) that if the social preferences parameter λ is close to 12 , expected joint payoffs under decentralization can be higher that under centralization for any value of γ. Moreover, the difference in payoffs under centralization and decentralization is maximized at γ ≈ 0.45. Our experimental design is not optimized to capture differences in payoffs, and we leave this task to a future study. 24
This has no immediate implication for welfare because an increase in γ also implies that adaptation losses have a lower impact on welfare. 25 When γ ∈ [0, 1) and players only care about their own earnings, dividing the payoff functions (3.1) γ by (1 − γ) and setting 1−γ = δ, our payoff functions coincide with those of Alonso et al. (2008) for the case in which Player 1 and Player 2 only care about their own payoff. Alonso et al. (2008) show in the proof of their Proposition 5 that ΠC − ΠD > 0 if, and only if, 33δ 3 + 32δ 2 + 3δ − 2 > 0. This inequality is equivalent to 6γ 3 + 20γ 2 + 9γ − 2 > 0. γ¯ is the only positive root of the equation 6γ 3 + 20γ 2 + 9γ − 2 = 0. Intuitively, when γ is large, the players mostly care about coordination, and it is much easier for Player 3 to coordinate d1 and d2 (he or she can always set the two decisions equal to each other) than it is for Player 1 and Player 2, who need to rely on each other’s messages.
15
5
Experimental Design
The experiment was conducted at Instituto Tecnol´ogico Aut´onomo de M´exico in Mexico City between October and November 2014 using the software z -Tree (Fischbacher, 2007). After entering the laboratory, sitting down at their computer terminals, and signing the consent forms, the subjects were distributed their treatment’s instructions.26 At the same time that the subjects were reading the instructions, a quiz was displayed on their computer screens. The subjects were informed that they have 20 minutes to read the instructions and complete the quiz. The quiz had 8 questions that were identical for all treatments, and 4 that differed across the decentralized and centralized treatments. It tested the subjects’ understanding of statistical independence, how they were to be matched in the experiment, the conversion of points to pesos, and the game’s basic structure. The answers to all of the quiz questions were incentivized: each subject gained one Mexican peso for each quiz question correctly answered. The questions are included in the online appendix. Each session consisted of two practice rounds followed by fifteen rounds which counted towards each subject’s earnings. Subjects were randomly and anonymously matched at the beginning of each round. The subjects’ earnings were determined as follows. Every subject was guaranteed a 30 Mexican pesos (≈ US$2) show up fee in addition to the earnings from the quiz. These earnings were called the subject’s “guaranteed earnings.” In addition, each subject was given 210 Mexican pesos (≈ US$15). In each (non-practice) round of the game, the decisions of each subject and his matched partners led the subject to lose a number of points. The subject’s “additional earnings” were determined as follows: Additional earnings = 210 − 3 × T otal points lost during the experiment. In addition to losing points from the game, the subjects lost points from their conjectures of other players’ states in accordance to a quadratic scoring rule. This “guessing game” is described in more detail below. Each subject’s total earnings were given by the sum of his/her guaranteed and additional earnings.27 It was explained to the subjects that any subject losing more than 50 cumulative points (150 Mexican pesos) would be excluded from further matches, and that in the event this happens, the remaining subjects will be rematched with each other, with some randomly chosen subjects sitting out in each subsequent match. In practice, this never happened during any of our sessions, although 26
See the online appendix at http://piotr-evdokimov.com/Appendix-Instructions.pdf. While the sample instructions are in English, the actual instructions were administered in Spanish. 27 The instructions provided subjects with several examples of final earnings as a function of points lost, and the quiz tested their understanding of the payment rules with yet another example.
16
the program we used allowed for the contingency.
5.1
Treatments
Our first experiment included four treatments: Decentralized-High, Centralized-High, Decentralized-Low, Centralized-Low.28 In the two Decentralized treatments, decision d1 was made by Player 1, decision d2 was made by Player 2, and each match consisted of two players. In the two Centralized treatments, each match had three players, and the decisions d1 and d2 were made by Player 3, whose points lost in every round were given by the average of the points lost by Player 1 and Player 2: π3 =
π1 + π 2 . 2
In the High treatments, the points lost by Player 1 and Player 2 in each round of the game were determined by the following formula: πi = −(di − θi )2 − 3 · (di − dj )2
i = 1, 2, i 6= j.
(5.1)
Thus, the High treatments placed a higher weight on coordinating d1 and d2 than on adapting to each state θi . The Low treatments placed a high weight on adaptation to θi : πi = −3 · (di − θi )2 − (di − dj )2
i = 1, 2, i 6= j.
(5.2)
In all of our treatments, the state, message, and decision spaces were restricted to be equal to each other. The θi , mi , and di variables were all selected in increments of 0.01 from the set {−1, −0.99, −0.98, ..., 0.98, 0.99, 1}. The decision space was restricted because allowing the decisions to be elements of R would have made it possible for a player who behaves randomly to sustain enormous losses, making the experiment infeasible. As several previous experimental studies, such as Cai and Wang (2006), we restricted the message space to be equal to the state space. This restriction can be motivated from the observation that subjects tend to interpret messages in cheap talk games using a natural language (see, e.g., Blume et al., 2001). Note that this restriction leaves the predictions described above unchanged. The timing in the Decentralized treatments proceeded as follows. First, the players i = 1, 2 were asked to simultaneously send their messages. Second, they were asked to 28
The second experiment is described below in Section 8.
17
simultaneously make their decisions. After the decisions were made, the players simultaneously made incentivized conjectures of each other’s states. At the end of each match, the players received feedback about the other player’s state, the other player’s decision, own points lost due to the decisions made, own points lost from the conjecture about the other player’s state, points lost in the round, points lost so far, and pesos lost so far. In the Centralized treatments, Player 1 and Player 2 started each match by sending their messages to Player 3. The screens they saw at this stage were identical to those displayed in the decentralized treatments. While these players decided what messages to send, Player 3 waited. After the messages were sent, Player 3 made his/her decisions. After the decisions were made, all players made conjectures about the states: Player 1 guessed θ2 ; Player 2 guessed θ1 , and Player 3 guessed both θ1 and θ2 . As in the decentralized treatments, these conjectures were incentivized. Finally, all players received feedback about the unknown state(s), Player 3’s decisions, own points lost from the decisions made, own points lost from the conjecture(s) about the other state(s), points lost in the round, points lost so far, and pesos lost so far. Subjects’ conjectures of each other’s states were obtained with quadratic scoring rules (Nyarko and Schotter, 2002). In both the centralized and the decentralized treatments, Player 1 guessed the state of Player 2, and Player 2 guessed the state of Player 1. For these players, the points lost for the guesses were equal to the square of the distance between the conjecture and the true value of the state being guessed. Formally, denote Player i’s conjecture about θj , conditional on having received message mj , by p(θj |mj ). The points lost for the conjecture are given by (p(θj |mj ) − θj )2 . In the centralized treatments, Player 3 also guessed the states of both Player 1 and Player 2. For this player, the points lost were equal to the average of the two squared distances to ensure that his losses for the guesses do not differ in magnitude from those of Player 1 and Player 2. Quadratic scoring rules incentivize risk-neutral subjects to report their mean beliefs truthfully. While there have been attempts at incentive compatible belief elicitation under risk-aversion, such scoring rules are more complicated and the evidence regarding their performance is mixed (Schotter and Trevino, 2014). In our analysis, we take the conjectures made by the subjects about their partners’ states to be proxies of the posteriors, E[θ1 |m1 ] and E[θ2 |m2 ], described in Section 3. We refer to Section 6.2.1 for a discussion of robustness checks supporting the validity of our belief elicitation procedure. Recall that subjects’ decision rules are in theory functions of their posterior beliefs. This allows us to obtain implied (implicit) equilibrium beliefs by inverting the decision rules. 18
In Section 6.1.2, we test the predictions about communication quality using both explicit and implicit beliefs and show that the results are qualitatively similar. Belief elicitation, however, is a necessary component of our experimental design because it allows to analyze how subjects’ decision rules deviate from equilibrium conditional on communication. Moreover, we argue in Section 6.1.2 that the implicit beliefs–which assume equilibrium–are much less likely to reflect subjects’ true beliefs in the experiment than the elicited ones.
6
Results
For the first experiment, we collected data from 238 undergraduate students recruited from introductory level classes. A total of 14 experimental sessions were conducted with a minimum of 11 students and a maximum of 21 students in each session. An average session lasted for 75 minutes with an individual average payment of 163.5 Mexican pesos (≈ US$11), excluding the show-up fee and the payment from the quiz. The distribution of these subjects among treatments is shown in Table 1. Note that more subjects participated in the centralized treatments. This was to ensure that the amount of observations (e.g., for d1 and d2 ) is not too unbalanced.29
Decentralized Centralized
Low γ 3 sessions N = 48 4 sessions N = 66
High γ 3 sessions N = 56 4 sessions N = 68
Table 1: Subjects per treatment.
Our results are summarized in Table 2. We find that the main prediction of Alonso et al. (2008) and Rantakari (2008) regarding the response of communication quality to a need to coordinate cannot be rejected: the quality of horizontal communication is significantly lower than that of vertical communication if and only if the need to coordinate is low. 29
The number of participants in the Centralized-High treatment is not divisible by three. This is because one of the subjects experienced a health issue while the instructions were being administered and had to leave the room. We re-calibrated the program in this session to accommodate 11 subjects rather than 12. This was accomplished by matching 9 people in every round of the game, with two remaining participants sitting out randomly. We also informed the participants in this session about the new rematching procedure. Our results do not significantly change if this session is excluded from the analysis.
19
Moreover, subjects’ decision rules respond to changes in incentives for coordination largely as theory predicts. We also find, however, that subjects in centralized (decentralized) treatments coordinate (adapt) less than predicted. In Section 7, we hypothesize that this is caused by uncertainty and provide a theoretical discussion of possible underlying channels for the effect of uncertainty on subjects’ decisions. In Section 8, we present the results of an additional experiment without incomplete information in which subjects’ behavior is significantly more consistent with equilibrium. The remainder of this section describes the results of the first experiment in detail.
Communication Quality Decision rules
Adaptation and Coordination
Prediction Prediction Prediction Prediction Prediction Prediction Prediction Prediction Prediction
1 2 3 4 5 6 7 8
Equilibrium Concept MIE MIE PBE PBE MIE MIE MIE MIE
Evidence?
Where
Yes
Result 2
Partial
Result 3
Partial
Result 5
Table 2: Predictions and results.
6.1 6.1.1
Communication Summary of the Data
While 63.78% of the messages in the experiment are truthful, only 44.48% of messages are believed. Counting each decision as an observation, we find that 32.67% of observations are associated with both a truthful sender and a trusting receiver. 22.27% of subjects are always truthful in their messages; the rest misrepresent the true value of the state at least once.30 These results suggest that the subjects are neither completely honest nor completely trusting. In theory, all of our treatments provide participants with incentives to exaggerate their states. The standard argument is that other players care less about θi than Player i does. This leads Player i’s partners to make decisions that are not sufficiently extreme from i’s perspective, and, to induce a more extreme decision, i reports mi > θi if θi > 0 and mi < θi 30
Every subject in the dataset had at least one opportunity to be a message sender.
20
if θi < 0.31 We test the hypothesis of exaggeration by regressing mi against θi (Table 3, column 1). Contrary to exaggeration, we find that subjects’ messages are significantly biased toward zero, i.e., the coefficient on θi in the regression is significantly smaller than one (P < 0.001 with H0 : β = 1). Note that while this result is inconsistent with the exaggeration hypothesis, it does not provide evidence against the theory because theory is agnostic about what language is used by the players. Importantly, the coefficient on θi is highly significant and close to one in magnitude. When we regress subjects’ guesses against their received messages (Table 3, column 2), we also find a highly significant coefficient close to one. This suggests that subjects’ guesses are highly correlated with the unobserved states. In a regression of subjects’ guesses against the states being guessed, the R2 is approximately equal to 0.7. This suggests that variation in states explains about 70% of the variation in the elicited beliefs. We interpret this as strong evidence of the efficacy of our belief elicitation procedure. In Table 18 in the appendix, we modify the regressions in Table 3 to include treatment dummies and all relevant interactions. As reported there, subjects in Centralized-Low send more truthful messages (P < 0.05) than subjects in Decentralized-Low, while subjects in DecentralizedHigh believe their messages more than subjects in Decentralized-Low (P < 0.001). Both of these results are consistent with our predictions about communication quality and the results in Section 6.1.2.
State
Message 0.890**** (0.0235)
Message
Constant Observations
Guess
Guess 0.805**** (0.0358)
0.902**** (0.0213) 0.0149* (0.00758) 2880
0.0169** (0.00566) 2880
0.0303*** (0.00997) 2880
Session-clustered standard errors in parentheses * p < 0.10, ** p < 0.05, *** p < 0.01, **** p < 0.001
Table 3: Subjects’ guesses and messages are biased toward zero. 31
Note that this is not an equilibrium argument. E.g., a noninformative equilibrium is a possibility in every treatment.
21
Recall that communication in a partitional equilibrium is noisier when the state is further away from zero. We test this prediction by regressing the absolute deviation between the message and the state, |mi − θi |, against the absolute value of the state, with treatment dummies used as explanatory variables. The results are reported in Table 4. We do not find evidence that less information is transmitted the larger |θi | in any of our treatments. These findings cast doubt on the ability of partitional equilibria to rationalize communication in our data. The coefficients involving the treatment dummies are significant with signs consistent with Prediction 1 and Prediction 2. Thus, when the incentives to coordinate are low, communication is less noisy under centralization than under decentralization (P < 0.01). When these incentives are high, |θi | is not significantly different under centralization and decentralization (P = 0.863). We test predictions about communication quality more rigorously in Section 6.1.2 below.
6.1.2
Communication Quality
The quality of communication is theoretically measured as the residual variance of the message receiver’s posterior, νi = E[θi |mi ], around the sender’s privately known state, that is, E [(θi − νi )2 ]. Our experimental data permits the construction of the empirical analogue of the residual variance of communication by exploiting subjects’ elicited posterior beliefs ν¯i as follows: N 1 X [θn − ν¯n ]2 . N n=1 Here, n denotes an observation and θn the true value of the state being guessed.32 Figure 2 shows how the quality of communication differs across treatments. Note that it is not significantly higher than predicted in any of the treatments. Moreover, in one of the treatments (Decentralized-Low), it is significantly lower than the MIE prediction (P < 0.01).33 This suggests an important counterpoint to Cai and Wang (2006), who find 32
Note that the way in which the centralized treatments were designed allowed us a choice of which observations to include when computing our measure of residual variance. Since all players made guesses about the state(s) unknown to them, we could use (a) only guesses of Player 3, (b) only guesses of Player 1 and Player 2, or (c) everybody’s guesses. To be closest to theory, we take the first approach and use Player 3’s guesses in our analysis of the centralized treatments. The results, however, are qualitatively the same regardless of which of these approaches is used. 33 In Section 6.4, we show that Decentralized-Low is the only treatment in our data where miscommunication led to statistically significant losses in welfare.
22
|mi − θi | 0.00894 (0.0378)
|θi |
High (dummy = 1 if γ = 34 )
-0.0622** (0.0241)
Vertical (dummy = 1 if centralized)
-0.0808*** (0.0221)
High × Vertical
0.0692** (0.0278)
High × |θi |
0.0631 (0.0556)
Vertical × |θi |
0.0214 (0.0482)
High × Vertical × |θi |
0.0195 (0.0830)
Constant
0.103**** (0.0220) 2880
Observations
Session-clustered standard errors in parentheses * p < 0.10, ** p < 0.05, *** p < 0.01, **** p < 0.001
Table 4: Noisiness in communication. The Decentralized-Low treatment serves as a baseline.
23
Centralized
.3
Decentralized
.2
**
0
.1
**
Low gamma
High gamma Low gamma
Data
High gamma
Theory
Figure 2: Residual variance of communication and MIE predictions. The double stars denote a significant difference at P < 0.05. overcommunication in cheap talk games relative to the MIE benchmark.34 Result 1. Communication quality is not higher than MIE predicts in any of our treatments, and significantly lower than predicted in Decentralized-Low. As the importance of coordination increases, the quality of communication improves in decentralized games and worsens in centralized ones, although the high standard error in the Centralized-High treatment suggests that the latter effect may not be statistically significant.35 The figure also suggests that the quality of communication in decentralized treatments is lower than that in centralized ones when the importance of coordination is low. To test these hypotheses more rigorously, we compare mean residual variances across treatments with the residual variance of horizontal communication in the DecentralizedLow treatment as a baseline. The results are reported in Table 5, column 1. The regression results are consistent with several of our theoretical predictions. First, 34
Another study that finds undercommunication relevant to MIE is Wilson (2014). That the standard error in Centralized-High is higher than in all other treatments provides some evidence that subjects find incentives in Centralized-High more difficult to understand. We also find that their answers to the quiz are significantly worse in this treatment than in any other. 35
24
(1) All periods Decentralized-High -0.0703** (0.0325)
(2) Last 8 periods -0.0508 (0.0446)
(3) Implicit Beliefs -0.0696 (0.0508)
(4) Implicit Beliefs (all) 1.055 (1.109)
Centralized-Low
-0.0807** (0.0351)
-0.0684** (0.0288)
-0.302**** (0.0478)
-4.381**** (0.728)
Centralized-High
0.0132 (0.0797)
0.00732 (0.0789)
-0.123 (0.0734)
5.503**** (1.207)
0.142**** (0.0191) 2880
0.121**** (0.0107) 1536
0.446**** (0.0470) 1636
4.694**** (0.727) 2880
Constant Observations
Session-clustered standard errors in parentheses * p < 0.10, ** p < 0.05, *** p < 0.01, **** p < 0.001
Table 5: Treatment effects on residual variance of communication. The Decentralized-Low treatment serves as a baseline. the coefficient on Decentralized-High is negative (P < 0.05), suggesting that the quality of horizontal communication increases with the importance of coordination. The coefficient on Centralized-Low is also negative (P < 0.05), suggesting that the quality of communication in Decentralized-Low is significantly lower than that in Centralized-Low. Third, there is no significant difference between coefficients on Decentralized-High and Centralized-High (P = 0.326), although the residual variance in the latter case is greater. This is also consistent with the theory, since it predicts that residual variances of horizontal and vertical communication are close when the importance of coordination is high. We highlight these results below: Result 2. The quality of vertical communication is higher than that of horizontal communication if and only if the incentive to coordinate is low. Inconsistent with the predictions, we find no significant difference between the coefficients on Centralized-Low and Centralized-High (P = 0.277). We interpret this to mean that it is difficult for subjects in centralized coordination games to see incentives become misaligned as the importance of coordination grows. This observation is consistent with
25
the finding that the average distance between messages and states is insignificantly different in the centralized treatments. We also find that the effect of incentives to coordinate on the quality of horizontal communication is less robust to learning than its effect on the quality of vertical communication. The second column of Table 5 re-estimates the same regression using only observations from the last eight periods of the experiment. Note that the coefficient on Decentralized-High is not significant in this specification (P = 0.276).36 One interpretation is that the difference between Decentralized-Low and DecentralizedHigh is more difficult for subjects to grasp than the difference between Decentralized-Low and Centralized-Low. Note, however, that our other results remain significant even if the observations are restricted to the last eight periods. Our findings in regard to communication quality can be summarized as follows. First, contrary to many other studies in the experimental literature, we find that subjects do not overcommunicate information in any of our treatments. To the extent that we find undercommunication in Decentralized-Low, we expect this to lead to a loss in welfare. In Section 6.4, we explore the welfare consequences of miscommunication in detail. Second, Alonso et al. (2008) and Rantakari (2008) provide a good if imperfect description of how communication quality responds to incentives to coordinate. In particular, we find that as the importance of coordination increases, the difference in quality of vertical and horizontal communication tends to zero. Thus, the main theoretical predictions of Alonso et al.(2008) and Rantakari (2008) about communication quality are consistent with the data.
6.1.3
Robustness to Implicit Beliefs
While our focus so far has been on subjects’ elicited beliefs, our experiment allows us to derive subjects’ implicit beliefs under the assumption of equilibrium behavior. Treating the posterior beliefs νi and νj as unknowns, this can be done using Equation 4.4 for the decentralized and using Equation 4.5 for the centralized treatments. When we do this, we find that 43.19% of implicit beliefs lie outside the interval [−1, 1]. While the correlation between subjects’ explicit guesses and the true values of the states being guessed is 0.83, the correlation between implicit beliefs and the true states is 0.40. This suggests that elicited beliefs are a much better proxy for participants’ true beliefs than the implicit equilibrium 36
It also loses significance in late periods of the interaction if we estimate a regression which uses all observations but interacts the treatment dummies with the period variable.
26
ones.37 Nevertheless, our main results regarding communication quality are robust to using implicit beliefs in place of elicited ones. The third column of Table 5 shows the results of a regression in which the residual variance of communication, formed using implicit beliefs, is regressed against the treatment dummies. Because subjects knew that the true value of the state was contained in the interval [−1, 1], we only include observations with beliefs in this range. The results, same as those in the first two columns of the table, show a negative and highly significant coefficient on Centralized-Low (P < 0.001). This shows that the residual variance of communication is significantly higher under decentralization when the importance of coordination is low. When the importance of coordination is high, we find no significant difference in communication quality under decentralization and centralization (P = 0.3853). This is consistent with Prediction 1, Prediction 2, and Result 2. Results of the same regression that includes all observations are reported in the fourth column of Table 5. As in all other regressions, the coefficient on Centralized-Low is negative and significant. While the results show that the residual variance of communication is higher under decentralization when the importance of coordination is low (P < 0.001), it is instead higher under centralization when the importance of coordination is high (P < 0.01). Note, however, that the coefficients are extremely large in magnitude. It is unlikely that subjects’ beliefs differed from the states’ true values by such large amounts. We interpret implicit beliefs outside of the [-1,1] range as not reflecting subjects’ true beliefs and therefore inappropriate for an analysis of communication quality. One possibility is that such beliefs are miscalculated by the inversion procedure, e.g. because of the failure of the equilibrium assumption. As we argue in Section 6.2, our data is also consistent with equilibrium behavior and systematic distortions of decision weights.
6.2
Decisions
We now explore how well the predicted decision rules (Equation 4.4 and Equation 4.5) correlate with those observed in the data. To this end, we calculate the equilibrium predicD 38 tions for dC i and di conditional on the realized states as well as subjects’ reported beliefs. 37
In Section 6.2, we provide a possible justification for the observed dispersion in implicit beliefs using subjects’ behavior in the decision making stage. 38 Note that this exercise would have been impossible without belief elicitation. In particular, it is impossible if we restrict our analysis to implicit beliefs, since these take equilibrium decision weights as given.
27
Centralized, Low gamma (P=0.8685)
Decentralized, High gamma (P=0.6913)
Centralized, High gamma (P=0.6829)
−1
−.5
0
.5
1
Data
−1
−.5
0
.5
1
Decentralized, Low gamma (P=0.7068)
−1
−.5
0
.5
1
−1
−.5
0
.5
1
Theory
Figure 3: Decision rules: theory and data. Figure 3 provides a scatter plot of what theory predicts the decisions should be and the decisions observed in the data. The figure suggests that the correlation in every treatment is high. Nevertheless, subjects’ decisions also show substantial deviations from the theoretical optima. As we show below, these deviations are in large part systematic. Using the decision rule described by Equation 4.4, we estimate the decision weights by regressing di against θi , ν¯i , and ν¯j .39 To accommodate the effect of γ on subjects’ decisions, the variables θi , ν¯i , and ν¯j are interacted with a dummy that takes on the value of one for treatments with γ = 43 . We also place the restriction that the weights add up to 1 both when γ = 41 and when γ = 34 .40 Note that the results of this regression allow us to discuss deviation from equilibrium decision rules conditional on the communication rules used by the subjects. The results are reported in the first column of Table 6. Qualitatively, subjects respond to incentives to coordinate and adapt roughly as theory predicts. As the importance of coordination increases, the weight on θi decreases, and the weight on ν¯j increases (P < 0.05 and P < 0.001, respectively). While the change in the weight on ν¯i is not statistically 39 40
Recall that νi = E[θi |mi ]. The results are qualitatively similar if this restriction is removed.
28
significant, the sign is in the right direction. Thus, subjects’ decisions put a smaller weight on the state and a larger weight on posterior beliefs when the incentive to coordinate is greater. We estimate an analogous regression for the centralized treatments. Using Equation 4.5, we regress the principals’ decisions against their elicited posterior beliefs with the restriction that the weights sum up to one. The results are reported in the second column of Table 6. These results also suggest that subjects’ decision rules respond as predicted to incentives to coordinate. Thus, the weight on ν¯i is significantly lower and the weight on ν¯j significantly higher when the importance of coordination is high (P < 0.001 in both cases). I.e., as predicted, the principal weighs the belief about i’s state less, and the belief about the state of i’s partner more, when the importance of coordination is high. We summarize these findings below: Result 3. Qualitatively, subjects’ decision rules respond to incentives to coordinate as theory predicts. Importantly, the estimated decision weights show significant and systematic deviations from those implied by the theory. Table 7 compares the estimated and predicted decision weights under decentralization. In Decentralized-Low, subjects underweigh their own states (P < 0.001), overweigh their partners’ posteriors (P < 0.001), and overweigh their own posteriors about their partners’ states (P < 0.001). In Decentralized-High, subjects overweigh their own posteriors (P < 0.001). Thus, subjects in the decentralized treatments underweigh their own states and overweigh their own and their partners’ beliefs. Table 8 compares the estimated and predicted decision weights under centralization. In both Centralized-Low and Centralized-High, the principal puts too much weight on the belief about θi and too little weight on the belief about θj when making the decision di (all P < 0.001). The deviations reported above are consistent with the agent overweighing the importance of coordination under decentralization and the principal overweighing the importance of adaptation under centralization. To quantify the degree to which coordination is over- or underweighed, we structurally estimate the γ implied by the subjects’ decisions under the null hypothesis of equilibrium. We do this separately for each of the experimental treatments using non-linear least squares. The estimation results are shown in Table 9. The estimated γ’s are significantly higher than what they should be (i.e., those specified in the instructions) in both Decentralized-Low and Decentralized-High (P < 0.05 in P < 0.01, respectively). For example, when γ = 43 , the agent acts as if adaptation is almost irrelevant. 29
Decentralized
Centralized
-0.00735 (0.0131)
0.0149 (0.0149)
High (dummy=1 if γ = 34 )
θi
0.493**** (0.0775)
ν¯i
0.162**** (0.0224)
0.946**** (0.0196)
ν¯j
0.345**** (0.0572)
0.0544*** (0.0196)
θi ×High
-0.270** (0.115)
ν¯i ×High
0.0576 (0.0939)
-0.290**** (0.0323)
ν¯j ×High
0.213**** (0.0598)
0.290**** (0.0323)
Constant
0.0197 (0.0123) 1560
0.00615 (0.0118) 1320
Observations
Session-clustered standard errors in parentheses * p < 0.10, ** p < 0.05, *** p < 0.01, **** p < 0.001
Table 6: Estimated decision weights. As the importance of coordination increases (High=1), agents in decentralized games put a smaller weight on their own state and a larger weight on their posterior belief about the state of the other agent. This is consistent with Prediction 4. In centralized treatments, when the importance of coordination is high, the weight on the posterior belief about the local conditions of the agent for which the decision is made is smaller, and the weight on the posterior belief about the other agent is larger. This is consistent with Prediction 3.
30
Decentralized Treatments Low (Predicted) Low (Actual) High (Predicted) High (Actual)
θi 0.75 0.49**** 0.25 0.22
νi 0.05 0.16**** 0.32 0.22
νj 0.2 0.34** 0.43 0.56****
* p < 0.10, ** p < 0.05, *** p < 0.01, **** p < 0.001
Table 7: Predicted equilibrium weights and actual weights in the decision rules of decentralized treatments. The significance levels refer to the difference between actual and predicted weights. Centralized Treatments Low (Predicted) Low (Actual) High (Predicted) High (Actual)
νi 0.71 0.95**** 0.54 0.66****
νj 0.29 0.05**** 0.46 0.34****
* p < 0.10, ** p < 0.05, *** p < 0.01, **** p < 0.001
Table 8: Predicted equilibrium weights and actual weights in the decision rules of centralized treatments. The significance levels refer to the difference between actual and predicted weights. In Centralized-Low and Centralized-High, the weights are significantly lower than what they should be (P < 0.001 in both treatments). For example, when γ = 41 , the principal acts as if the weight on coordination were almost zero. We summarize these results below and postpone a discussion of how the distortions in decision weights relate to uncertainty to Section 7. Result 4. Agents in decentralized coordination games overweigh, and principals in centralized coordination games underweigh, the importance of coordination.
6.2.1
Robustness Checks
In principle, risk-aversion might have affected belief elicitation. If this were the case, the documented distortions in decision rules might simply be a result of estimation based 31
γˆ when γ = 0.25
γˆ when γ = 0.75 Observations
Decentralized 0.517 >∗∗ 0.25 (0.103)
Centralized 0.0296 ∗∗∗ 0.75 (0.0375) 1560
0.356 0.1 on the test of the constant plus any of the indicator variables being equal to zero). This is consistent with the results on communication quality reported in Section 6.1.2, where we find that the quality of communication is significantly different from MIE only in Decentralized-Low.
7 7.1
Discussion Social Preferences
Under centralization, social preferences cannot explain our observed distortions in the principal’s decision rules. Conditional on the messages received, the principal is already maximizing the agents’ joint payoffs. Social preferences therefore leave the theoretical predictions about the principal’s decisions unaltered. Under decentralization, it is in theory possible that social preferences lead the agents to over-coordinate relative to the selfinterested benchmark. If social preferences caused the distortions in Decentralized-Low and 40
Decentralized-High, similar distortions should be observed in otherwise identical treatments where both agents observe each other’s states. We highlight this below as a prediction: Prediction 9. If social preferences caused the distortions in Decentralized-Low and Decentralized-High, we should observe similar distortions in analogous treatments with complete information. We test this prediction in an additional experiment whose results are reported in Section 8.
7.2
Risk Aversion
Unlike social preferences, risk aversion can generate distortions in the directions we observe both under centralization and decentralization. Thus, under centralization, a risk-averse principal will under-coordinate relative to the risk-neutral baseline. Under decentralization, a risk-averse agent will under-adapt. To get some intuition for these results, consider the case of centralization. Note that the principal’s payoff can be decomposed into two adaptation losses (one for each agent) and a coordination loss. The coordination loss is non-stochastic, while the adaptation losses depend on unknown states. If the agents report truthfully and the principal believes the messages received, the adaptation losses are also non-stochastic–conditional on the information received–and risk preferences have no bite. Noise in messages introduces uncertainty about the principal’s adaptation losses. While a risk-neutral decision maker only cares about the posterior expectations of the states, a risk-averse principal also cares about the variance of her conditional expectation, and she will under-coordinate to decrease this variance.46 Introducing risk preferences into the model described in Section 3 and deriving the associated predictions is a difficult task because the decision rules have no closed-form solutions.47 Nevertheless, we can get a sense for the effect of curvature on optimal decisions by considering a simple reduced-form version of the baseline model. We describe such a model in Section A.4 for the centralized and Section A.5 for the decentralized case and perform simulations to calculate average degrees of under-coordination and underadaptation for different values of the risk aversion parameter. Both under centralization 46
In the simulations discussed below, we are agnostic about how subjects’ beliefs are related to the realized states. We therefore abstract away from the issue of over-adaptation under centralization. 47 We know of no extensions of Alonso et al. (2008), Rantakari (2008), or similar models to risk-averse agents.
41
and decentralization, we find that risk aversion generates distortions that go in the right direction, although the model performs somewhat better under decentralization. Thus, even if we allow risk aversion to be unreasonably high, the simulations can only account for 13%-34% of the under-coordination observed in the centralized treatments. For reasonable degrees of risk aversion, we can explain only 4%-17% of the observed under-coordination. In contrast, under decentralization, reasonable degrees of risk aversion can explain 28%-50% of the under-adaptation observed in the data. Consider now the following modification of our original experiment. Under decentralization, both agents are informed of each other’s states, and the game is otherwise identical. Under centralization, both agents and the principal are informed of the agents’ states. As we discuss in Section 8, the centralized and decentralized games with complete information have unique equilibria and hence no uncertainty under the null hypothesis of equilibrium behavior. This suggests the following prediction: Prediction 10. If risk aversion caused the distortions in Centralized-Low, CentralizedHigh, Decentralized-Low and Decentralized-High, we should observe no distortions in analogous treatments with complete information.
7.3
Risk Seeking
In our experiment, subjects only incur losses, and a large literature starting from Kahneman and Tversky (1979) found subjects to be risk-seeking in the loss domain.48 For example, with the utility function used in our simulations, U (x) = −(−x)α , x ≤ 0, Tversky and Kahneman (1992) have estimated the curvature of subjects’ utility function to be around 0.88, which implies risk seeking over the loss domain and risk aversion otherwise. Our simulation results in Section A.4 and Section A.5, however, show that risk seeking generates distortions in the opposite direction to those observed in the data both under centralization and decentralization. I.e., a risk-seeking principal over-coordinates under centralization, while a risk-seeking agent over-adapts under decentralization. There are several possible explanations for why our results differ from those of Tversky and Kahneman (1992). First, our experiment differs substantially from the choice-theoretic experiments in the prospect theory literature or related experiments such as Myagkov and Plott (1997). The noise in our game is not objective but derived from other subjects’ communication rules. Second, it’s possible that subjects’ reference points are not zero (Kah48
See also Myagkov and Plott (1997), which documents risk-seeking with losses in a market experiment.
42
neman and Tversky, 1979). Indeed, incurring no loss is difficult in our environment and subjects might have formed expectations accordingly (K˝oszegi and Rabin, 2006). Third, as we show below, ambiguity-aversion can generate distortions in the right direction even with moderate risk-seeking preferences.
7.4
Strategic Uncertainty and Ambiguity
If the players fail to coordinate on the same equilibrium, they face strategic uncertainty. To study the effect of strategic uncertainty on behavior, consider first the case of centralization with incomplete information. Here, the principal potentially faces uncertainty about the communication rules used by the agents. She therefore needs to form subjective beliefs about how states are communicated through subjects’ messages. Assuming ambiguity-neutral preferences, the principal can easily form posterior beliefs E(θi |mi ), and the relevant theoretical (resp., numerical) predictions are in Section 4.2 for risk-neutral, Section 7.2 for risk-seeking, and Section 7.3 for risk-averse preferences. This shows that strategic uncertainty about communication rules has no bite under ambiguity-neutral preferences. In the presence of ambiguity aversion, however, the predictions change. In Section A.6, we perform simulations to solve the problem of an ambiguity-averse principal under centralization. The simulation assumes that the principal has preferences of the max min sort, where the min is taken over different beliefs about the agents’ states. We find that ambiguity aversion substantially amplifies the distortions due to risk aversion. Thus, with α = 2 and ambiguity-averse preferences, the model accounts for 17% to 55% of the undercoordination observed in the data. Moreover, if the principal is ambiguity-averse, the model can generate distortions in the right direction even if her utility function is moderately risk-seeking. If, following Van Huyck et al. (1990), we interpret strategic uncertainty as arising from multiplicity of equilibria, then strategic uncertainty is not a concern under complete information regardless of whether the game is centralized. Under centralization, this is because–conditional on her beliefs–the principal faces an optimization problem with a unique solution. Under decentralization, the equilibrium is unique if both agents know each other’s states. This suggests the following prediction: Prediction 11. If ambiguity aversion caused the distortions in Centralized-Low, CentralizedHigh, Decentralized-Low and Decentralized-High, we should observe no distortions in analogous treatments with complete information.
43
8
The Second Experiment
To test Prediction 9, Prediction 10, and Prediction 11, we ran an additional experiment in September 2015. We collected data from 60 subjects for two treatments: DecentralizedComplete Info and Centralized-Complete Info (see Table 14). For both of these treatments, we chose γ = 34 because the observed distortions in this case were most robust to learning. The treatments were identical to Decentralized-High and Centralized-High in all respects but the following. First, every player observed every state θi (i = 1, 2) before making any decision. Second, the players did not make any guesses about the states of other players. The instructions and quizzes were modified accordingly.49
Decentralized-Complete Info Centralized-Complete Info
High γ 2 sessions N = 30 2 sessions N = 30
Table 14: Subjects per treatment in the second experiment.
The predictions for these treatments are simple extensions of the results in Section 4. In particular, the complete information decision rules are the same as those in Equation 4.4 and Equation 4.5, with the modification that each posterior belief νi is replaced by the true value of the corresponding state θi . While the decision rule under decentralization is the unique equilibrium solution, the decision rule under centralization is the unique solution to the principal’s decision problem:
dD i =
dC i =
1 γ θi + θj , 1+γ 1+γ
1+γ 2γ θi + θj , 1 + 3γ 1 + 3γ
i = 1, 2,
i = 1, 2,
i 6= j.
i 6= j.
(8.1)
(8.2)
In principle, given complete information, subjects’ messages can be interpreted as suggestions.50 I.e., under decentralization, the agents who understood the equilibrium solution 49
See the online appendix. While we could have removed communication from the complete information treatments, we avoided doing this so that uncertainty and communication are not manipulated at the same time. Thus, the only 50
44
can send messages to other agents to inform them what this solution is. Under centralization, the agents can send messages to the principal to inform them of the unique solution to the principal’s problem. From a theoretical point of view, however, these messages are irrelevant. We therefore make no predictions about communication quality for CentralizedComplete Info and Decentralized-Complete Info. The predictions about coordination and adaptation losses are simple extensions of Prediction 5, Prediction 6, Prediction 7, and Prediction 8. Thus, both under centralization and decentralization the decisions are better coordinated and less adapted to the states when γ is higher. The adaptation loss remains greater under centralization, and the coordination loss remains greater under decentralization. Intuitively, it can be seen from Equation 8.1 and Equation 8.2 that both decision rules decrease the weight on θi when γ is greater. It can also be seen that the decision rule under centralization puts a greater weight on coordination both when γ = 41 and γ = 34 . We re-estimated the models discussed in Section 6 using data from treatments with both complete and incomplete information. Our first set of results deals with adaptation and coordination losses. These are reported in Table 15. First, we find that coordination losses are greater under decentralization than under centralization in the complete information treatments (P < 0.001). This stands in stark contrast to our results with incomplete information, where the opposite was true. Second, we find that adaptation losses are not significantly different under decentralization and centralization with complete information (P = 0.801). This also stands in contrast to the incomplete information results, where we found greater adaptation losses under decentralization than under centralization. We highlight these results below: Result 8. With complete information, centralized treatments show more coordination than decentralized treatments and as much adaptation. Result 8 suggests that subjects in centralized treatments overweighed adaptation less and subjects in decentralized treatments overweighed coordination less with complete information. To quantify the magnitudes of these distortions, we estimated subjects’ decision weights structurally taking the same approach as in Section 6.2. In particular, we augmented Table 9 to include estimation results for treatments with complete information.51 The results are reported in Table 16. The table shows that the estimated γˆ ’s for the comdifference between the complete and incomplete information treatments is uncertainty about θi . 51 To compare the distortions to their analogues with incomplete information, we carry out the estimation for all decentralized treatments together. Likewise, a second non-linear regression is estimated for all centralized treatments.
45
(1) (d1 − d2 )2 -0.0935* (0.0528)
(2) (di − θi )2 0.117** (0.0424)
Centralized-Low
0.204*** (0.0695)
-0.145*** (0.0399)
Centralized-High
-0.0596 (0.0465)
0.0590 (0.0735)
Decentralized-Complete Info
-0.122** (0.0472)
0.00881 (0.0453)
Centralized-Complete Info
-0.220**** (0.0422)
0.00105 (0.0394)
Constant
0.319**** (0.0420) 1815
0.220**** (0.0366) 3630
Decentralized-High
Observations
Session-clustered standard errors in parentheses * p < 0.10, ** p < 0.05, *** p < 0.01, **** p < 0.001
Table 15: Degrees of adaptation and coordination in different experimental treatments. The Decentralized-Low treatment serves as a baseline.
46
plete information treatments do not significantly differ from the optimal γ = 0.75. We highlight this result below: Result 9. With complete information, the hypothesis that subjects use the correct γ cannot be rejected.
γˆ when γ = 0.25
γˆ when γ = 0.75
γˆ when γ = 0.75 (Complete Information) Observations
Decentralized 0.517 >∗∗ 0.25 (0.1)
Centralized 0.0296 ∗∗∗ 0.75 (0.037)
0.356 ALiD . Also,
dALiC dγ
> 0 and
dALiD dγ
> 0.
Proof. From the proof of Proposition 2 in Alonso et al. (2008, pag. 174), it follows again that 1 (1 + γ)(1 + 6γ + γ 2 ) ALiC = − , (A.5) 3 (1 + 3γ)2 (3 + 4γ) 7γ 2 + γ 3 . 3(1 + γ)(4 + 3γ) Differentiating (A.5) and (A.6) with respect to γ gives ALiD =
(A.6)
dALiC 1 + 57γ + 131γ 2 + 67γ 3 = > 0, dγ (1 + 3γ)3 (3 + 4γ)2
(A.7)
dALiD γ 56 + 61γ + 14γ 2 + 3γ 3 > 0. = dγ 3 (1 + γ)2 (4 + 3γ)2
(A.8)
Finally, ALiC > ALiD , for any γ ∈ (0, 1), follows from Lemma 2 in Alonso et al. (2008). 49
A.2
Additional Analysis of Adaptation and Coordination Losses
Table 17 breaks point losses of teams in different treatments of the experiment into miscomunication and miscoordination components. Thus, for example, the miscommunication component of the relative coordination loss in the treatments with γ = 34 is calculated as:53 ) ( 2 X belief s belief s 2 IE 2 IE − dM 3 ∗ (dreported − dreported ) − (dM −i ) i i −i i=1
Note that this table can be used to recover the overall relative losses due to distortions or communication reported in Table 13. For example, to compute the relative losses due to distortions in Decentralized-Low (Table 13, constant term in the second column), add the relative coordination losses due to distortions in Decentralized-Low (Table 17, first column, second row) to the relative adaptation losses due to distortions in Decentralized-Low (Table 17, first column, fifth row). Recall from the second column of Table 13 that the relative welfare losses due to distortions were smaller in Centralized-Low than in any of the other treatments. Table 17 provides evidence for our conjecture that this was driven by coordination losses being smaller in Centralized-Low (where the over-weighting of adaptation was less costly) than in Centralized-High. Thus, while the coordination loss due to distortions was greater in Centralized-High than in Centralized-Low, distortions in decision rules did not lead to adaptation losses under centralization (all P > 0.1). The table also provides additional evidence for Result 7: very little of the significant loss in welfare is due to miscommunication. As discussed above, the only treatment showing significant welfare loss due to miscommunication is Decentralized-Low.
53
The sum is necessary in the expression because the analysis of the decompositions is carried out in terms of units of welfare and not individual payoffs.
50
D-L 0.125 (0.081)
D-H C-L C-H 1.236**** 0.810**** 1.533**** (0.191) (0.111) (0.120)
Relative coordination loss (Distortions)
0.111 (0.070)
1.233**** 0.826**** 1.533**** (0.200) (0.097) (0.120)
Relative coordination loss (Miscommunication)
0.014 (0.014)
0.003 (0.012)
-0.017 (0.016)
0.0001 (0.001)
Relative adaptation loss
1.173**** (0.220)
0.397**** 0.028 (0.038) (0.082)
0.219 (0.134)
Relative adaptation loss (Distortions)
1.157**** (0.221)
0.381**** -0.158 (0.042) (0.106)
0.088* (0.050)
Relative adaptation loss (Miscommunication)
0.016**** 0.016 (0.002) (0.012) 360 420
Relative coordination loss
Observations
0.187 (0.134) 330
0.131 (0.088) 330
Session-clustered standard errors in parentheses * p < 0.10, ** p < 0.05, *** p < 0.01, **** p < 0.001
Table 17: Decompositions of adaptation and coordination losses into a component due to distortions of decision weights and a component due to miscommunication. The standard errors are obtained by regressing each of the variables (e.g., relative coordination loss) against the treatment dummies.
51
A.3
Omitted Tables
State
Message 0.867**** (0.0183)
Message
Guess
Guess 0.794**** (0.0247)
0.921**** (0.00291)
Decentralized-High
-0.00211 (0.0153)
-0.00300 (0.0138)
-0.00483 (0.0157)
Centralized-Low
-0.0118 (0.0141)
-0.00503 (0.0168)
-0.0160 (0.0111)
Centralized-High
0.0214 (0.0259)
0.00792 (0.0173)
0.0245 (0.0297)
Decentralized-High × State
0.0215 (0.0477)
0.0402 (0.0506)
Centralized-Low × State
0.0762** (0.0341)
0.0721 (0.0558)
Centralized-High × State
-0.00111 (0.0683)
-0.0711 (0.112)
Decentralized-High × Message
0.0214**** (0.00465)
Centralized-Low × Message
-0.0139 (0.0321)
Centralized-High × Message
-0.0916 (0.0641)
Constant Observations
0.0133 (0.0136) 2880
0.0173 (0.0108) 2880
0.0296**** (0.00352) 2880
Session-clustered standard errors in parentheses * p < 0.10, ** p < 0.05, *** p < 0.01, **** p < 0.001
Table 18: Subjects’ guesses, messages, and interactions with the treatment dummies.
52
A.4
Simulations for Risk Preferences (Centralization)
To accommodate risk-seeking as well as risk-averse preferences, we assume that the decision maker in the experiment has a utility function of the form U (x) = −(−x)α over her point losses x < 0 (Tversky and Kahneman, 1992). The parameter α > 0 determines the risk attitudes of the decision maker, with α > 1 leading to risk-averse and α ∈ (0, 1) to risk-seeking behavior. Our goal is to explore how changes in this parameter affect decision making. Suppose that authority is centralized. Let νi be the principal’s posterior expectation about θi after having received a message about θi . Suppose that ν˜i ∈ {νi − , νi + }, with > 0, i = 1, 2. Let p = Prob(˜ νi = νi + ). Then, E[˜ νi ] = νi + (2p − 1) and 1 2 V ar(˜ νi ) = 4p(1 − p) . When p is close to 2 , the distribution of νi is a proxy for a uniform posterior distribution around the posterior mean νi , as it would be in communication equilibria with risk-neutrality.54 The parameter can be interpreted as a measure of uncertainty about the posterior expectation νi . The problem of the principal can therefore be written as: max −E
d1 ,d2 ∈R
(1 − γ)(d1 − ν˜1 )2 + (1 − γ)(d2 − ν˜2 )2 + 2γ(d1 − d2 )2
α
.
(A.9)
If the principal were risk-neutral (α = 1), she would choose di =
2γ 1+γ E[˜ νi ] + E[˜ νj ], 1 + 3γ 1 + 3γ
i = 1, 2, i 6= j.
(A.10)
Note that the decision rules are exactly those used by the principal in our baseline model. We perform simulations to calculate the average distance between the principal’s decisions, |d1 − d2 |, for different values of ν1 , ν2 , α, and .55 In the simulations, we assume that p = 21 56 and consider νi ∈ [−0.6, 0.6].57 Figure 6 shows the simulated average distances RS RN − dRN for different values of α and , with D(, α) ≡ M ean(ν1 ,ν2 ) |dRS 2 | 1 − d2 | − |d1 α ∈ [0, 1].58 The figure shows that the simulated average distance is negative, which means that the decisions are on average more coordinated under risk-seeking than risk-neutrality. If we average over ∈ [0, 0.4], and α ∈ [0, 1], we obtain that the average distance between 54
Although communication equilibria could have different features under risk aversion, we make this distributional assumption for tractability. 55 We set the grid sizes to 0.05 for νi , i = 1, 2, 0.02 for . 56 Robustness checks suggest that the magnitude of distortions is little affected by relaxing it. 57 The values of ν1 and ν2 are chosen in such a way that max {|νi − |, |νi + |} ≤ 1, i = 1, 2, given the simulated values of ∈ [0, 0.4]. Varying over a smaller interval leads to smaller distortions. 58 More precisely, we calculate the distance D for each vector (α, , ν1 , ν2 ) and, holding α and fixed, average the distances obtained for different values of (ν1 , ν2 ). The grid size for α is set at 0.05.
53
decisions under risk-seeking is -0.04 for γ = 1/4 and -0.01 for γ = 3/4.59 In the experiment, the average absolute distance between the observed and risk-neutral equilibrium decisions, Eq |dObserved − dObserved | − |dEq 1 − d2 |, is approximately 0.29 for Centralized-Low and 0.22 for 1 2 Centralized-High. Based on these simulation results, we conclude that risk-seeking cannot explain the over-coordination observed in the centralized treatments in the data. Simulation results for risk-averse preferences are shown in Figure 7. The figure gives us a rough idea of how much risk aversion is necessary to generate distortions of the order observed in the experiment. With α ∈ [1, 5],60 we obtain that the average difference in the distances is 0.05 for γ = 1/4 and 0.01 for γ = 3/4. Although the simulated distortions go in the same direction as what we observe in our data, the magnitudes are of a different order even with highly unreasonable degrees of risk aversion. For example, averaging over α ∈ [10, 20] only raises the average between distances to 0.098 for γ = 1/4 and 0.029 γ = 3/4. We performed simulations with alternative, standard, utility functions such as the log and CRRA and obtained similar results. This shows that risk aversion can partly explain the distortions observed in the centralized treatments with incomplete information but cannot fully accommodate them.
We also performed simulations with a larger number of states, namely, νi − , νi − 3 , νi + 3 , νi + , i = 1, 2. We found similar qualitative and quantitative results. 60 The grid size for α was increased to 0.1 due to the larger parameter interval. 59
54
(a) Comparison of average distances between optimal risk-seeking and risk-neutral decisions, RS RN − dRN | for γ = 1 . Each point corresponds to the average over (ν , ν ) ∈ |dRS 1 2 1 − d2 | − |d1 2 4 2 [−0.6, 0.6] with a grid of size 0.05. The grid size for α is 0.05.
(b) Comparison of average distances between optimal risk-seeking and risk-neutral decisions, RS RN − dRN | for γ = 3 . Each point corresponds to the average over (ν , ν ) ∈ |dRS 1 2 1 − d2 | − |d1 2 4 2 [−0.6, 0.6] with a grid of size 0.05. The grid size for α is 0.05.
Figure 6: The effect of risk-seeking on coordination behavior in centralized coordination games. 55
(a) Comparison of average distances between optimal risk-averse and risk-neutral decisions, |dRA 1 − 1 RA RN RN d2 | − |d1 − d2 | for γ = 4 . Each point corresponds to the average over (ν1 , ν2 ) ∈ [−0.6, 0.6]2 with a grid of size 0.05.
(b) Comparison of average distances between optimal risk-averse and risk-neutral decisions, |dRA 1 − 3 RA RN RN d2 | − |d1 − d2 | for γ = 4 . Each point corresponds to the average over (ν1 , ν2 ) ∈ [−0.6, 0.6]2 with a grid of size 0.05.
Figure 7: The effect of risk aversion on coordination behavior in centralized coordination games.
56
A.5
Simulations for Risk Preferences (Decentralization)
Under decentralization, we can without loss of generality consider the decision problem of Player 1. Player 1 observes her own local conditions θ1 and needs to make a single decision without knowing the decision made by Player 2. Let us reformulate the problem assuming that the decision of Player 2, d˜2 , is random from Player 1’s perspective and could take on the value d2 + with probability p, or d2 − otherwise, where d2 ∈ (−1, 1) and ∈ (0, 1 − |d2 |). We interpret d2 as the expected decision of Player 2 from Player 1’s perspective. Given a risk aversion coefficient α, Player 1’s decision problem can be written as h α i max −E (1 − γ)(d1 − θ1 )2 + γ(d1 − d˜2 )2 . (A.11) d1 ∈R
If Player 1 were risk-neutral (α = 1), she would choose d1 = (1 − γ)θ1 + γE[d˜2 ].
(A.12)
Note that this decision rule is the same as the one used by Player 1 in the baseline model, given our interpretation of d2 . We perform simulations to calculate the degree of adaptation, |d1 − θ1 |, for different values of θ1 , d2 , α, and . In the simulations, we assume that p = 21 and consider values of θ1 ∈ [−1, 1], d2 ∈ [−0.6, 0.6], and ∈ [0, 0.4].61 Figure 8 shows the simulated average RN distances D(, α) ≡ M ean(θ1 ,d2 ) |dR − θ | − |d − θ | for different values of α and . 1 1 1 1 The figure shows that the decisions are on average more adapted under risk-seeking than risk neutrality, and more adapted under risk neutrality than under risk aversion. More precisely, averaging over ∈ [0, 0.4] and α ∈ {0.2, 0.4, 0.6, 0.8}, for a risk seeking decision maker, we obtain that the average distances are approximately -0.02 for γ = 1/4, and -0.05 for γ = 3/4. For degrees of risk aversion in the set {2, 3, 4, 5}, the same average leads to 0.036 for γ = 1/4, and 0.05 for γ = 3/4. For comparison, the average distance between decisions and states in the data is approximately 0.128 for γ = 1/4, and 0.10 for γ = 3/4. Thus, risk aversion explains the direction of the observed distortions under decentralization. It is also provides quantitative benchmarks that are closer to the data than their counterparts in the centralized case.
61
The grids for θ1 , d2 , and are 0.01, 0.01, and 0.02, respectively.
57
(a) Comparison of average distances between optimal risky and risk neutral level of adaptation, RN − θ | for γ = 1 , different attitudes toward risk, α ∈ {0.2, 0.4, 0.6, 0.8, 1, 2, 3, 4, 5}, |dR 1 1 − θ1 | − |d1 4 and different values of ∈ [0, 0.4] with grid of size 0.02. Each point corresponds to the average over (θ1 , d2 ) ∈ [−1, 1] × [−0.6, 0.6] with a grid of size 0.01.
(b) Comparison of average distances between optimal risky and risk neutral level of adaptation, RN − θ | for γ = 3 , different attitudes toward risk, α ∈ {0.2, 0.4, 0.6, 0.8, 1, 2, 3, 4, 5}, |dR 1 1 − θ1 | − |d1 4 and different values of ∈ [0, 0.4] with grid of size 0.02. Each point corresponds to the average over (θ1 , d2 ) ∈ [−1, 1] × [−0.6, 0.6] with a grid of size 0.01.
Figure 8: The effect of attitudes toward risk on the degree of adaptation in decentralized coordination games. 58
A.6
Simulations for Ambiguity Preferences (Centralization)
We now use simulations similar to those described in Sections A.4 and A.5 to argue that strategic uncertainty about communication rules combined with ambiguity-aversion can generate distortions of larger magnitudes than those generated by risk-aversion alone. Moreover, ambiguity-aversion can generate distortions in the right direction even with risk-seeking preferences. To see this, assume that the principal solves the following optimization problem: max
min
d1 ,d2 ∈R µ∈{(p,1−p),(1−p,p)}
−Eµ
(1 − γ)(d1 − ν˜1 )2 + (1 − γ)(d2 − ν˜2 )2 + 2γ(d1 − d2 )2
α
.
Here, µ indexes the principal’s belief system, which specifies beliefs both about ν1 and about ν2 .62 The belief system can be either (p, 1 − p) or (1 − p, p). If µ = (p, 1 − p), p is the probability that ν1 is high as well as the probability that ν2 is low.63 If µ = (1 − p, p), then p is the probability that ν1 is low as well as the probability that ν2 is high. Thus, for any p 6= 1/2, the principal considers two belief systems: one in which the probability that ν1 is high is greater than the probability that ν2 is high, and another in which the probability that ν2 is high is greater than the probability that ν1 is high. Intuitively, for any (ν1 , ν2 ), the principal posterior beliefs can take on one of four values: (ν1 − , ν2 − ), (ν1 − , ν2 + ), (ν1 + , ν2 − ), or (ν1 + , ν2 + ). The principal will use one of two belief systems (p, 1 − p) and (1 − p, p) to compute her expected utility. Ambiguity-aversion will make the principal select the belief system under which “bad” posteriors–posteriors where beliefs about ν1 and ν2 are further apart–are more likely. In the simulation, we consider three possible values for p ∈ {0.1, 0.3, 0.6}. To complete the description of the simulation, we assume that both belief systems are equally likely, so that an ambiguity neutral decision maker will have a posterior belief equal to 1/2 for any of our possible values of p. Our simulation results with different values of p and α are reported in Figure 9 in the appendix. These results show that introducing ambiguity-aversion amplifies the distortions caused by risk aversion considerably. Thus, even with risk neutrality, that is, α = 1, we obtain that the average difference in the distances, over our simulated values of p, is 0.1369 for γ = 1/4 and 0.031 for γ = 3/4. Increasing the risk aversion coefficient to α = 2 increases the average difference in distance to 0.1547 for γ = 1/4 and 0.039 for γ = 3/4, thus more than tripling the average distances compared to an ambiguity neutral but risk averse agent 62 63
Recall that we assume ν1 and ν2 are independent. Formally, p = Prob(˜ ν1 = ν1 + ) and p = Prob(˜ ν1 = ν1 − ).
59
with the same attitudes toward risk. We conclude that reasonable degrees of risk aversion (i.e., α = 2), coupled with extreme aversion to ambiguity, can account for 17% to 55% of the distortions observed in the data. Moreover, note that when p is either sufficiently low or sufficiently high, the simulated distortions are quantitatively close to those for an ambiguity neutral decision maker for values of α in the upper part of the interval [0, 1]. This suggests that ambiguity-aversion can generate a reasonable fit to the data even with moderate risk-seeking preferences.
60
(a) Comparison of average distances between optimal maxmin and risk/ambiguity-neutral deciM − dM M | − |dRN − dRN | for γ = 1 , different degrees of risk aversion α, and different sions, |dM 1 2 1 2 4 distribution parameters p. Each point corresponds to the average over (ν1 , ν2 ) ∈ [−0.6, 0.6]2 with a grid of size 0.05. The grid size for the decisions is 0.05, and the value of = 0.4.
(b) Comparison of average distances between optimal maxmin and risk/ambiguity-neutral deciM − dM M | − |dRN − dRN | for γ = 3 , different degrees of risk aversion α, and different sions, |dM 1 2 1 2 4 distribution parameters p. Each point corresponds to the average over (ν1 , ν2 ) ∈ [−0.6, 0.6]2 with a grid of size 0.05. The grid size for the decisions is 0.05, and the value of = 0.4.
Figure 9: The effect of ambiguity-aversion on coordination behavior in centralized coordination games. 61
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