An Experimental Study of Information Revelation Policies

An Experimental Study of Information Revelation Policies in Sequential Auctions Timothy N. Cason∗

Karthik N. Kannan†

Ralph Siebert‡

October 2009

Abstract Theoretical models of information asymmetry have identified a tradeoff between the desire to learn and the desire to prevent an opponent from learning private information. This paper reports a laboratory experiment that investigates if actual bidders account for this tradeoff, using a sequential procurement auction with private cost information and varying information revelation policies. Specifically, the Complete Information Policy, where all submitted bids are revealed between auctions, is compared against the Incomplete Information Policy, where only the winning bid is revealed. The experimental results are largely consistent with the theoretical predictions. For example, bidders pool with other types to prevent an opponent from learning significantly more often under a Complete Information Policy. Also as predicted, the procurer pays less when employing an Incomplete Information Policy only when suppliers’ cost structures are highly competitive. Observed deviations from the quantitative theoretical predictions appear to be consistent with risk aversion and bounded rationality. JEL: C91, D44, D82. Keywords: Complete and Incomplete Information Revelation Policies, Laboratory Study, Procurement Auction, Multistage Game. Acknowledgments: We thank Dan Kovenock, Juan Carlos Escanciano and seminar participants at the Universities of Washington and Melbourne, and Carnegie Mellon, Ohio State and Purdue Universities for helpful comments. Manish Gupte, Justin Krieg and Jingjing Zhang provided valuable research assistance. All errors are our own.

∗ Purdue University, Krannert School of Management, Department of Economics, 403 West State Street, West Lafayette, IN 47906-2056, USA, Email: [email protected]. † Purdue University, Krannert School of Management, 403 West State Street, West Lafayette, IN 47907-2056, USA, Email: [email protected]. ‡ Purdue University, Krannert School of Management, Department of Economics, 403 West State Street, West Lafayette, IN 47906-2056, USA, Email: [email protected].

1

Introduction

In multistage non-cooperative games with information asymmetry, previous theoretical studies (e.g., Anand and Goyal, 2009, Kannan, 2008) have considered two learning aspects exhibited by the players – the incentive to extract and the incentive to obscure private information – as well as the trade-off between these incentives from the perspective of the market-maker. The managerial insights offered by these analyses depend on steep informational and rationality requirements for the players. Therefore, the relevance of these insights to practical, real-life situations remains an open question. In this paper, we experimentally investigate these learning aspects in one particular setting: procurement auctions. Auctions are one of the most commonly-used mechanisms for procurement. General Dynamics, GE, Sears Logistics, and Staples, are among the many organizations that have employed auction technologies for procurement (Chandrashekar et al., 2007). These technologies enable a marketmaker to easily alter the information policy for an auction i.e., the extent to which information about bids are revealed at the beginning, during, and at the end of the auction. In fact, one of the important problems in the procurement context is the choice of the information policy (Elmaghraby and Keskinocak, 2000, highlights this issue). For example, in Freemarkets,1 a firm that specializes in convening electronic procurement markets, the buyer who convenes the market has a wide range of policy choices. At one end of the spectrum, the buyer can accept sealed bids and simply notify sellers individually whether each of them won or not. At the other end, all bids can be revealed as they are submitted, allowing bidders to respond in real time. Different information policies have also been adopted in traditional marketplaces. In federal and some state procurement auctions, the government is legally mandated to disclose only the winner’s bid at the end of the auction (Milgrom and Weber, 1982). By contrast, in municipal construction contracting all bids are often revealed after the winner is selected (Thomas, 1996). Note that the procurement contexts typically feature repeated competition among the same suppliers across different auctions (e.g., Milgrom and Weber, 1982). Because of this repeated competition, the information revealed offers the bidders an opportunity to learn about their opponents’ private information (such as their costs) across auctions. This opportunity can lead bidders to alter their behavior, which in turn affects the 1

Freemarkets has merged with Ariba.

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buyer’s procurement costs. In this context, a procurement setting is modeled as a sequential private value auction in which winners do not drop out from subsequent auctions. Experimental analyses of this setting are quite limited. Our experiment focuses on the learning-related bidder behaviors which a buyer should consider when choosing procurement auction information policies. Specifically, we consider the following two policies in a first price sealed bid procurement auction: (i) the Incomplete Information Revelation Policy (IIP), in which only the winner’s bid is revealed at the end of every auction, similar to the federal government mandated revelation policy; and (ii) the Complete Information Revelation Policy (CIP), in which all bids are made public at the end of every auction, similar to that in municipal construction auctions. We evaluate how well the theoretical insights from Kannan (2008) explain behavior in a laboratory experiment. The perfect Bayesian Nash equilibrium involves two key learning effects which are consequences of the information policies. The extraction effect, which occurs under IIP, refers to the bidders’ desire to alter their bids so as to learn about the opponents’ private information. The faking effect, which occurs under CIP, refers to the bidders’ desire to prevent their opponents from learning about their own private information by pooling with other cost types. Although both these effects arise from a bidder’s desire to maintain a relative informational advantage over her competitors, their manifestations are different. Both effects lead to higher prices, but either may have a more dominant impact on procurement costs, depending on the distribution of the suppliers’ costs. (The details regarding the two effects and their relative importance are provided later.) Some of the theoretical insights from Kannan (2008) are counter to the practices in the industry. For example, as Elmaghraby and Keskinocak (2000) note, there is a perception in the industry that bid transparency leads to aggressive bidding behavior and, therefore, lower buyer procurement costs. However, the perception does not take into account obfuscation strategies which the bidders may adopt when the buyer completely reveals all bids. One may wonder if the differences between theory and industry practice can be attributed to the steep informational and rationality requirement imposed to compute the two stage Bayesian Nash equilibria, which could limit the practical value of the theory. This highlights the need to test directly the predictive power of the theoretical results. One obvious way to test the model would be through an empirical analysis of field data. However, private information about costs is typically not available, making such an analysis difficult. The 2

choice of the information policy is also endogenous in the field, which complicates causal inferences. An experimental study, through its use of a controlled setting and exogenous manipulation of information policies, can overcome these problems. Our experiment allows us to study bidding behaviors under different, exogenously-imposed information policies. Our experimental results are broadly consistent with the comparative static predictions of the theoretical model based on risk neutral bidders. For example, bidders pool with other types to prevent opponents from learning significantly more often under CIP, and the procurer pays less under IIP only when suppliers’ cost structures are highly competitive. This indicates that subjects behave as if they can compute a perfect Bayesian-Nash equilibrium of this dynamic game, or at least appreciate the intuition of the extraction and deception effects. Nevertheless, we document several deviations from the theoretical model with risk-neutral bidders. Observed bids are lower than the prediction in all treatments and the observed faking rates are lower than expected in the CIP treatments. We first examine the extent to which risk aversion can explain these deviations. For this, we analyze a theoretical model which allows for risk averse bidders. Our analysis reveals a surprising result. While increasing risk aversion in single stage private value auctions leads to a lower weight placed on high-priced bids, this is not always the case in CIP. The equilibrium probability of faking by pooling with high cost types actually increases with risk aversion. This occurs because faking in the first stage increases the likelihood of winning in the subsequent stage and reduces bidders’ overall risk. Building on this new set of theoretical results, we also estimate the degree of bidders’ risk aversion using a structural approach. Risk aversion appears to be an important factor in explaining bidding behavior, although it appears that cognitive limitations prevent some subjects from understanding the benefits of faking. We further analyze how the subject’s propensity to fake evolves over time. Our analysis reveals that once a subject discovers the benefit from the faking strategy, he repeatedly uses it. We also find that, in the more competitive environment, subjects learn the faking behavior from observing their opponents using it. Our paper is distinct in the literature in several ways. To the best of our knowledge, our paper is the first one to experimentally study information revelation policies in a procurement auction setting. Moreover, the notion of learning we focus on is across multiple stages within a game. It is different from prior research which has extensively analyzed learning across multiple iterations of 3

the same game (e.g., Camerer and Ho, 1999). Furthermore, the paper features an experimental, a theoretical, and a structural-estimation component. Thus, by drawing upon a diverse set of research methodologies, we provide a comprehensive analysis of the information policy choice problem. The analysis offers actionable managerial insights into the interaction between information disclosure policies and bidder learning effects. The rest of the paper is organized as follows. Section 2 reviews the related literature, and Section 3 summarizes the theoretical model and results. Section 4 presents the research hypotheses and the experiment designed to test them. Section 5 reports the basic results and main hypothesis tests, and Sections 6 and 7 discuss the main deviations of our findings from the theoretical predictions of the model. Section 8 concludes.

2 2.1

Literature Review Sequential Auctions

The two main streams of research in the literature on sequential auctions differ in whether or not the winning bidder drops out of future auctions. Much of the literature has focused on environments where the winner drops out (e.g., Krishna, 2009, Maskin and Riley, 1989). The literature where the winning bidder does not drop out is much less developed, as Klemperer (2004) notes, and whether drop outs occur leads to substantial differences in the results. For example, Krishna (2009) shows that when the winner drops out, the first-price and second-price auctions are revenue equivalent. In contrast, Hausch (1988) shows that this equivalence fails when the winner does not drop out. As mentioned earlier, suppliers in procurement auctions frequently compete against the same set of opponents across different markets or in different auctions held sequentially by the same buyer. This repeated competition should be modeled as the case where the winner does not drop out. One of the first papers to study information revelation policies in this type of context is Hausch (1986). He extends the CIP model specification in Ortega-Reichert (1968), which employed a common value setup, and compares sequential and simultaneous auctions. The procurement context, which is the present focus, arguably corresponds more closely to a private value auction framework. Two other closely related papers are Thomas (1996) and Tu (2005). Thomas (1996) primarily focuses on studying mergers, although part of his analysis compares policies that are similar to CIP and IIP. The model is similar to the case in which two bidders repeatedly compete and each

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is equally likely to be a low- or a high-cost type and their own cost type information is private. He concludes that CIP always generates a higher buyer surplus and lower bids than IIP. This result is different from the conclusion in Kannan (2008), which shows that the surplus ranking of IIP or CIP depends on the distribution of cost types. Tu (2005) also analytically studies a two bidder game where bidders have private information about their cost types, except that their cost is drawn from a continuous distribution. He explicitly imposes an assumption that suppliers cannot fake their types and determines that CIP generates higher buyer surplus than IIP. However, faking is an important aspect we focus on. 2.2

Auction Experiments

Auctions have been studied extensively using experiments, including very practical applications such as for the design of FCC spectrum auctions (e.g., Goeree et al., 2006). However, few experiments have focused on comparing the outcomes of different information revelation policies in sequential auctions, and only two studies has focused on contexts similar to ours. Dufwenberg and Gneezy (2002) analyze the importance of information disclosure policy in a common value setup like Hausch (1986). The main difference, however, is that while Hausch (1986) considers first-price sealed-bid auctions, Dufwenberg and Gneezy (2002) consider a setup where bidders agree to share the royalty with the buyer. They consider three types of information revelation policies, including both of our policies where all bids, or all winning bids, are announced by the auctioneer between stages. They consider a common value setting, however, and bidders have no private value that could be potentially revealed from earlier bids, and competing bidders do not interact repeatedly. Theory does not predict any difference in bid prices between the bid revelation policies, but they find that when bidders are informed about the losing bids in previous stages, prices are significantly higher than the theoretical prediction. Bidders become more competitive when this information is not revealed, which moves bids closer to the theoretical prediction. In a different study, Adomavicius et al. (2008) use a laboratory experiment to compare the information policies in a combinatorial procurement auction setting, where each stage is a multi-round sealed bid auction. In one policy, the provisional allocation (e.g., the current rank) is the feedback; in another, task-related information (e.g., the price or level of quality needed to win the auction) is the feedback; and the third policy offers the bidder a choice of prices along with the possible ranks and the profits generated.

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The analysis finds that the buyer generates maximum surplus with providing task-related feedback. The paper also discusses the changes in bidder behaviors observed with the varying levels of feedback. 2.3

Risk Aversion

Risk aversion is important in environments such as auctions, where agents face situations characterized by uncertainty related to the value of the object, the strategies used by others, and the private information possessed by their opponents. Auctions with risk-averse bidders have been studied by Maskin and Riley (1984) and Maskin and Riley (1987); see also Perrigne and Vuong (1999) for a survey in this area. Risk aversion induces bidders to bid more aggressively in private value auctions. The phenomenon of overbidding relative to the risk neutral Bayesian-Nash equilibrium in first price auctions with competing buyers is well-documented in experimental research; see for instance Cox et al. (1988), Harrison (1989) and Kagel and Roth (1995). Risk aversion has also been proposed as an explanation for such deviations in non-experimental auction data, even with large firms as bidders (e.g., Campo et al., 2003). In a procurement auction context similar to ours, risk averse agents are expected to bid lower than the risk neutral Bayesian Nash equilibrium because it increases the likelihood of selling the unit, although these lower bids reduce the profit conditional on winning. In our paper, as mentioned earlier, we examine the extent to which risk aversion explains deviations from risk neutral predictions. For this, we estimate the degree of risk aversion of the subjects using a structural approach. Many empirical studies on auctions have extended the structural approach from Paarsch (1992) to estimate risk aversion. See, for example, Athey and Levin (2001), Bajari and Hortacsu (2005); Cox et al. (1988) and Cox et al. (1992); Goeree et al. (2002); Campo et al. (2003) among others. We explain later why our analysis requires a different approach.

3

Theoretical Model, Equilibrium and Insights: Summary

We initially build on Kannan’s (2008) two-stage, private-value, sequential auction model with no winner drop outs to design the experimental framework. In this setup, bidders are assumed to be risk neutral. Later, in Section 6.1, we extend the model to allow for risk averse bidders (the details of the analysis on risk aversion are provided in Appendix A). While the original model in Kannan (2008) considers an arbitrary number of potential competitors, we examine the special 6

case of two bidders (or suppliers) in our experiment. Bidders have private costs which are drawn from a discrete distribution of two cost types. Prior work has also used a two-type framework to study information policies in other settings, in part because some interesting aspects of learning across auction rounds do not exist in the continuous cost distribution case (Jeitschko, 1998). The main advantage of the two-type model is that the second stage is a relatively straightforward game and the analysis can focus on the learning effects in the first stage. This advantage is lost even in a three-cost type model, where the nature of the second stage equilibrium varies significantly depending on the first stage outcome, which must also be considered by bidders in the first stage. Let cl be the marginal cost of production for a low-cost supplier and ch for a high-cost supplier, and θ be the probability with which a bidder is a low-cost type. We assume common knowledge of all three variables. Although θ is common knowledge, the outcome of the draw for each supplier is private information. Since we focus on bidders’ learning across auctions, we model a two stage game, each stage corresponding to an auction initiated by a buyer. The cost type for a bidder is determined before the beginning of the first stage and remains the same for both stages. In each stage, both suppliers simultaneously submit a sealed bid, p ∈ [0, ∞). Only one winner is picked in each stage and he is the bidder with the lowest bid price in that stage (ties are broken randomly). The winner’s payoff is his bid price minus the marginal cost, and the loser’s payoff is zero. Therefore, the auction in each stage is a first-price sealed bid type. Between the first and the second stages, information is revealed according to the policy. Under CIP, all bids are revealed at the end of the first stage; while in IIP, only the winner’s bid is revealed. We are interested in comparing the impact of the different information policies on bid prices and procurer surplus. The model specified above does not always have an equilibrium. Even single stage games, similar to the ones we encounter in the second stage, may not have an equilibrium. We follow Maskin and Riley (1985) to overcome the non-existence problem by implicitly assuming discrete bids in infinitesimal increments.2 The game then has a unique Bayesian Nash equilibrium under CIP and IIP for each stage. We focus on the bidding behavior of the low-cost types since the bidding behavior of the high-cost type is uninteresting. A high-cost type always bids ch at equilibrium 2 Maskin and Riley’s footnote 2 is directly applicable to our context: “As our model is formulated, an equilibrium in the sealed-bid auction may not exist. The nonexistence problem, however, is an artifact of our allowing literally a continuum of possible bids. In fact, we can restore existence even with a continuum by allowing the possibility of positive but infinitesimal bids, which we implicitly assume in our analysis.”

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independent of the policy. In the rest of this section, we summarize the results from Kannan (2008) for the case with two bidders. The results can also be obtained as a special case of the risk aversion extension shown in Appendix A. Scenarios CIP Separating equilibrium case: θ < 31

Stage Stage 1 Stage 2

CIP semipooling equilibrium case: θ > 31

Stage 1

IIP

Stage 1

Stage 2

Stage 2

Equilibrium bids for a low-cost type  (ch −cl ) − 1 F CIP-sep,θ (p) = 1 − 1−θ θ (p−cl ) Bid equal to own cost if the opponent is also of the same type; otherwise, bid just below opponent’s cost. √ −2(1−θ)+ (1−θ)(3−θ) ; Faking probability θ γ=  (1−θ+θγ)(ch −cl ) 1 CIP-semi,θ F (p) = θ 1 − . p−cl If both reveal low-cost type in the first stage: bid cl ; θγ If one reveals his type: set α = 1−θ+θγ and β = 0 in SSG; θγ If both bid ch : set α = β = 1−θ+θγ in SSG. Solution to: (1 − θ)(1 − log (1 − θ))(ch − cl ) = (1 − θF IIP,θ (p))(p − cl ) − (1 − θ)(ch − cl ) log(1 − θF IIP,θ (p)). θγ If pw is the first stage winner’s bid, set α = 1−θF IIP,θ (pw ) and β = 0 in SSG. F single,θ (p) = 1θ (1 −

Single stage

(1−θ)(ch −δ−cl ) ) (p−cl )

Table 1: Results from the theoretical model. The superscript ‘sep’ refers to the separating equilibrium and ‘semi’ refers to the semipooling equilibrium. F l is the cdf of the first stage bid distribution under the case where l represents the different cases for a given θ. Since the equilibrium is derived by backward induction, consider first the second stage. This stage is similar in structure for the two policies, although beliefs will differ. Let Rα be a lowcost bidder who believes that his opponent is a high-cost type with a probability of α while his opponent Rβ believes that Rα is a high-cost type with a probability of β. Let β and α be common knowledge, and β ≤ α. Then, the equilibrium price is given by a mixed strategy distribution for Rβ as FRβ (p) = 1 − point of

α−β 1−β

α 1−α



ch −cl p−cl





− 1 and for Rα as FRα (p) = 1 −



α 1−β



ch −cl p−cl



−

β 1−β



with a mass

at the price infinitesimally smaller than ch . Using this generic second stage game,

which we refer to as SSG in Table 1, we can obtain the second stage equilibrium under each policy by substituting for α and β depending on the first stage equilibrium as well as the bid information available at the end of the first stage. The second stage equilibrium is straightforward if the bidders are aware of each other’s type, which can occur in the second period in our context only if the first stage bids reveal the bidders’ type. The game then simply corresponds to a Bertrand game. This occurs only when both bids 8

are revealed, i.e., in CIP, and the first stage has a separating equilibrium. The corresponding equilibrium is described at the top of Table 1. The separating equilibrium exists in the first stage under CIP only when θ < 13 . For higher θ values, low-cost bidders may prefer to bid like a high-cost type. In equilibrium, they mix between prices less than ch and submitting a faking bid of ch . By preventing their opponent from learning about their type, bidders gain an information advantage for the second stage. To see this, consider bidder A who fakes and pools with the high-cost type in the first stage. Then, a Bayesian-updating bidder B will lower his belief in the second stage that bidder A is a low-cost type. This, in turn, allows bidder A to undercut bidder B in the second stage. We refer to this pooling strategy in the first stage as the faking effect. The equilibrium corresponding to this case is shown as the second panel in Table 1. Under IIP, only a separating equilibrium exists because bidders do not have an incentive to pool with the high-cost type to prevent the other bidder from learning. Since only the winner’s bid is revealed in IIP, even if a low-cost bidder bids like a high-cost type and loses the first stage, his bid is not revealed to the winner. Therefore, the losing bidder cannot bias his opponent’s belief. This makes faking undesirable under IIP. Note that the separating equilibrium under IIP is also different from that under CIP. Since only the winner’s bid is revealed under IIP, a low-cost bidder gains more information about his opponent if he loses. A winner, however, gain less information than the losing bidder. So, to gain some information for use in the second stage, a low-cost bidder may risk losing the first stage game. This bidding behavior exhibited to extract information about the opponent is referred to as the extraction effect, and it raises bids relative to the benchmark of a single stage equilibrium. The equilibrium corresponding to this case is shown as the last case in Table 1. The bottom of Table 1 shows the cdf of the bid distribution for the benchmark of the single stage game. This can be used to compare against the first stage bids under CIP and IIP in order to understand how the multi-stage auction and information policy affects the bids. The comparisons are illustrated in Figure 1, which shows the equilibrium bid distributions for the single stage game and the first stage games of IIP and CIP for two different values of θ. Figure 1(a) corresponds to a case when a separating equilibrium exists under CIP. Note that the bid distribution under CIP for θ = 0.25 is identical to that of the single stage game. Figure 1(b) displays a case when a 9

(a) CIP (θ = 0.25)

(b) CIP (θ = 0.9)

Figure 1: CDF for first stage bid distributions and single stage bid distribution for a low-cost type when θ = 0.25 and θ = 0.9

semipooling equilibrium exists under CIP. Note that the bid distributions are skewed to the right under CIP (θ = 0.9). Also, notice the mass point corresponding to the faking bid. Under IIP, the bids in both figures are skewed to the right compared to the single stage game because of the extraction effect. Comparing the equilibrium characterizations for different cost distributions, the faking effect can be shown to dominate the deception effect in the more competitive settings i.e., when θ is high. This lowers procurer surplus under CIP relative to IIP. However, in the less competitive settings, the deception effect dominates the faking effect and leads to lower procurer surplus under IIP. When θ ≈ 0.87 and bidders are risk neutral, the expected procurer profits under both cases are equal.

4

Testable Hypotheses and Experiment Design

Our interest is in understanding the tradeoff between the desire to learn about the opponent (extraction effect) and the desire to prevent an opponent from learning (faking effect), and how these incentives depend on the information disclosure policy of the auction. Given the equilibria under the different conditions, it is appropriate to study the tradeoff only when a semipooling equilibrium exists under CIP. Therefore, the experiment only considers the case of θ >

1 3;

in

particular two specific θ values θ = {0.5, 0.9}, one when the extraction effect dominates and the

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other when the deception effect dominates. We first present four hypotheses and then describe the experiment design to test them. 4.1

Hypotheses

Note that the learning effects of interest are behaviors exhibited in the first stage by the low-cost types. Our first hypothesis is obvious and intuitive from the functional forms of the equilibrium bid distributions. Hypothesis 1 As the probability of observing a low-cost opponent increases, the average price bid by the low-cost suppliers decreases. One of the key results of the model is that the desire to learn about the opponent (the extraction effect) dominates the desire to prevent opponent from learning (the faking effect) if the probability of facing a low-cost opponent is low, and the faking effect dominates the extraction effect in more competitive conditions when the probability of facing a low-cost opponent is high. Specifically, in our experiment, the bids are expected to be higher on average in CIP than in IIP for the highly competitive (θ = 0.9) treatment, but lower bids in CIP for the less competitive (θ = 0.5) treatment. This leads to the following hypothesis: Hypothesis 2 When the probability of facing a low-cost opponent is low, the average price bid in the first stage by the low-cost suppliers is higher under IIP than CIP and vice-versa. For reasons discussed earlier, bidders fake with a non-zero probability only under CIP and no faking incentive exists under IIP. Therefore, Hypothesis 3 The probability of faking by a low-cost bidder is higher under CIP than under IIP. The first three hypotheses focus on the variations in bidding behavior while the final hypothesis concerns the procurer surplus. Kannan (2008) shows that the extraction and the faking effect also influence the total buyer payments across both periods. That analysis leads to the following hypothesis, which parallels the bidding Hypothesis 2 above. Hypothesis 4 When the probability of facing a low-cost opponent is low, the total buyer payment across both periods is higher under IIP than CIP and vice-versa. 11

In other words, the procurer pays lower prices and obtains greater surplus on average by choosing the Incomplete Information Policy in more competitive conditions that feature similar costs among suppliers. 4.2

Experimental Design

To test these hypotheses we employ an experimental framework that implements the stylized model summarized in Section 3. We compare the information revelation policies for two different values of θ (0.5 and 0.9), both of which correspond to the semipooling equilibrium case under CIP. From the theoretical results summarized in Table 1, it should be clear that the policy variations could produce qualitative changes in bidder behavior since the policy variations affect how bidders process the information revealed. The variations because of changes to θ, however, are merely quantitative since the type of equilibrium is unchanged. For this reason, we choose to vary θ within the experimental sessions we conduct and vary the policies across sessions. We conduct 8 experimental sessions in total, with four sessions devoted to each information policy. Each session employs 12 subjects and involves 50 periods, and each period consists of two stages. A Bernoulli process is conducted in each period to determine the cost type for every subject, and this type remains the same for both stages of that period. Since we are testing a noncooperative equilibrium of a one-shot, two-stage game, we reduce repeated game incentives by randomly repairing the subjects with new opponents in each period. Although subjects interact repeatedly across the 50 periods, their interactions are anonymous and the identity of the interacting subject is never revealed. In each session, 25 periods are conducted with one value of θ, followed by 25 periods for the other θ. We vary the sequence of the θ treatments across sessions in order to control for possible order effects. Thus, we have 4800 bids in each stage from 96 different participants across the 8 sessions. The experimental sessions were conducted in the Vernon Smith Experimental Laboratory at Purdue University. The subjects were recruited by email from the undergraduate student population and each subject was limited to participate in one session. Upon arrival at their experimental session, subjects were randomly assigned to individual computers and no communication between subjects was permitted throughout the session. At the start of the experiment, the instructions were read orally by an experimenter while the subjects followed along on their own copy. A sample

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of the instructions is provided in Appendix B. One important design requirement for the experiment is to identify bids from low-cost bidders that can be unambiguously interpreted as faking. This is needed to investigate the effect of using this pooling strategy to prevent an opponent from learning. During our initial pilot sessions, however, without any explicit restriction on the maximum bid submitted the high-cost types often submitted a bid greater than their equilibrium bid of ch . This provided incentive for low-cost bidders who may have adopted the faking strategy to submit bids equal to or higher than ch , which is problematic since it complicates any inference regarding which bids from low-cost bidders should be interpreted as faking bids. To overcome this problem we modified the experiment software for the sessions reported here to restrict the maximum possible bid to be ch . This serves the purpose of clearly identifying faking bids from the low-cost bidders.3 This restriction on feasible bids can be justified by an assumption that the buyer has access to the same common information about the cost distribution as the bidders. Since she knows that the maximum possible cost is ch , she is not willing to consider any offer greater than ch . In any case, our focus is not on the bidding behavior of the high-cost type. In the theoretical equilibrium analysis bidders use Bayes’ Rule to update their beliefs. Since beliefs play such an important role in the theory, we adopt a procedure (now common in the experimental economics literature) to elicit beliefs directly. In each period at the end of the first stage, every subject is asked to state, based on the information revealed, his belief (expressed as a probability) that his opponent has a cost of cl . Monetary incentives are provided for making accurate guesses. Specifically, for eliciting the beliefs, we applied a quadratic scoring rule, which is incentive compatible for players to state their true their beliefs (Selten, 1998, Nyarko and Schotter, 2002, Costa-Gomes and Weizsacker, 2008). To reduce the likelihood that the belief elicitation reward significantly affects bidding behavior, the maximum reward for the beliefs (20 experimental Francs) is kept low relative to the difference in ch and cl . The sequence of the experiment is as follows: Each bidder submits his bid for the first stage. After all bids are submitted, information about bids is revealed according to the chosen policy. Each subject then enters his belief about the opponent’s cost type and his bid for the second stage. 3

Alternatives based on fitting the distribution of high-cost bids would add noise unnecessarily and were considered an inferior approach.

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At the end of the second stage, the computers display the opponent’s bids in the two stages and also the opponent’s costs and the subject’s own earnings. Subjects record all this information on hard copy record sheets so their personal history was easily accessible. Across all the 8-sessions, the earnings for the 96 subjects ranged from $18.00 to $32.00 with an average of $24.00 per subject. Sessions typically lasted 90 to 100 minutes in total, including instruction time.

5

Results

All 25 periods included

Only periods 11-25 included Theory: risk neutral Theory: risk aversion r = 0.42

Mean of the first stage bids Stdev % of faking bids in the first stage Mean of the second stage bids Stdev Observations Mean of the first stage bids Stdev % of faking bids in the first stage Mean of the second stage bids Stdev Observations Expected first stage bid Expected probability of faking Expected first stage bid Expected probability of faking

CIP θ = 0.5 θ = 0.9 355.37 298.14 44.94 60.02 8.16% 11.78% 338.22 272.70 55.91 54.82 588 1,112 358.73 299.46 39.87 64.76 4.83% 15.36% 340.52 273.69 53.43 57.88 352 664 366.08 339.33 23.95% 28.85% 360.69 318.79 39.17% 31.59%

IIP θ = 0.5 θ = 0.9 368.33 287.06 41.52 44.42 2.21% 0.99% 360.19 262.25 47.61 51.31 588 1,112 372.00 277.12 35.38 41.47 1.14% 0.45% 361.91 255.03 44.91 52.47 352 664 385.77 309.97 0% 0% 375.27 277.29 0% 0%

Table 2: Descriptive statistics of bids submitted by low-cost bidders under CIP and IIP. Table 2 provides the summary statistics of the bids submitted by the low-cost bidders for the first and the second stages. Consider the upper panel, which is based on data from all 25 periods of each treatment sequence. The average bids are lower, as predicted, when θ = 0.9. Moreover, the average bids are also lower in IIP than CIP when θ = 0.9, and higher in IIP than CIP when θ = 0.5, again consistent with the equilibrium. Note that sellers sometimes adopt the faking strategy, the key feature of the semipooling equilibrium. Roughly 8 to 12% of the first stage bids from the low-cost types are equal to 400 in the CIP treatment. By contrast, the model predicts that no low-cost bidders should bid 400 in the IIP treatment, and a much smaller fraction (about 1 to 2%) of such bids were observed. Recall from the theoretical model that low-cost bidders never submit

14

bids of 400 in the second stage. This prediction is supported in the data, as these bidders submit second stage bids of 400 only 0.18% of the time in CIP. This provides additional evidence that the first stage bids of 400 are not mistakes, but are submitted in order to deceive the opponent from learning about his low-cost type, and therefore part of the faking strategy. 5.1

Hypotheses Testing

Since the initial periods under each treatment involve considerable learning and adjustments to the treatment conditions, for tests of the theoretical equilibrium hypotheses we exclude the first ten periods under each treatment condition and present the results from analyzing the rest of the data. The middle panel of Table 2 presents the corresponding summary statistics. Our results are robust and they remain qualitatively unchanged when using all 25 periods of each treatment. The first three hypotheses concern the first stage bids from the low-cost type sellers. In the following regression models, we include control variables: (i) the inverse of period (Inv P eriod) to account for nonlinear time trends, and (ii) treatment order effects (T reat Seq), a dummy variable to differentiate the first and second treatments run within a session. These control variables allow for learning or other time series adjustments in behavior that are unrelated to the hypotheses of interest. In addition, we employ independent variables that are relevant for the respective hypothesis. The regressions account for unobserved subject heterogeneity as random effects in order to control for additional factors not captured by our independent variables. Table 3 shows the coefficient estimates. For the regressions testing Hypothesis 1, the dependent variable is the bid submitted by a lowcost bidder. Since the maximum bid is 400, we employ a Tobit model with a 400 upper-bound. We consider the bids submitted by low-cost bidders under each policy separately. The base case is when θ = 0.5, and a dummy variable, Dummy θ 0.9, is set to one for θ = 0.9. The left column in Table 3 shows that the regression results. The coefficients on Dummy θ 0.9 are negative and highly significant, consistent with Hypothesis 1. The data thus provide evidence that as the fraction of low-cost bidders increases, the first stage bids by low-cost bidders decrease. To test Hypothesis 2, the regressions again use the bid submitted by a low-cost bidder as the dependent variable and a Tobit model for estimation. Since the focus here is on the information policy, we consider the bids from low-cost bidders corresponding to each θ separately but pool the

15

Dependent Variable Intercept Inv P eriod T reat Seq Dummy θ 0.9

Hyp. 1 – Tobit IIP CIP First period bid from a low-cost type 362.91** 358.29** (4.15) (6.76) 7.02* -21.02** (3.56) (6.85) 18.94** 17.30** (1.84) (3.53) -96.90** -58.14** (1.84) (3.54)

Dummy CIP Observations Log Likelihood

1,016 -4,856.04

1,016 -4,988.29

Hyp. 2 – Tobit θ = 0.5 θ = 0.9 First period bid from a low-cost type 373.82** 262.87** (5.69) (7.01) -5.92* -7.51 (3.77) (4.84) 4.28 31.73** (6.51) (7.99)

Hyp. 3 – Probit θ = 0.5 θ = 0.9 If a low-cost type faked in the first period -10.03** -4.78** (1.18) (1.59) 0.98 -1.97** (0.89) (0.49) -0.37 0.28 (0.92) (0.53)

-14.87** (6.51) 704 -3,252.38

3.76** (1.11) 704 -41.18

26.02** (7.99) 1,328 -6,550.25

2.96** (1.34) 1,328 -193.67

Table 3: Regressions to test Hypotheses 1, 2, and 3. Numbers in parentheses indicate the standard errors. Note that ** indicates a significance level of 1%, and * a significance level of 5%. data across the policies. The base case corresponds to IIP, and a dummy variable equal to one for CIP is the independent variable of interest. The regression results shown in the middle of Table 3 are consistent with these predictions of Hypothesis 2. The CIP dummy variable is negative and significant for the less competitive treatment and positive and significant for the more competitive treatment. This indicates that the desire to learn dominates if there are fewer low-cost sellers (i.e., a lower θ value), so switching to a complete information policy lowers average bids. For the regressions to test Hypothesis 3 we estimate a probit model with the dependent variable being a binary indicator that takes on a value of one when the low-cost bidder bids 400 in the first stage. This hypothesis concerns the impact of the policy on the propensity to submit faking bids. So, as before, we consider low-cost bids from each θ separately but pool the data across policies. The base case corresponds to IIP, and the dummy for CIP is the independent variable of interest. The last two columns of Table 3 provide support for Hypothesis 3: the likelihood of observing faking is significantly higher in CIP than IIP. Bidders apparently perceive correctly that bidding to prevent an opponent from learning one’s cost type is feasible and useful in CIP, but not in IIP.4 Finally, consider the procurer prices paid under each policy, the subject of Hypothesis 4. The dependent variable in the regressions shown in Table 4 is the sum of the prices paid by the buyer 4

A Hausman test confirms that fixed effect results are not statistically different from the random effects results.

16

Dependent Variable Intercept Inv P eriod T reat Seq Dummy CIP Observations Log Likelihood

Hyp. 4 – Tobit θ = 0.5 θ = 0.9 Total price paid by the procurer across both stages 382.42** 248.96** (2.02) (1.66) -13.52** -19.65** (4.29) (3.55) 4.91** 16.43** (2.05) (1.69) -14.53** 13.45** (2.05) (1.69) 2,880 2,880 -12,646.09 -14,987.03

Table 4: Regression to test Hypothesis 4. Numbers in parentheses indicate the standard errors. Note that ** indicates a significance level of 1%.

across the two stages. The base case is IIP and we again measure the impact of switching to CIP through the independent variable Dummy CIP . A Tobit model accounts for both lower and upper bounds on payments.5 Consistent with Hypothesis 4, the Dummy CIP coefficient is negative when θ = 0.5, but positive when θ = 0.9. Thus, the procurer pays less when adopting CIP only when the environment is less competitive. In more competitive environments, the procurer pays more when employing CIP since the low-cost bidders more frequently bid high prices to hide their type from the other bidder.6

6

Risk Aversion

The previous section shows that the experimental data are broadly consistent with the comparative static predictions of the theoretical model with risk neutral bidders. Sellers’ bids shift in the correct direction in response to changes in the information policy and changes in the degree of competition. However, bidder behavior and market outcomes are not completely explained by the model and several important quantitative deviations exist. In particular, the observed average bids in Table 2 (Page 14) are lower for all treatments than the theoretical expectations. Table 2 also shows that the 5

Unlike the previous cases, these regression models do not include random subject effects because this market level performance measure depends on both sellers in each market and sellers are randomly re-paired each period. 6 Empirical tests based on the directly elicited bidder beliefs indicated that belief updating is also broadly consistent with theory. Bids observed in the first stage correctly influence the beliefs held by the bidders regarding their opponent’s type, and beliefs for the second stage game correctly influence bids submitted in the second stage.

17

(a) CIP (θ = 0.5)

(b) CIP (θ = 0.9)

(c) IIP (θ = 0.5)

(d) IIP (θ = 0.9)

Figure 2: First stage bid distributions for a low-cost type under CIP and IIP observed frequencies of faking are lower than those implied by the model. Figure 2 indicates that the observed bid distributions place a greater weight on the lower bids compared to the equilibrium distributions with risk neutral bidders. The question we are interested in addressing in this section is: to what extent does risk aversion explain the observed aggressive bidding behavior? We first extend our original theoretical model by allowing for risk averse bidders. We then use the theoretical results and follow a structural approach to estimate the degree of risk aversion of our subjects.

18

6.1

Results from a Theoretical Model with Risk Aversion

We extend the theoretical model summarized in Section 3 to include a bidder utility function which exhibits a constant degree of relative risk aversion with coefficient r ∈ (0, 1). This type of utility function is commonly used in the auction literature. When r = 0 this corresponds to the risk neutral case, and risk aversion increases with r. The mathematical details of the equilibrium analysis are shown in Appendix A. In the following paragraphs, we discuss how bidder behavior varies with r.

Figure 3: Variation of the average first stage bid with risk aversion parameter r. Figure 3 shows the average first stage bids for low-cost bidders under CIP and IIP as r changes for the θ values considered in our experiment. Note that, as in single stage games, the average price in each treatment decreases with risk aversion. Note also that average prices are ordered across the treatment conditions in the same pattern as the risk neutral case. Thus, the main comparative static predictions for average bids are robust to risk aversion. The equilibrium first stage bid distributions under IIP (Figures 4(c) and 4(d)) reveals that with risk aversion, the mixed strategy equilibrium places a higher probability weight on lower bid prices. Under CIP, Figures 4(a) and 4(b) illustrate that, as before, semipooling equilibria exist in both treatments for various degrees of risk aversion r. Unlike the IIP case, however, the bid distributions for the risk neutral case do not first-order stochastically dominate the risk averse cases. As risk aversion increases, the bid distributions in CIP shift leftwards only in the lower and intermediate bid range. This shift explains why the average bids decrease very slowly with r in Figure 3. In the higher bid ranges, the bid distributions intersect because the faking rate increases with risk 19

(a) CIP (θ = 0.5) First Stage Distribution

(b) CIP (θ = 0.9) First Stage Distribution

(c) IIP (θ = 0.5) First Stage Distribution

(d) IIP (θ = 0.9) First Stage Distribution

Figure 4: Bid distributions aversion. This observation may be surprising at first glance since faking increases the likelihood of losing in the first stage and therefore appears to be a risky strategy. However, in this multi-stage environment, one can show that faking increases the likelihood of winning in the second stage, and, hence, reduces the overall risk across the two stages. Figure 5 compares the faking probabilities under both CIP treatments. The faking rates are lower in the less competitive environment only if r is smaller than approximately 0.16. Note that the faking rates are more sensitive to the degree of risk aversion in the less competitive (θ = 0.5) environment. (As in the risk neutral case, no faking occurs in IIP.)

20

Figure 5: Faking rates under CIP with different degrees of risk aversion

6.2

Structural Approach for Estimating Risk Aversion

In this section, we apply a structural approach to estimate the degree of risk aversion of our subjects. Prior structural work on risk aversion typically estimates the density of the bid distribution, recovers the pseudo private values, and finally estimates the bidders’ private values. Such a strategy exploits the monotonicity property of bids with valuations, i.e., each private value can be expressed as a function of the corresponding bid. We, however, cannot rely on the property since in this game the equilibrium is in mixed strategies. Moreover, we observe bidders’ valuations and, hence, do not need to estimate them. We account for mixed equilibrium strategies and apply a structural approach that relies on the assumption that observed bids are generated from the theoretical equilibrium model allowing for risk aversion. Specifically, we estimate the degree of risk aversion that minimizes the distance between the theoretical and the empirical bid distributions. Let bl1 , . . . , bln be the observed first stage bids from low-cost suppliers, where l represents the treatment, i.e., l ∈ L = {CIP, IIP } × {θ = 0.5, θ = 0.9}, and X l be a set such that all the observed first stage bids from low-cost types are included distinctly (only once). The cdf of the observed bid distribution for an arbitrary bid x is

1 Pn n i=1 I





bli ≤ x ≡ F˜ l (x), where I is the indicator function.

Let the bids originate from an underlying distribution with cdf for x being F l (x, rb), where rb is the risk aversion coefficient of the subjects. For any arbitrary risk coefficient r, F l (x, r) indicates the cdf of the theoretically generated bid distribution. For any x, the squared difference between the cdfs of the theoretical distribution computed at a risk aversion coefficient of r and the empirical

21

distribution is given by lr (x) =



2

F˜ l (x) − F l (x, r)

.

The objective is to find the risk aversion parameter that minimizes the sum of squared differences between the theoretical and empirical bid distributions:7 X X

rb = arg min r

lr (x).

(1)

l∈L x∈X l

The differences lr (x) can be interpreted as the errors arising from the agent’s optimization, or from considering a subset of the whole population. Since no closed form solution exists for the equilibrium, we apply a grid search over r in increments of 0.01. In order to test for statistical significance as well as equality of risk aversion estimates between treatments we compute the standard errors using the non-parametric bootstrap method (with replacement) as suggested by Efron (1982). Our estimate of the standard error is obtained using 500 bootstrap samples from the empirical distribution of the data.8 6.3

Results from the Estimation Procedure

We estimate the risk aversion coefficient using the dataset with the first 10 periods in each treatment omitted in order to reduce the influence of learning effects and adjustments to the treatment conditions.9 The risk aversion coefficient that best fits the 2, 032 observations pooled across all treatments is r = 0.42 and the bootstrapped standard error is 0.008.10 Note that this estimate is significantly different from zero (risk neutrality) at level 0.05 and is roughly consistent with previous estimates reported in the literature (e.g., Goeree et al., 2002). Figure 2 (Page 18) also shows the theoretical first stage bid distributions for the different 7

In order to check for the robustness of this metric, we also use tests based on the supremum distance norm as suggested by Romano (1988) and Romano (1989). We use a Kolmogorov-Smirnov test using the supremum distance between the theoretical and empirical bid distributions and search over the different risk aversion parameters according to the minimum distance principle. The optimal risk aversion parameter satisfies b r = arg minr sup F˜ l (x) − F l (x, r) . The results are not significantly different for most of the treatments, and they are available from the authors upon request. 8 For tests at level 0.05, Efron and Tibsharani (1993) and Davidson and MacKinnon (2000) recommend 200 and 399 samples, respectively. 9 This is the same subsetting to later periods employed in the initial equilibrium hypothesis testing in Section 5. The estimation results using all periods are similar to the reported ones. 10 As a robustness check, we also weighted the sum of the squared errors in each treatment by the number of observations in the corresponding treatment. The estimated risk aversion coefficient is 0.43.

22

Treatment l CIP Differences of Distributions

Separate risk aversion estimates for each treatment

1 |X l | 1 |X l |

lr=0 (x) P l x∈X l r=0.42 (x) % improvement with risk aversion (r = 0.42) Risk aversion (r) P

x∈X l

Observations

θ = 0.5 0.001 0.022 −120.83%

θ = 0.9 0.095 0.017 81.65%

IIP θ = 0.5 θ = 0.9 0.021 0.083 0.011 0.004 47.18% 95.27%

0.00 (0.01) 352

0.54** (0.01) 664

0.40** (0.09) 352

0.39** (0.01) 664

Table 5: Differences of distributions in monetary values. Results for estimated risk aversion coefficients. Numbers in parentheses indicate the bootstrapped standard errors. Note that ** indicates a significance level of 1%.

treatments when r = 0.42. The figure indicates that the risk averse specifications provide a better fit to the empirical bid distributions than the corresponding risk neutral cases. For the same r, the last two rows in Table 2 (Page 14) show the expected first stage bid and the probability of faking obtained using the theoretical model. Notice that the average bid prices are closer to those implied by the risk averse case. In order to quantity the fit for the different treatments, we calculated the accumulated absolute differences between the empirical and theoretical bid distributions. The upper panel of Table 5 shows that the differences are largest for the competitive treatments under risk neutrality. Those treatments benefit the most from allowing for risk aversion. Incorporating risk aversion reduces the differences in CIP when θ = 0.9 by 81.65% and in IIP when θ = 0.9 by 95.27%. Risk aversion also explains a fair amount of deviations in IIP for the θ = 0.5 treatment, reducing the errors by 47.18%. The θ = 0.5 treatment in CIP is the only case in which risk aversion does not capture additional differences between the theoretical and empirical distribution. We seek additional insights by estimating the risk aversion coefficient for every treatment separately using a specification similar to Equation 1. The lower panel of Table 5 shows those estimates for each treatment. Using t-tests we can confirm that the risk aversion parameters are not significantly different between treatments, with the exception of the CIP (θ = 0.5) treatment. In this treatment, the data appear most consistent with risk neutrality (r = 0). Figure 2(a) shows that the empirical bid distribution in the CIP (θ = 0.5) treatment fits better

23

χ 300 320 340 360 380 400

CIP,0.5 x 0. We characterize the equilibrium for the two stage game by considering the second stage first. For the sake of simplicity of discussion, we normalize the low- and the high-costs to be 0 and 1, respectively. The probability that a supplier is low-cost is θ, which is common knowledge. The random draws realized for each supplier are, however, private information. The information is revealed according to the policy. The analysis of this appendix will also implicitly assume the presence of the discontinuity in the price space as mentioned in Section 3 of the manuscript to ensure equilibrium existence. As before, the equilibrium strategy for a high-cost type is to always bid his cost, 1. We are interested in characterizing the equilibrium strategies for the low-cost types. A.1

Second Stage Game

The second stage games under the two policies have some commonality, which one can capture through the following specification. Let one bidder, say A, believe that his opponent B is high-cost type with a probability of α, while B believes that A is high-cost type with a probability of β. Suppose α ≥ β. The profit expressions for A and B when they are of low-cost type and bidding a price of q are: πA (q) = (α + (1 − α)(1 − FB (q)))(q − cl )1−r and πB (q) = (β + (1 − β)(1 − FA (q)))(q − cl )1−r , where FA (q) and FB (q) are the cumulative density functions (cdfs) of the bid distributions from players A and B. Note that the bid price infinitesimally smaller than ch is the supremum of the strategy space and, in a mixed strategy equilibrium, payoffs from any price q should yield the same payoff. Based 1−α

on that, we can compute FB (q) =

(ch −cl )1−r (q−cl )1−r

(1−α)

. We find that the infimum of FB (q) is cl +

1

( (ch −cαl )1−r ) 1−r . This price will also correspond to the infimum of FA (q). Using that information, 1−α

we compute FA (q) =

(ch −cl )1−r (q−cl )1−r

(1−β)

and a masspoint of

α−β 1−β

at the price infinitesimally smaller than

ch . In this case, the expected profits for both players A and B is α(ch − cl )1−r . Using these cdfs, one can directly obtain the pdf of the bid distributions. Let the respective probability density

32

functions (pdfs) be fA (q) and fB (q). A.2

First Stage Game: CIP

Recall that under CIP, all bids are revealed. In this case, when solving for the equilibrium, we find that the first stage game has an equilibrium which is one of two types depending on θ and r. We characterize both types of equilibria below. In computing the Bayesian Nash equilibrium, we first consider a separating equilibrium and then consider deviations. A.2.1

Separating Equilibrium

Suppose a separating equilibrium exists in the first period. It implies that the first stage reveals the type of the bidder. As a result, the second period game simply corresponds to a Bertrand game. The payoff across both stages from bidding p in the first stage is π CIP-sep,θ (p) = (1 − θ)(ch − cl + p − cl )1−r + θ(1 − F CIP-sep,θ (p))(p − cl )1−r . Note that the price infinitesimally smaller than ch is in the supremum of the strategy set and the expected profit from any p in the mixed strategy equilibrium strategy set should be the same. Based on that, we can obtain the expected profit from any p is (2(ch − cl ))1−r (1 − θ) and F CIP-sep,θ (p) = 1 −

1−θ θ



(2(ch −cl ))1−r −(ch −cl +p−cl )1−r (p−cl )1−r



. As expected, the

bids tend to be lower with increasing risk aversion (=r) and higher degrees of competition (θ). The separating equilibrium is only sustained under a certain condition. To determine when the separating equilibrium becomes violated, consider deviations by one of the bidders. Suppose a bidder pretends to be a high-cost type in the first stage. The expected profit for that bidder is: High-cost rival

z z

Bidder wins both stages

}|

}| { z

Bidder wins second stage

}|

{ {

Low-cost rival

z

}| }|

{

Bidder wins second stage

1 1 π f ake = (1 − θ) (2(ch − cl ))1−r + (1 − θ) (ch − cl )1−r + θ(ch − cl )1−r . 2 2 z

{

We can observe that π f ake > (2(ch − cl ))1−r (1 − θ), i.e., faking is more profitable, if θ > find that

∂ ∂r (

21−r −1 21−r +1

21−r −1 . 21−r +1

We

) < 0, implying that the threshold degree of competition (i.e., θ) beyond which

faking occurs decreases with r. A.2.2

Semipooling Equilibrium

A pooling equilibrium where both types bid the same price cannot be sustained and only a semipooling equilibrium exists. Let γ be the probability in the first stage with which a bidder fakes. In the semipooling equilibrium case, three different possibilities exist for the second stage game. When 33

both bidders reveal their type to be low-cost, the second stage game is a Bertrand game. When 1−θ 1−θ(1−γ)

both bidders fake, we can set α = β =

in the second stage game and determine the equi-

librium. We represent the cdf of the bid distribution in that case to be Fs (q) where the subscript represents the symmetric case. When one bidder fakes but the other does not, the asymmetric game is such that β = 0 and α =

1−θ 1−θ(1−γ) .

The winner, whose belief is α regarding his opponent,

bids according to the cdf Fwa (q), and the loser according to Fla (q), where the subscript a represents the asymmetric case. The corresponding pdfs are fs (q), fwa (q) and fla (q). We next characterize the expected payoffs under two scenarios: (i) when the bid in the first stage reveals the type, and (ii) when the low-cost bidder fakes his type. Consider scenario (i). If p < ch , which reveals the low-cost type, is submitted in the first stage, F CIP-semi,θ (p) is the cdf and f CIP-semi,θ (p) is the pdf of the first stage bid distribution, and q is the bid in the second stage, then the expected profits are: High-cost rival

π

CIP-semi,θ

(p, q) =

z }| {

(1 − θ) f CIP-semi,θ (p)fwa (q)(p − cl + q − cl )1−r Bidder wins stage 1; Both low-costs revealed =⇒ 0 profit in stage 2

z

}|

{

θ(1 − F CIP-semi,θ (p) − γ)

+

f CIP-semi,θ (p)(p − cl )1−r

Bidder wins both stages; low-cost rival hides

z

}|

{

θγ(1 − Fla (q))

+

f CIP-semi,θ (p)fwa (q)(p − cl + q − cl )1−r

Hiding low-cost rival and bidder wins stage 2

z

+

}|

{

f CIP-semi,θ (p)fla (q)(q − cl )1−r

θγFla (q)

By rearranging and simplifying, we obtain the conditional profit of bidding p as: π

CIP-semi,θ

1−r

(p) = (1 − θ)(ch − cl )

Z 1 1−θ 1−θF CIP-semi,θ (p)

(p − cl + q − cl )1−r − (p − cl )1−r (q − cl )1−r

!

fwa (q)dq

+(1 − θF CIP-semi,θ (p))(p − cl )1−r The expected profit from any bid p can be obtained by setting F CIP-semi,θ (p) for a p price infinitesimally smaller than ch to be 1 − γ. Next, consider the profit from submitting a faking bid:

π CIP-semi,θ fake =

High-cost rival but bidder loses stage 1 High-cost rival but bidder wins stage 1  z }| { z }| {   (1 − θ) (1 − θ)   (q − cl )1−r + (ch − cl + q − cl )1−r  fs (q)    2 2

34

(2)

Revealing low-cost rival and bidder wins stage 2

z

}|

θγ(1 − Fla (q))(q − cl )

+

{

1−r

fwa (q)

Hiding low-cost rival and bidder wins both stages

z

}|

{

θγ(1 − Fs (q)) + (ch − cl + q − cl )1−r fs (q) 2 Hiding low-cost rival and bidder wins stage 2

z

+

}|

{

θγ(1 − Fs (q)) (q − cl )1−r 2

fs (q)

Hiding low-cost rival and bidder wins stage 1

z

}|

{

θγFs (q) (ch − cl )1−r 2

+

fs (q)

By simplifying and re-arranging, we have π

CIP-semi,θ fake

=

(1 − θ) 2 +

Z 1 1−θ 1−θ(1−γ)





(ch − cl + q − cl )1−r − 1 + (q − cl )1−r fs (q) dq

(1 − θ + θγ) θ(1 − θ)(1 − γ) + . 2 1 − (1 − γ)(1 − θ)

(3)

We solve for the two variables F CIP-semi,θ (p) and γ by using two equations, each of which is obtained by setting any two of the following expressions equal: expected profit in Equation 2 for any p < ch ; the expected profit computed at the supremum of the strategy set, which is infinitesimally smaller than ch ; and Equation 3. The equilibrium bid distribution does not have a closed form expression for any arbitrary r but is available for r = 0. Again, when r = 0, the equilibrium strategies are similar to those in Kannan (2008). For a given r, the equilibrium bid distribution can be obtained numerically. A.3

First Stage Game: IIP

When determining the first stage equilibrium, we compute the expected profit across both stages. We determine a Bayesian Nash equilibrium first by assuming that a separating equilibrium exists in the first stage and then for deviations from that assumption. Recall that only the winner’s bid price is revealed in the first stage. When considering the separating equilibrium in the first stage, the bid reveals the winner’s type creating an information asymmetry in the second stage. We represent F IIP,θ (p) and f IIP,θ (p) as the cdf and the pdf of the first stage bid distribution; Fw (q) and fl (q) as those for the first stage winner in the second stage; and Fl (q) and fl (q) for the first stage loser in the second stage. Note that the second stage equilibrium bid distributions for the winner and the loser can be obtained by setting β = 0 35

and α =

1−θ , 1−θF IIP,θ (pw )

where pw is the first stage winning bid. Note that the expected profits

are the same across both types given α. However, the expectation of α varies with the price bid by the winner and that distinction has to be taken into account when considering the first stage equilibrium. The expected payoff from bidding p in the first stage and q in the second stage is: High-cost rival

πIIP,θ (p, q) =

z }| {

(1 − θ) f IIP,θ (p)fw (q)(p − cl + q − cl )1−r Low-cost rival but bidder wins both stages

z

+ θ(1 − F

}|

IIP,θ

{

(p))(1 − Fl (q)) f IIP,θ (p)fw (q)(p − cl + q − cl )1−r

Low-cost rival but bidder wins stage 1

+

z

}|

{

θ(1 − F IIP,θ (p))Fl (q)

f IIP,θ (p)fw (q)(p − cl )1−r

Low-cost rival but bidder wins stage 2

+

z

}|

{

θF IIP,θ (p)(1 − Fw (q))

f IIP,θ (p)fl (q)(q − cl )1−r

When computing the expectation, note that the last expression involves beliefs held by the opponent conditional on q < p. From that, we obtain the unconditional profit of bidding p: πIIP,θ (p) = ((1 − θF IIP,θ (p))(p − cl )1−r − (1 − θ) log (1 − θF IIP,θ (p)))f IIP,θ (p)(ch − cl )1−r 1−r

+(1 − θ)(ch − cl )

Z

1−θ 1−θF IIP,θ (p)

pl

(p − cl + q − cl )1−r − (p − cl )1−r (q − cl )1−r

!

fw (q) dq

which we can readily simplify using the pdf fw (q) (remembering to also account for the masspoint). We have not presented the expanded expression for ease of exposition. Note that any price in the strategy space yields the same profit in the mixed strategy equilibrium and that a price infinitesimally smaller than ch is the supremum of the strategy space. Hence, the total profit across the two stages under IIP is πIIP,θ = (1 − θ)(ch − cl )1−r (21−r − log (1 − θ)) and, from that, one can obtain F IIP,θ (p). The bid distribution does not have a closed form but can be numerically computed.

36

Appendix B: Experiment Instructions (Complete Information Policy, θ=0.9) This is an experiment in the economics of strategic decision making. Purdue University has provided funds for this research. If you follow the instructions and make appropriate decisions, you can earn an appreciable amount of money. The currency used in the experiment is francs. Your francs will be converted to U.S. Dollars at a rate of _____ francs to one dollar. At the end of today’s session, you will be paid in private and in cash. It is important that you remain silent and do not look at other people’s work. If you have any questions, or need assistance of any kind, please raise your hand and an experimenter will come to you. If you talk, laugh, exclaim out loud, etc., you will be asked to leave and you will not be paid. We expect and appreciate your cooperation. The experiment consists of 50 decision making periods. Each period you will be grouped with one other person in the experiment. At the beginning of each decision making period you will be randomly re-grouped with another person. Since the groupings change randomly every period, you will be grouped with a new person in almost every period. These instructions are for Part 1, which lasts for 25 periods. You will receive additional instructions for Part 2. Your Offer Prices and Profits During each period, you can sell units of a fictitious commodity. If you sell a unit, then you will have to incur that unit’s production cost. Each period you and all other participants will make two choices—an offer price in stage 1 and an offer price in stage 2. Each represents an offer price to sell a unit of a fictitious good to the experimenter. You can sell one unit in each of the two stages. If you sell your unit, then you will earn profits (in experimental francs) equal to Your profits = Your offer price – Your production cost If you do not sell your unit in a stage, then your profit for that stage is 0. This will happen frequently, since only one of the two people in your group can sell a unit in each stage. For example, suppose your production cost is 200 and your offer price is 322, and you sell a unit in this stage. Then your profit would be 322 – 200 = 122 for this stage. Note that you only incur your production cost if you sell a unit. Costs are Determined Randomly Your costs and the costs of the other person in your group are determined randomly by the computer at the start of each period. Everyone’s costs remain unchanged for both stage 1 and B-1

stage 2 within a period, but then they are randomly determined again at the start of each period. There is a 90% chance that your cost is 200 and a 10% chance that your cost is 400. Which cost you have this period is determined through a (virtual) “ball draw” from a bingo cage containing 10 balls, comprised of 9 red and 1 black balls. If a red ball is drawn then your cost is 200 and if a black ball is drawn then your cost is 400. The other person in your group will have a separate ball draw (with replacement) to determine his or her cost. Everyone will have a new ball draw to determine cost at the start of every period. Everyone always simply has a 90% chance (that is, a 0.9 probability) of having a 200 cost. Remember, everyone’s cost also remains unchanged over the 2 stages of each period.

Submitting Your Offer Prices You will submit your offer prices using your computer. An example screen for Stage 1 is shown above. As you can see on this screen, you will know your cost for the period before you B-2

submit your offer, but you will not know the cost (or the offer price) of anyone else at this stage. Up to two decimal places are permitted for any price offer. Determining Who Makes the Sale The computer determines whether you or the other person in your group makes the sale each stage following a very simple rule: The lowest offer price in your two-person group sells the unit, as long as this lowest offer price is not greater than 400. Since the computerized buyer will not pay more than 400 for a unit, any offer that you submit that is greater than 400 will be automatically lowered to 400. If the two offer prices are equal, then the person who sells the unit is determined randomly. The buyer will buy at most one unit in each stage from each two-person group.

B-3

Stage 2: New Offer Prices You will learn the Stage 1 offer price submitted by the other person in your group at the start of Stage 2, as shown above. This screen will also indicate who sold a unit in Stage 1. At this point you will submit a Stage 2 offer price, at the same time the other person submits her price. Your cost does not change between the two offer stages. Guessing the Cost for the Other Person The other person’s cost also does not change between the offer stages. At the same time that you submit your Stage 2 offer price, you will also enter a guess about the chances that the other person has a cost of 200. (Remember, we already told you that costs are determined randomly at the beginning of the period, and everyone always has a 90-percent chance of having the cost of 200 for the period.) What you enter on your screen is the probability that the other person has a cost of 200 this period. For example, if you think that she has a 50-percent chance of having a cost of 200, then you enter 0.5. Or, if you think that she is three times as likely to have a cost of 200, rather than the cost of 400, then you enter 0.75. (Up to two decimal places are allowed.) Or, if you think that she certainly does not have a cost of 200, then you enter 0. Your guess can earn you additional money. At the end of the period, we will show you the cost of this other person, and compare it to your guess. We will then pay you for the accuracy of your guess as follows: Suppose you guess that the person you are grouped with has a cost of 200 with a 75% chance and a cost of 400 with a 25% chance (as in one example above). Suppose further that this person actually has a cost of 400. In that case your Guess Payoff = 20 – 10(1-0.25)2 – 10(0.75)2 = 8.75 francs. In other words, we will give you a fixed amount of 20 francs from which we will subtract an amount that depends on how inaccurate your guess was. To do this we use the cost of the person you are grouped and we will take the probability you assigned to that cost, in this case 25% on 400, subtract it from 100% and square it. We will then take the probability you assigned to the wrong cost, in this case the 75% you assigned to 200, and square it also. These two squared numbers will then be multiplied by 10 and subtracted from the 20 points that we initially gave you, to determine your final guessing payoff (which is 8.75 francs in this example).

B-4

Note that you get the lowest payment under this payoff procedure when you state that you believe that there is a 100% chance that the other person has a particular cost when it turns out that she actually has the other cost. In this case your guessing payoff would be 0, so you can never lose earnings from inaccurate guesses. You get the highest payment if you guess correctly and assign 100% to the cost that turns out to the actual cost of the person you are grouped with; in this case your guessing payoff would be 20 francs. Note that since your guess is made before you know the cost of the person you are grouped with, you maximize the expected size of your guessing payoff by simply stating your true beliefs about what you think this other person’s cost is. Any other guess will decrease the amount you can expect to earn from your guessing payoff. The End of the Period After everyone has submitted offer prices for both stages of the current period you will be shown the final results screen, as shown on the next page. This screen displays your offer prices as well as the offer price and cost of the person you are grouped with for the current decision making period. It also shows your total earnings for this period and your cumulative earnings for the experiment so far. Once the outcome screen is displayed you should record your offer prices, cost, and the other person’s offer prices and cost on your Personal Record Sheet. Also record your current and cumulative earnings. Then click on the continue button on the lower right of your screen. Remember, at the start of the next period all participants are randomly re-grouped, and you are randomly re-grouped each and every period of the experiment.

B-5

Personal Record Sheet – Part 1 Period

Your Your Stage 1 Cost this Offer Period Price

Other Seller’s Stage 1 Offer Price

Your Stage 2 Offer Price

1 2 3 4 … 24 25 B-6

Other Seller’s Stage 2 Offer Price

Other Seller’s Cost this Period

Your earnings this period

Total earnings in Part 1 so far