Learning to bid: The design of auctions under uncertainty ... - CiteSeerX
Learning to bid: The design of auctions under uncertainty and adaptation Thomas H. Noe Tulane University [email protected] Michael Rebello Tulane University [email protected] Jun Wang Baruch College jun [email protected] Abstract We examine auction design in a context where symmetrically informed agents with common valuations learn to bid for a good. We show that bidder strategies, even in the long run, do not converge to the Bertrand–Nash strategy of bidding the expected value of the good. Although individual agents learn Nash bidding in isolation, the learning of each agent, by flattening the best reply correspondence of other agents, blocks common learning. These negative externalities are more severe in second-price auctions, auctions with many bidders, and auctions where the good has an uncertain value ex post. For this reason, uncertainty, auction structure, and the number of bidders matter, even absent private valuations, asymmetric information, or risk aversion. These results suggest that parametrically very parsimonious auction models, requiring only information regarding the statistical properties of the auctioned good’s payoffs, the number of bidders, and the auction mechanism, can yield a rich set of predictions regarding auction outcomes.
First draft: May 2005; Key words: auction design, adaptive learning, genetic algorithm.
From the beginning of economic science, auctions have been the archetype for allocating scarce resources through the price mechanism. The image of the “Walrasian auctioneer” clearing markets is as old as the concept of equilibrium. Each generation of economists has examined the auction mechanism through the analytical lenses fashionable at its time. Some assumed, without clearly specifying the exact nature of the auction design, a sort of mechanical or adaptive tatonnment process whereby auctions would clear and goods would be allocated efficiently (see, for example, Samuelson (1947)). Others such as Bertrand made a fundamental advance by noting that in the simplest complete information setting, price competition should drive bidder profits to zero, allowing auctioneers to capture the surplus from the auction. Later, pioneering researchers such as Vickery (1961) used the tools of expected utility and probabilistic analysis to develop formal models of auctions under the assumption that bidders participating in the auction maximize utility given subjective probability distributions over the preferences of the other agents and the value of the auctioned good. This body of research was continued by studies such as Milgrom and Weber (1982) that have not only produced a series of elegant and subtle characterizations of agent behavior in auctions, but also provided fundamental insights for many other areas of economic research. Such models of bidder behavior prescribe complex and somewhat non-intuitive bidding strategies. Thus, one expects that bidders in actual auctions will only reach equilibrium outcomes through learning. However, it is not clear that bidders can learn Nash equilibrium bidding strategies through experience. As Harrison (1989) points out, because a bidder’s best response function often has a “flat maximum,” the cost of deviating from equilibrium bidding strategies is small relative to the value of the good being auctioned. This small cost of deviation attenuates the feedback effect that is essential for learning. Kagel (1995) argues that, when the marginal cost of deviating from equilibrium behavior is low, nonmonetary considerations may attain ascendancy in governing the behavior of bidders in experimental auctions. However, even if agents are not diverted by nonmonetary incentives, if feedback is sufficiently weak, there is little likelihood that human agents can ever learn equilibrium bidding strategies through experience. One natural approach to studying the learnability of auction equilibria while abstracting from nonmonetary incentives is to study learning by artificial agents. Gode and Sunder (1993) adopt such an approach and demonstrate that purely adaptive zero-intelligence agents can produce efficient clearing in a double auction market; Andreoni and Miller (1995) employ artificial agents to examine learning in a variety of auction formats, including the common value firstprice auction, and conclude that agents learn Nash equilibrium bidding strategies in common value first-price auctions.1 1 There
is now a growing body of literature that attempts to consider the economic effect of adaptive behavior
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In this paper, we also use artificial agents to examine learning in auctions. However, we take the Harrison “flat maximum” critique as a starting point for an independent investigation. We model symmetric information auctions with adaptive agents in which the good produces the same value regardless of which bidder wins the auction. We consider deterministic and stochastic values and both first-price and second-price auctions. In all these designs we remove private information and thus lift the winner’s curse. Hence, the standard Nash equilibrium solution predicts that all bidders “bid value,” that is, submit a bid equal to (or one tick less than) the expected value of the auctioned good. Our findings contrast strongly with this prediction. In fact, when the auctioned good’s value is stochastic, on average, our artificial agents bid less than half the expected value of the auctioned good in first-price auctions. Thus, even absent any winner’s curse or risk aversion, artificial agent bidders do not usually play Nash strategies. Although both first-price and second-price auctions produce deviations from the Nash solution, the reasons for these deviations depend on the auction design. In first-price auctions with deterministic values, marginal bidders submit bids close to the auctioned good’s value and thus revenue closely approximates the Nash equilibrium predictions. However, nonmarginal bidders “underbid,” that is, submit bids less than the value of the auctioned good. This pattern emerges because, as soon as any bidder bids close to the good’s value, the payoffs received by other bidders become highly nonlinear in their bids. Overbidding, that is, bidding higher than the value of the auctioned good, leads to a sure loss that increases 1-1 with the bid; in contrast, underbidding leads to a zero payoff that is constant as the bid falls. Thus, once the equilibrium bidding strategy of bidding value is reached by any agent, the remaining agents face a highly concave bid response function that features a flat maximum below the good’s value and a steep downward slope above the good’s value. This concavity leads to seemingly risk averse behavior for nonmarginal bidders and thus, when the number of bidders is large, low bids on average. Yet, the marginal or winning bid, and thus revenue of the auctioneer, remains close to the Nash prediction. For second-price auctions, even with nonstochastic values, not only strategies but also auction revenues diverge strongly from the prediction of the Nash dominant strategy solution which requires that all agents bid the good’s value.2 The failure of the dominant strategy through artificial agent simulations. For example, Allen and Karjalainen (1999) investigate the use of artificial agents in identifying profitable trading strategies in financial markets; Arifovic (1996) simulates exchange-rate determination; Arifovic (1994) and Chen and Yeh (1996) study market clearing demand and supply functions; Noe and Pi (2000) investigate how free-riding behavior evolves in a corporate takeover setting; Routledge (2001) studies the features of simulated information acquisition in rational expectations economies; and Noe, Rebello, and Wang (2003) attempt to discover how adaptive learning affects corporate financial policy in a perfectly competitive capital market. 2 While the fact that evolution leads away from a dominant strategy solution should not be surprising given
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solution to survive in our learning framework follows from certain features of a second-price auction. In a second-price auction, as long as one bidder overbids, other bidders’ payoffs are weakly decreasing in their own bids. At the same time, as long as all other bidders underbid, the payoff of the remaining bidder is weakly increasing in her bid. Moreover, in the secondprice auction, the effect of an agent’s bid on her own payoff is less than in a first-price auction as the bid does not affect how much the bidder pays for the auctioned good. These facts about the payoff function in the second-price auction underlie the convergence to asymmetric strategy profiles, where one agent overbids while the remaining agents underbid. This outcome renders agent payoffs very insensitive to agents’ own strategies, thereby dampening learning. Consequently, strategies are very unstable, having 10 or more times the variance of strategies in first-price auctions. This variability leads the second-price auction to clear on occasion at a very high price, especially when there are many bidders and thus many chances for one high outlier bid. However, on average, second-price auction revenue is lower than first-price revenue, with the difference being very large (more than 15%) when the number of bidders is small. The importance of learning externalities for obtaining this result is highlighted by the fact that, when an artificial agent is matched up with a fixed bidding strategy rather than other artificial agents, that agent converges to the dominant strategy Nash solution 100% of the time. This failure of revenue in second-price auctions to reach the predicted level contrasts with first-price auctions under certainty, where revenues but not strategies conform to the Nash prediction. However, even this revenue conformity in first-price auctions fails when the auctioned good’s value becomes uncertain. With uncertainty, overbidding results in expected losses, but on occasion is rewarded, while underbidding may sometimes produce a high payoff. These effects make learning much more difficult in the first-price auction with value uncertainty and greatly increase (more than threefold) the variance of agent strategies. To win the auction, a bidder must only beat the maximum bid of the other bidders. Hence, when the number of bidders is small and bidder strategies are highly volatile, the maximum bid of the other bidders will frequently be substantially below value. When this occurs, underbidding is rewarded and agents learn to underbid. In fact, we see that in the two-bidder case, the average bid is about 20% below the value for the good. As the number of bidders increases, because of the volatility in bidding strategies, the highest bid increases toward the good’s value even as individual bids remain low. However, Nash revenues are again not attained. Once the expected winning bid approaches bid value, the expected payoff to nonmarginal bidders becomes flat at bids below value and stochastic yet with a negative expected slope of one for bids above the expected value the work of Gale, Binmore, and Samuelson (1995) on dominance and evolutionary stable strategies (ESS), our results are very different from those an ESS model such as Gale, Binmore, and Samuelson would produce. For example, in contrast to the ESS solution, which by definition has symmetric players playing symmetric strategies, our adaptive agent simulations lead to highly asymmetric strategies being played by different artificial agents.
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of the auctioned good. Among nonmarginal bidders, this result produces strategies that drift around in the region below value; Bids that reach above value are usually but not always punished by losses. Because of the possibility that bids above auction value will not be punished, there is a small probability that a bidder will learn to bid more than the value of the auctioned good. Consequently, when the number of bidders is very large, say 50, because of the small chance that any one bid is above value, there is a large chance that the highest bid, which wins the auction, is above value. Thus, we find that in such cases, the first-price auction clears on average at about 10% more than the value of the auctioned good. Hence, in first-price auctions under uncertainty, auction revenue is substantially lower than predicted by theory when the number of bidders is small and substantially higher than predicted by theory when the number of bidders is very large. Given that even under certainty, second-price auctions do not conform with the dominant strategy solution, it is not surprising that conformity is also not observed under payoff uncertainty. In fact, for second-price auctions, the introduction of payoff uncertainty has little effect on strategies or revenues: Agents underbid but less than in the case of first-price auctions, and bid variance remains very high. In summary, our analysis shows that even absent private information, private valuations, or risk aversion, learning dynamics combined with the highly nonlinear structure of auction payoffs can produce sharp predictions regarding auction outcomes. For example, • Bidders will on average shade, bidding below the value of the auctioned good in both first- and second-price auctions. • In the absence of payoff uncertainty, auction revenues will be higher and less risky in first-price auctions. • Bidder payoff uncertainty leads to lower and more variable bids in first-price auctions but has little effect on second-price auctions. • With payoff uncertainty, second-price auctions are on average both riskier and more profitable for sellers. These results have a number of implications for experimental, theoretical, and simulation research on auction design. First, note that we show that without private information regarding the common value of the auctioned good, artificial agents underbid. Such underbidding without private information is inconsistent with the Nash solution. However, it is similar to the underbidding behavior documented by Andreoni and Miller (1995) who parameterize auctions using the Kagel and Levin (1986) framework and study bidding by artificial agents who receive private signals regarding the common value of an auctioned good. Andreoni and Miller find 4
that adaptive agents converge to the approximate Kagel and Levin solution of bidding at the lower end point of the support of the conditional distribution of the good’s value and conclude that this result demonstrates that agents learn Nash equilibrium bidding strategies in common value first-price auctions. Our results, however, provide an alternative explanation for their results—the general tendency of artificial agents to underbid in auctions. A second prediction of our model is that the behavior of human subjects bidding in auctions where learning is important and the underlying value distribution for the auctioned good is poorly understood, for example when agents are bidding in an auction for new technology regarding which there is Knightian uncertainty, should exhibit specific cross-sectional differences based on the auction mechanism. For example, in second-price auctions with a large number of bidders most subjects should have great difficulty fixing on a single bidding strategy. This should result in significant uncertainty regarding auction revenue. Our analysis also shows that when uncertainty is low or the auctioneer is risk averse and the number of bidders is not too large, differences in the distributions of revenue from first- and second-price auctions might lead auctioneers to prefer first-price auctions. The first-price design “trains” bidders better and this frequently helps the auctioneer. Hence, our result may explain why most sealed bid auctions are first-price auctions, despite the lower equilibrium revenue from first-price auctions in many situations. Moreover, our results support, albeit from a very different perspective, the conclusions of Klemperer (2003) that designing auctions to maximize participation is more important than maximizing extractions from a fixed set of auction participants. In a very general sense our approach to analyzing auction behavior is consistent with Freidman and Sunder (2004), which argues that non-linearity and learning dynamics can parsimoniously account for many phenomena that economists have typically analyzed through complex models of risk preferences. Having a simple theory allows for sharp implication based directly on measurable auction characteristics such as auction rules, the unconditional distribution of payoffs from the auctioned good, and the number of auction participants. For example, in first-price auctions, we predict that a high degree of uncertainty coupled with low bidder participation leads to auction prices substantially below true value, while with high participation auction prices exceed value. This suggests that, for high volatility assets in thin markets, auction prices will be low and thus the realized returns from holding the auctioned good will be supernormal. When markets are thick, prices will be high and returns will be subnormal. Perhaps this insight partially explains the underperformance of newly issued stocks, which tend to be bid on by many market participants and feature highly uncertain payoffs. As well as having value for predicting human agent behavior, our artificial agent model looks forward to a future where markets are populated by artificial agents. This future may be closer than we think. For example, the DEAL logistics consortium has developed artificial bid5
ding agents that bid for cargo in online auctions.3 Dave Cliff, former head of Hewlett Packard’s Digital Media Systems Laboratory, predicts that in the near future“some or possibly all major financial markets will be implemented as e-marketplaces populated by autonomous softwareagent traders” (Cliff 2003). Thus, as well as being an attempt to explain some familiar, and well-studied features of human auction behavior, this paper also represents an exploratory examination of the behavior of the human agents’ new rival, the artificial agent. The remainder of the paper is organized as follows: The next section describes the structure of our experiment. Section II describes our central results. In Section III, we present results from an experiment employing an alternative specification of our genetic algorithm. The paper concludes with a discussion in Section IV. I
Implementation of the Auctions via a Genetic Algorithm
Consider an agent attempting to sell one unit of a good at an auction. Every bidder participating in the auction knows that the good will generate a payoff, x, ˜ after the auction is complete. In simulations where there is no uncertainty regarding the value of the auctioned good x˜ = 0.5. Otherwise x˜ ∈ [0, 1]. The seller can choose to employ either a first-price or a second-price auction. Regardless of the type of auction selected, all bids are sealed and every bidder can enter only one bid. The good is awarded to the bidder submitting the highest bid, with ties being broken by splitting both the payment for the good and the good’s payoffs among the highest bidders. Let P1 and P2 be the highest and the next highest bid in the auction, respectively. If the seller employs a first-price auction, the bidder placing the bid P1 receives the good and pays the seller P1 . This results in a realized payoff of π1B = x − P1 to the winning bidder and a revenue of π1S = P1 to the seller. All other bidders receive a payoff of 0. If the second-price auction is employed to sell the good, the highest bidder is once again selected as the winner. However, in this case, he pays the seller P2 . Thus the winning bidder receives a realized payoff of π2B = x − P2 and the seller receives revenue π2S = P2 . We simulate the bidder behavior by playing a game between virtual bidders. Agent actions are determined by a genetic algorithm. Each run of the genetic-algorithm simulation consists of a number of rounds. In each round, the virtual agents select strategies and receive payoffs. The auction design is fixed by the seller at the beginning of every run. Conditional on the auction design, bidders submit a bid for the good being auctioned. Each bidder’s strategy is drawn at random from a strategy pool associated with that auction design. For each auction design, each bidder has a strategy pool consisting of 80 chromosomes. Each chromosome is encoded with a strategy. The strategies are binary strings of 0s and 1s that decode to a ten-place, base-two 3 See
http://www.ercim.org/publication/Ercim News/enw56/t hoen.html.
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representation of a number, z, from zero to 1023. For example, the string 0000000010 decodes to the number 2. If the strategy encoded by the drawn chromosome decodes to the number z, z the bid by the bidder is 1023 . Initially, the 80 chromosomes in each pool are randomly encoded, using a uniform distribution over strategies.4 After each round, chromosome pools are updated. This involves selection, crossover, mutation, and election.5 Selection is performed first and involves estimation of the
profitability of each strategy. Selection takes the form of a lottery where chromosomes are selected at random to form a new pool of chromosomes. The probability that a chromosome will be selected into the new pool is proportional to the profit from employing the strategy encoded by the chromosome. Bidders’ profits are transformed by adding 1 to the profit for each strategy. Because bidder profits are bounded from below by −1, the transformed profits, πnaB , are always positive. Specifically, after each round of an auction of design a, the probability that chromosome n is selected into the new pool is proportional to the transformed profitability of chromosome n, πnaB . This probability equals πnaB . ∑m πm aB Selection continues until the new pool consists of 80 chromosomes. A new chromosome pool is produced from the post-selection pool as follows. First, two chromosomes are randomly chosen from the pool, and, with a probability of 0.6, they are
crossed. A crossing of two chromosomes involves splitting each chromosome into two parts at a randomly selected position and then swapping the parts between the pair of chromosomes. For example, suppose that two chosen chromosomes are encoded with 0000010101 and 0000111000, respectively. If the randomly selected position for splitting the chromosomes is seven, two new chromosomes are generated such that the chromosome encoded with 0000010101 is recoded with 0000011000, and the chromosome encoded with 0000111000 is recoded with 0000110101. The two new chromosomes are then subject to mutation with probability 0.003.6 Mutation involves replacing the chromosome with a new chromosome that is randomly encoded using a uniform probability measure over the 1024 permitted strategies. The new chromosomes generated through crossover and mutation, together with the two original chromosomes, are then subjected to election; that is, only the two chromosomes encoding the 4 In
simulations not reported in this paper, we constructed the initial bidder chromosome pools using alternate procedures. The simulations generated results that are qualitatively similar to the ones reported in this paper. 5 The process for selection, mutation, crossover, and election employed here is similar to the process used in Arifovic (1996). 6 Results from simulations not reported in this paper indicate that, qualitatively, our results are not sensitive to material changes in mutation rates employed in the reported simulations.
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most profitable strategies are added to the new chromosome pool. Each time this process of crossover, mutation, and election is repeated, two new chromosomes are added to the new pool. The process continues until the new pool contains 80 chromosomes. This completion of the new chromosome pool marks the end of a round. A run of the genetic algorithm starts with the initial encoding of chromosome pools for the N bidders. A run ends when 1,000 rounds have elapsed.7 Each experiment consists of 1,000 runs. II
Results In this section, we characterize the results of simulations implementing the algorithm de-
scribed above. First, we present results from simulations in which the value of the auctioned good is certain and fixed at 0.5. Next, we present results from simulations where the auctioned good’s value is uniformly distributed over the interval [0, 1]. A
Fixed value auctioned good
In Table 1, we present the outcomes of simulations where the value of the auctioned good was fixed at 0.5 and bidder behavior was permitted to evolve. The first set of columns presents results from runs where the good was auctioned using a first-price auction while the second set of columns presents results from runs where the good was auctioned using a second-price auction. Each row presents the outcomes of auctions with a given number of bidders. Results are reported for auctions with 2, 4, 8, 20, and 50 bidders. In simulations using first-price auctions, average seller revenue was always close to 0.5. Only in auctions with two and four bidders did revenues diverge from 0.5, and only in auctions with two bidders did the standard deviation of auction revenue exceed 1% of the value of the auctioned good. This result contrasts strongly with the results from simulations employing second-price auctions where average revenue was perceptibly lower than the value of the auctioned good. As can be seen in Figure 1, average revenue in second-price auctions increased with the number of bidders because of a shift in the probability mass from the lower tail of the revenue distribution to its upper tail, with the modal revenue remaining close to 0.5. For example, average revenue was 36% lower than the value of the auctioned good when only two bidders participated, and 1% lower when 50 bidders participated. The second-price auction revenues also displayed higher standard deviations, which we term “revenue volatility.” With only two bidders, revenue volatility was 30% of the value of the auctioned good. However, 7 Results
from simulations not reported in this paper indicate that 1,000 rounds provide enough time for the bidders to learn. Our results are not sensitive to material changes in the number of rounds per run when we extend all runs to 10,000 rounds.
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revenue volatility dropped as the number of bidders increased, falling to 4% of the value of the auctioned good when the number of bidders increased to 50. The difference between first-price and second-price auctions was also reflected in agents’ strategy pools. We characterize these strategy pools statistically as follows: For each agent we can compute an average bid and a bid variance using the bids in that agent’s strategy pool weighted by their representation in the pool. We call the average across all agents of their average bids, the “mean bid,” and the average of the standard deviations of the agents’ bids “bid volatility.” We also compute the extent of “bidder heterogeneity” as captured by the standard deviation of the average bids. As we can see from Figure 2, average bids in first-price auctions never exceeded 0.5, the value of the good. Average bids that were less than 0.5 were approximately uniformly distributed between zero and 0.5. In marked contrast, when only two bidders participated, average bids in second-price auctions varied across the entire range of feasible bids, [0, 1]. As the number of bidders increased, the incidence of average bids in second-price auctions above 0.5 fell dramatically. However, even with 50 bidders, there was a non-trivial fraction of bidders whose average bid exceeded 0.5. These patterns are captured by bidder pool statistics presented in Table 1. In first-price auctions, the mean bid was 2% lower than good value when only two bidders participated. The mean bid decreased with the number of bidders and was approximately 19% lower than the value of the auctioned good for auctions with 50 bidders. Mean bids in the second-price auctions displayed a steeper decline, falling from approximately 0.5 when only two bidders participated to approximately 0.25 when 50 bidders participated. In first-price auctions, bid volatility also tended to increase with the number of bidders, from 0.003 in first-price auctions with two bidders to 0.023 when 50 bidders participated. Because of this increase, bid volatility exceeded revenue volatility when four or more bidders participated in first-price auctions. The pattern of bid volatility in second-price auctions was quite different. Bid volatility in second-price auctions was much higher, ranging from 30 times higher in auctions with two bidders to 3 times higher in auctions with 50 bidders. Further, in contrast to the trend in first-price auctions, volatility declined with the number of bidders, dropping from approximately 0.085 when only two bidders participated to 0.067 with 50 bidders. However, the even faster decline in revenue volatility led to higher bid volatility than revenue volatility with eight or more bidders. The statistics in Table 1 also suggest that in both first- and second-price auctions, changes in the number of participants had contrasting effects on bidder heterogeneity. In first-price auctions, bidder heterogeneity increased from approximately 0.02 when only two bidders participated to 0.15 with 50 participating bidders. In contrast, in second-price auctions bidder heterogeneity dropped from approximately 0.22 when only two bidders participated in a second-price 9
auction to 0.13 when 50 bidders participated. To provide further insights into the evolution of bidder strategies, we now present information on bidder strategy pools that had converged to a pure strategy, that is, bidder strategy pools where a single strategy accounted for all bids in the pool. Statistics describing these pools are presented in Table 2. In excess of 90% of bidder pools converged in first-price auctions when only two bidders participated, and the frequency with which bidder pools converged declined as the number of bidders rose. Further, in first-price auctions, the vast majority of bidder pools tended to converge to bids close to the true value of the auction good. In auctions with more than eight bidders, a small fraction of bidder pools tended to converge to bids distant from 0.5, and the frequency of such behavior declined as the number of bidders increased. This contrasts strongly with the evolution of bidder pools in the second-price auctions. Between one and two percent of bidder pools converged, and these pools never converged to a value close to 0.5. Table 3 presents summary statistics for bidder pools that did not converge to a pure strategy. In the first-price auction designs with 8, 20, and 50 bidders, where a large fraction of bidders failed to converge to pure strategies, the average bid in the non-converged strategy pools was approximately 0.23, and the average value of the standard deviation of bids was approximately 0.012. In the corresponding second-price auctions, the behavior of non-converged bidders exhibited more dependence on the auction parameters. The average non-converged bid and the standard deviation of these bids both declined as the number of bidders rose. The auction behavior documented above has a relatively simple explanation based on the nonlinear payoff structure of auctions and learning dynamics. In first-price auctions, overbids lead to sure losses to at least one of the overbidders. Such overbidding strategies are quickly weeded out. Moreover, topping a winning bid that is substantially below the value of the auctioned good generates a large reward. Thus, some of the bidders quickly converge to bidding close to the value of the auctioned good. Once a few bidders are bidding the value of the auctioned good, the payoffs to other bidders become flat below the good’s value and fall steeply with bids above value. Thus, the remaining bidders no longer receive enough feedback to fix on a specific bidding strategy, and so become a group of nonmarginal underbidders whose bids wander aimlessly over the underbidding range. Thus, in first-price auctions, expected revenue is not affected by the auction design (i.e., number of bidders) and equals the value of the auctioned good. However, as the number of bidders increases, so does the likelihood that a given bidder becomes an untrained underbidder, boosting bid volatility and lowering the mean bid. The feedback mechanism in second-price auctions is quite different. The key feature of such auctions from the perspective of our simulation is that a lone overbidder will not be penalized with losses and may lose by lowering his bid. Moreover, such an agent’s bid will cause 10
other agents to realize either losses or zero profits if they overbid, and realize the same zero payoff for all underbids. Thus, a lone overbidder will condition other bidders to underbid, but will not condition them to pick any particular underbidding strategy. In turn, the underbidders will condition a lone overbidder to continue to overbid, but will not condition any particular high bid. Under this pattern of behavior, agents are unlikely to converge to any pure strategy and thus bid volatility will be high. Moreover, as the number of bidders increases, the fraction of bidders who become underbidders increases. Thus, more and more bids are crowded into the underbidding range of 0.00 to 0.50. This reduces both the bid mean and bid volatility. Revenue, which equals the second highest bid, will tend to equal the maximum bid of the underbidders. Thus, as the number of auction participants increases, despite a decline in the average bid, this maximum bid from the underbidders, and thus auction revenue, increases. The pattern of bidding in second-price auctions contrasts with the dominant strategy solution, which calls for all bidders to bid the value of the auctioned good. The problem with the dominant strategy solution is that, once it is played by more than one agent, the payoffs to all other agents are independent of their strategies so long as they do not top the dominant strategy bid. This lack of feedback blocks learning. Thus, learning externalities are key to the failure of the dominant strategy solution to emerge in our artificial agent simulation. To verify this assertion, we simulated a two-bidder auction in which the bids from one of the agents are drawn from a uniform distribution over [0, 1]. In this simulation, the incentives for the other bidder are identical to those produced by the Becker, DeGroot, and Marschak (1964) procedure for incentive compatible elicitation of valuations. In this simulation, the bidders’ strategies converged in every simulation to the dominant strategy solution of bidding value, 0.50. Thus, individually the artificial agents learn to play the dominant strategy solution; collectively, however, they converge to a very different pattern of behavior. Our result in second-price auctions is consistent with experimental behavior in which a core group of subjects overbids, and this demonstrates that such behavior can result even in the absence of perceptual biases to which the literature frequently appeals (see, e.g., Kagel 1995). B
Uncertain value for the auctioned good
In Table 4, we present the outcomes of simulations where the value of the auctioned good was uniformly distributed over the interval [0, 1] and bidder behavior was permitted to evolve. The first set of columns presents results from runs where the good was auctioned using a first-price auction; the second set of columns presents results from runs where the good was auctioned using a second-price auction. Each row presents the outcomes of auctions with a given number of bidders. Results are reported for auctions with 2, 4, 8, 20, and 50 bidders. In the first-price auction simulations the introduction of uncertainty lowers revenue substan11
tially when only a small number of bidders participates and raises revenue when 20 or more bidders participate. Further, revenue volatility increases dramatically, with the increase most marked in auctions with few bidders. Revenue in second-price auctions was not affected as dramatically. However, the introduction of uncertainty raised revenue, especially when many bidders participated. The variance of revenue also increased, especially when there were many bidders. This increase, however, was not as marked as the increase in revenue volatility in first-price auctions. These effects on revenue of introducing uncertainty are also illustrated in Figure 3. The introduction of uncertainty also affected bidding strategies in first-price auctions more dramatically than bidding strategies in second-price auctions. In first-price auctions, as can also be seen in Figure 4, uncertainty resulted in lower average bids which tended to vary between 0.225 and 0.300. The introduction of uncertainty also boosted bid volatility; however, the bid volatility tended to be lower than the volatility of revenue. Bidder heterogeneity also increased with the introduction of uncertainty, with the largest increase occuring for auctions with two bidders. In second-price auctions, the introduction of uncertainty increased bid averages, especially when the number of bidders was from 4 and 20. The volatility of bids was lower than the volatility of auction revenues. Further, the introduction of uncertainty resulted in a marked decline in bid volatility for auctions with a large number of participants. Bidder heterogeneity also tended to increase, most significantly in auctions with 20 or more bidders. Table 5 presents information on the strategy pools of bidders who have converged to a pure strategy, and Table 6 presents this information for the remaining bidders. Once again, the effect of the introduction of uncertainty is very dramatic in the case of first-price auctions. There is no longer any convergence to bids close to 0.5. Further, there is a dramatic decline in the incidence of convergence, with less than 30% of participants converging to any pure strategy in auctions with more than two bidders. In addition, in first-price auctions with more than 20 participants, the average bid of participants who converged to a pure strategy are similar to the average bids of the remaining participants. In the second-price auctions, the rate of convergence remained low with the exception of auctions with more than 20 participants. In these auctions, the average bids of participants who converged to a pure strategy was similar to the mean value of the bids of the remaining participants. From the above data we see that value uncertainty has a profound effect on the outcomes of auctions, especially for first-price auctions. The key driver for this effect is that with uncertainty, overbids do not always result in losses; thus, overbidding is not subject to such intense selection pressure and consequently remains in some agents’ strategy pools. At the same time, simply bidding the expected value of the auctioned good and winning the auction can generate losses; thus, bids below value are sometimes selected into the pool. These two effects greatly 12
increase the volatility of the marginal bid. This increased volatility also blocks learning, further increasing the randomness of bidding strategies. This increased randomness, in turn, raises the gain from placing low bids. Consequently, the distribution of mean bids shifts left and becomes more left skewed. Nevertheless, because of the greater bid volatility, auction revenue still increases because each agent has some probability of overbidding. In the case of second-price auctions, convergence to pure strategies was almost nonexistent even in the absence of uncertainty. Thus, the introduction of uncertainty, which further impedes learning, has less effect on bidder strategies. However, under certainty, overbidding by more than one bidder generates sure losses while under uncertainty it does not. Thus, overbidding by the second highest bidder is frequent and hence auction revenue increases, especially when there are many bidders. III
Conclusion
Auction-like mechanisms have historically been employed to allocate high value assets such as oil leases, antiques, equity in corporations, government contracts, and so on. Recently, with advances in communication technology which have reduced the costs of organizing and participating in auctions, auctions are emerging as an important means of retailing a wider range of goods and services to both businesses and consumers. Consequently, more and more auctioneers face bidders who are both unsophisticated and likely participants in the auctioneer’s future auctions. For both these reasons auctioneers need to consider the influence of the auction mechanism on bidder learning. This paper provides, in a symmetric information setting, guidance to these auctioneers. First we show that the behavior of unsophisticated bidders is unlikely to conform to the predictions of auction theory and thus rational expectations theoretical models may not provide a useful guide to the modern day auctioneer. Further, we identify the learning externalities embedded in two common auction designs—first-price and second-price auctions. Our results demonstrate that second-price auctions generate greater negative learning externalities, resulting in more uncertain auction revenue. Unless the number of bidders is large and uncertainty about the auctioned good’s value is high, this increased randomness lowers auction revenue. These results may explain why goods are almost never sold using sealed-bid second-price auctions and why, in cases where the good’s value is highly uncertain, winning bidders overbid even when private information regarding the common value of the good is not salient.8 Given that our effort is one of the first to examine the potential of agent learning to explain auction behavior, it has featured very simple specifications of auction design, the pref8 For
example, many winning bidders in the 1994 U.S. Federal Communication Commission spectrum auction were bankrupted by their aggressive bids despite the fact that they had similar valuations for bandwidth and believed that common value information was not important in determining their bidding strategies (see, e.g., Cramton 1995).
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erences of bidders, and the learning mechanism. However, even this very simple framework has yielded some fairly powerful insights. We believe that future research projects within this research program, by enriching our specification with more complex designs and agent preference structures, could yield even greater insights into the tradeoffs that underlie real-world auction design. For example, while theory identifes an isomorphism between second-price sealed bid auctions and open outcry English auctions, the sequential nature of English auctions would offer a very different training environment for agents. Thus, we would not be surprised if these two designs yielded very different outcomes. Further, it would appear that multistage auctions, by weeding out poorly trained bidders who submit low bids, may improve the learning of the remaining bidders. Finally, in the learning framework, heterogenous valuation for the auctioned good would have an additional effect not present in a rational expectations setting. High valuation bidders would submit marginal bids much more often thus such bidders would become better trained. Low valuation bidders, receiving little feedback from their mostly intramarginal bids, would be poorly trained and thus have high variance bidding strategies. This competition from “noise bidders” could force up the bids of high valuation bidders, perhaps explaining, without relying on the hypotheses of risk aversion, the observed overbidding in private value auctions.
14
REFERENCES Andreoni, J., and J.H. Miller, 1995, Auctions with Artificial Adaptive Agents. Games and
Economic Behavior 10, 39-64. Allen, F., and R. Karjalainen, 1999, Using Genetic Algorithms to Find Technical Trading Rules. Journal of Financial Economics 51, 245–271. Arifovic, J., 1994, Genetic Algorithm Learning and the Cobweb Model. Journal of Economic Dynamics and Control 18, 3–28. Arifovic, J., 1996, The Behavior of the Exchange Rate in the Genetic Algorithm and Experimental Economies. Journal of Political Economy 104, 510–541. Becker G., M. DeGroot, and J. Marschak, 1964, Measuring Utility by a Single-Response Sequential Method. Behavioral Science 9, 226-232. Carleton, W., D. Guilkey, R. Harris, and J. Stewart, 1983, An Empirical Analysis of the Role of the Medium of Exchange in Mergers. Journal of Finance 38, 813–826. Chen, S., and C. Yeh, 1996, On the Coordination and Adaptability of the Large Economy: An Application of Genetic Programming to the Cobweb Model. In P. Angeline and K. Kinnear Jr. (Eds), Advances in Genetic Programming II, Cambridge, MA: MIT Press. Cliff, D., 2003, Explorations in Evolutionary Design of Online Auction Market Mechanisms. Journal of Electronic Commerce Research and Applications 2, 162-175. Cox, J., V. Smith, and M. Walker, 1998, Theory and Individual Behavior in First-price Auctions. Journal of Risk and Uncertainty 1,61–99. Cramton, P., 1995, Money Out of Thin Air: The Nationwide Narrowband PCS Auction. Journal of Economics and Management Strategy 4(2), 267-343. Forsyth, R., R. Isaac, and T. Palfrey, 1989, Theories and Tests of Blind Bidding in Sealed-bid Auctions. RAND Journal of Economics 20, 214-238. Friedman, D., and S. Sunder, 2004, Risk Curves: From Unobservable Utility to Risk Opportunity Sets. Yale working paper. Gale, J., K. Binmore, and L. Samuelson, 1995, Learning to be Imperfect: The Ultimatum Game. Games and Economic Behavior 8, 56-90. Gode, D., and S. Sunder, 1993, Allocative Efficiency of Markets with Zero-intelligence Traders: Market as a Partial Substitute for Individual Rationality. Journal of Political Economy 101, 119-137. Ghosh, A., and W. Ruland, 1998, Managerial Ownership, the Method of Payment for Acquisitions, and Executive Job Retention. Journal of Finance 53, 785–798. Hansen, R., 1985, Empirical Testing of Auction Theory. American Economic Review 75, 156– 159. Harrison, G, 1989, Theory and Misbehavior of First-Price Auctions. American Economic Re15
view 79, 749-762. Holt, C., and R. Sherman, 1994, The loser’s curse and bidder bias. American Economic Review 84, 642–652. Kagel, J.H., 1995, Auctions: A survey of experimental research. In J.H. Kagel and A.E. Roth (Eds.), The Handbook of Experimental Economics, Princeton University Press, New Jersey. Kagel, J.H., and D. Levin, 1986, The winner’s curse and public infomration in common value auctions. American Economic Review 76, 894-920. Klemperer, P., 2003, Using and Abusing Economic Theory. Journal of the European Economic Association 1, 272–300. Milgrom, P., and R. Weber, 1982, A theory of Auctions and Competitive Bidding. Econometrica 50, 1089-1122. Noe, T., and L. Pi, 2000, Genetic Algorithms, Learning, and the Dynamics of Corporate Takeovers, Journal of Economic Dynamics and Control, 24, 189-217. Noe, T., M. Rebello, and J. Wang, 2003, Corporate Financing: An Artificial Agent-Based Analysis. Journal of Finance 63, 943–973. Routledge, R., 2001, Genetic Algorithm Learning to Choose and Use Information. Macroeconomic Dynamics, 303–325. Samuelson, P., 1947, Foundations of economic analysis, Cambridge, MA: Harvard University Press. Vickery, W., 1961, Counterspeculation, Auctions, and Competitive Sealed Tenders. Journal of Finance 16, 8–37.
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Table 1: Outcomes of Simulations When the Value of the Good is Fixed at 0.5 This table presents outcomes when the seller fixes on either first-price auction or second-price auction with the number of bidders (N). It presents the mean (RAVE) and standard deviation of the seller’s revenue (RSTD) in the last round of 1000 runs. The mean values of bid averages (BAVE), the average value of standard deviations of bidders’ chromosome pools (BSTD), and the standard deviation of bid averages across all bidders (STAVE) in the last round across all 1,000 runs are reported in the table as well. The results were generated by simulations employing crossover and mutation rates for bidders of 0.6 and 0.003, respectively. Bidders employ lottery-based selection. Each run lasts 1000 rounds.
N 2 4 8 20 50
RAVE 0.495 0.499 0.500 0.500 0.500
First-Price Auctions Second-Price Auctions RSTD BAVE BSTD STAVE RAVE RSTD BAVE BSTD 0.011 0.492 0.003 0.023 0.318 0.149 0.499 0.085 0.001 0.467 0.008 0.095 0.396 0.102 0.358 0.080 0.000 0.443 0.013 0.122 0.445 0.065 0.293 0.074 0.000 0.420 0.019 0.138 0.478 0.034 0.259 0.070 0.000 0.404 0.023 0.146 0.494 0.021 0.251 0.067
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STAVE 0.219 0.205 0.169 0.140 0.128
Table 2: Distribution of Converged Bidder Pools When the Value of the Good is Fixed at 0.5 This table presents the distribution of converged bidder pools when the seller fixes on either first-price auction or second-price auction with varying number of bidders (N). It presents the percentages of bidder pools that have converged to the highest or the second highest tick below 0.5 (CNV12), and any other tick (CNVE). The mean values of bid averages (BAVE) and the standard deviations of bid averages (STAVE) in the last round across all bidder pools that have converged to a tick other than the highest and the second highest below 0.5 are reported in the table as well. The results were generated by simulations employing crossover and mutation rates for bidders of 0.6 and 0.003, respectively. Bidders employ lottery-based selection. Each run lasts 1000 rounds. First-Price Auctions Second-Price Auctions N CNV12 CNVE BAVE STAVE CNV12 CNVE BAVE STAVE 2 45.10 47.30 0.493 0.012 0.05 1.15 0.406 0.200 4 85.63 1.93 0.454 0.111 0.00 0.98 0.335 0.200 8 79.03 0.66 0.228 0.152 0.03 1.25 0.283 0.165 20 70.25 0.64 0.233 0.143 0.00 1.50 0.269 0.167 50 63.83 0.82 0.242 0.140 0.01 1.76 0.253 0.149
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Table 3: Distribution of Non-Converged Bidder Pools When the Value of the Good is Fixed at 0.5 This table presents the distribution of non-converged bidder pools when the seller fixes on either first-price auction or second-price auction with varying number of bidders (N). It presents the percentages of bidder pools that are not converged (NOCNV). The mean values of bid averages (BAVE), the average value of standard deviations of bidders’ chromosome pools (BSTD), and the standard deviation of bid averages (STAVE) in the last round across all non-converged bidder pools are reported in the table as well. The results were generated by simulations employing crossover and mutation rates for bidders of 0.6 and 0.003, respectively. Bidders employ lottery-based selection. Each run lasts 1000 rounds. First-Price Auctions Second-Price Auctions N NOCNV BAVE BSTD STAVE NOCNV BAVE BSTD STAVE 2 7.60 0.450 0.064 0.044 98.80 0.500 0.219 0.086 4 12.45 0.250 0.127 0.063 99.03 0.358 0.205 0.080 8 20.31 0.231 0.120 0.064 98.73 0.293 0.169 0.075 20 29.12 0.233 0.119 0.066 98.50 0.258 0.139 0.071 50 35.36 0.236 0.120 0.065 98.24 0.251 0.128 0.068
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Table 4: Outcomes of Simulations When the Value of the Good is Uniformly Distributed Over the Interval [0, 1]. This table presents outcomes when the seller fixes on either first-price auction or second-price auction with varying number of bidders (N). It presents the mean (RAVE) and standard deviation of the seller’s revenue (RSTD) in the last round of 1000 runs. The mean values of bid averages (BAVE), the average value of standard deviations of bidders’ chromosome pools (BSTD), and the standard deviation of bid averages across all bidders (STAVE) in the last round across all 1,000 runs are reported in the table as well. The results were generated by simulations employing crossover and mutation rates for bidders of 0.6 and 0.003, respectively. Bidders employ lottery-based selection. Each run lasts 1000 rounds.
N 2 4 8 20 50
RAVE 0.300 0.409 0.455 0.502 0.546
First-Price Auctions Second-Price Auctions RSTD BAVE BSTD STAVE RAVE RSTD BAVE BSTD 0.077 0.279 0.019 0.074 0.318 0.160 0.499 0.095 0.066 0.305 0.043 0.118 0.433 0.129 0.374 0.091 0.053 0.279 0.056 0.137 0.500 0.102 0.314 0.093 0.050 0.247 0.031 0.149 0.552 0.073 0.268 0.048 0.052 0.233 0.035 0.149 0.596 0.067 0.255 0.050
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STAVE 0.218 0.201 0.176 0.173 0.164
Table 5: Distribution of Converged Bidder Pools When the Value of the Good is Uniformly Distributed Over the Interval [0, 1]. This table presents the distribution of converged bidder pools When the seller fixes on either first-price auction or second-price auction with varying number of bidders (N). It presents the percentages of bidder pools that have converged to the highest or the second highest tick below 0.5 (CNV12), and any other tick (CNVE). The mean values of bid averages (BAVE) and the standard deviations of bid averages (STAVE) in the last round across all bidder pools that have converged to a tick other than the highest and the second highest below 0.5 are reported in the table as well. The results were generated by simulations employing crossover and mutation rates for bidders of 0.6 and 0.003, respectively. Bidders employ lottery-based selection. Each run lasts 1000 rounds. First-Price Auctions Second-Price Auctions N CNV12 CNVE BAVE STAVE CNV12 CNVE BAVE STAVE 2 0.00 44.90 0.287 0.069 0.00 0.55 0.443 0.186 4 0.00 20.58 0.385 0.069 0.00 1.03 0.384 0.199 8 0.00 10.18 0.417 0.093 0.00 0.83 0.360 0.214 20 0.00 28.80 0.273 0.169 0.00 17.85 0.272 0.188 50 0.00 24.77 0.245 0.169 0.00 17.09 0.258 0.182
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Table 6: Distribution of Non-Converged Bidder Pools When the Value of the Good is Uniformly Distributed Over the Interval [0, 1]. This table presents the distribution of non-converged bidder pools when the seller fixes on either first-price auction or second-price auction with varying number of bidders (N). It presents the percentages of bidder pools that are not converged (NOCNV). The mean values of bid averages (BAVE), the average value of standard deviations of bidders’ chromosome pools (BSTD), and the standard deviation of bid averages (STAVE) in the last round across all non-converged bidder pools are reported in the table as well. The results were generated by simulations employing crossover and mutation rates for bidders of 0.6 and 0.003, respectively. Bidders employ lottery-based selection. Each run lasts 1000 rounds. First-Price Auctions Second-Price Auctions N NOCNV BAVE BSTD STAVE NOCNV BAVE BSTD STAVE 2 55.10 0.272 0.077 0.035 99.45 0.500 0.218 0.095 4 79.43 0.284 0.119 0.054 98.98 0.374 0.201 0.092 8 89.81 0.264 0.132 0.062 99.18 0.313 0.176 0.094 20 71.16 0.237 0.139 0.043 82.11 0.267 0.170 0.058 50 75.17 0.229 0.141 0.047 82.87 0.254 0.160 0.060
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(a) 2 bidders
(b) 4 bidders
(c) 8 bidders
(d) 20 bidders
(e) 50 bidders
Figure 1: Distribution of auction revenues when the value of the good is fixed at 0.5. Blank bars are for first-price auctions and dashed bars are for second-price auctions.
(a) 2 bidders
(b) 4 bidders
(c) 8 bidders
(d) 20 bidders
(e) 50 bidders
Figure 2: Distribution of bid pool averages when the value of the good is fixed at 0.5. Blank bars are for first-price auctions and dashed bars are for second-price auctions.
(a) 2 bidders
(b) 4 bidders
(c) 8 bidders
(d) 20 bidders
(e) 50 bidders
Figure 3: Distribution of auction revenues when the value of the good is random between 0 and 1. Blank bars are for first-price auctions and dashed bars are for second-price auctions.
(a) 2 bidders
(b) 4 bidders
(c) 8 bidders
(d) 20 bidders
(e) 50 bidders
Figure 4: Distribution of bid pool averages when the value of the good is random between 0 and 1. Blank bars are for first-price auctions and dashed bars are for second-price auctions.
First draft: May 2005; Key words: auction design, adaptive learning, genetic algorithm.
From the beginning of economic science, auctions have been the archetype for allocating scarce resources through the price mechanism. The image of the “Walrasian auctioneer” clearing markets is as old as the concept of equilibrium. Each generation of economists has examined the auction mechanism through the analytical lenses fashionable at its time. Some assumed, without clearly specifying the exact nature of the auction design, a sort of mechanical or adaptive tatonnment process whereby auctions would clear and goods would be allocated efficiently (see, for example, Samuelson (1947)). Others such as Bertrand made a fundamental advance by noting that in the simplest complete information setting, price competition should drive bidder profits to zero, allowing auctioneers to capture the surplus from the auction. Later, pioneering researchers such as Vickery (1961) used the tools of expected utility and probabilistic analysis to develop formal models of auctions under the assumption that bidders participating in the auction maximize utility given subjective probability distributions over the preferences of the other agents and the value of the auctioned good. This body of research was continued by studies such as Milgrom and Weber (1982) that have not only produced a series of elegant and subtle characterizations of agent behavior in auctions, but also provided fundamental insights for many other areas of economic research. Such models of bidder behavior prescribe complex and somewhat non-intuitive bidding strategies. Thus, one expects that bidders in actual auctions will only reach equilibrium outcomes through learning. However, it is not clear that bidders can learn Nash equilibrium bidding strategies through experience. As Harrison (1989) points out, because a bidder’s best response function often has a “flat maximum,” the cost of deviating from equilibrium bidding strategies is small relative to the value of the good being auctioned. This small cost of deviation attenuates the feedback effect that is essential for learning. Kagel (1995) argues that, when the marginal cost of deviating from equilibrium behavior is low, nonmonetary considerations may attain ascendancy in governing the behavior of bidders in experimental auctions. However, even if agents are not diverted by nonmonetary incentives, if feedback is sufficiently weak, there is little likelihood that human agents can ever learn equilibrium bidding strategies through experience. One natural approach to studying the learnability of auction equilibria while abstracting from nonmonetary incentives is to study learning by artificial agents. Gode and Sunder (1993) adopt such an approach and demonstrate that purely adaptive zero-intelligence agents can produce efficient clearing in a double auction market; Andreoni and Miller (1995) employ artificial agents to examine learning in a variety of auction formats, including the common value firstprice auction, and conclude that agents learn Nash equilibrium bidding strategies in common value first-price auctions.1 1 There
is now a growing body of literature that attempts to consider the economic effect of adaptive behavior
1
In this paper, we also use artificial agents to examine learning in auctions. However, we take the Harrison “flat maximum” critique as a starting point for an independent investigation. We model symmetric information auctions with adaptive agents in which the good produces the same value regardless of which bidder wins the auction. We consider deterministic and stochastic values and both first-price and second-price auctions. In all these designs we remove private information and thus lift the winner’s curse. Hence, the standard Nash equilibrium solution predicts that all bidders “bid value,” that is, submit a bid equal to (or one tick less than) the expected value of the auctioned good. Our findings contrast strongly with this prediction. In fact, when the auctioned good’s value is stochastic, on average, our artificial agents bid less than half the expected value of the auctioned good in first-price auctions. Thus, even absent any winner’s curse or risk aversion, artificial agent bidders do not usually play Nash strategies. Although both first-price and second-price auctions produce deviations from the Nash solution, the reasons for these deviations depend on the auction design. In first-price auctions with deterministic values, marginal bidders submit bids close to the auctioned good’s value and thus revenue closely approximates the Nash equilibrium predictions. However, nonmarginal bidders “underbid,” that is, submit bids less than the value of the auctioned good. This pattern emerges because, as soon as any bidder bids close to the good’s value, the payoffs received by other bidders become highly nonlinear in their bids. Overbidding, that is, bidding higher than the value of the auctioned good, leads to a sure loss that increases 1-1 with the bid; in contrast, underbidding leads to a zero payoff that is constant as the bid falls. Thus, once the equilibrium bidding strategy of bidding value is reached by any agent, the remaining agents face a highly concave bid response function that features a flat maximum below the good’s value and a steep downward slope above the good’s value. This concavity leads to seemingly risk averse behavior for nonmarginal bidders and thus, when the number of bidders is large, low bids on average. Yet, the marginal or winning bid, and thus revenue of the auctioneer, remains close to the Nash prediction. For second-price auctions, even with nonstochastic values, not only strategies but also auction revenues diverge strongly from the prediction of the Nash dominant strategy solution which requires that all agents bid the good’s value.2 The failure of the dominant strategy through artificial agent simulations. For example, Allen and Karjalainen (1999) investigate the use of artificial agents in identifying profitable trading strategies in financial markets; Arifovic (1996) simulates exchange-rate determination; Arifovic (1994) and Chen and Yeh (1996) study market clearing demand and supply functions; Noe and Pi (2000) investigate how free-riding behavior evolves in a corporate takeover setting; Routledge (2001) studies the features of simulated information acquisition in rational expectations economies; and Noe, Rebello, and Wang (2003) attempt to discover how adaptive learning affects corporate financial policy in a perfectly competitive capital market. 2 While the fact that evolution leads away from a dominant strategy solution should not be surprising given
2
solution to survive in our learning framework follows from certain features of a second-price auction. In a second-price auction, as long as one bidder overbids, other bidders’ payoffs are weakly decreasing in their own bids. At the same time, as long as all other bidders underbid, the payoff of the remaining bidder is weakly increasing in her bid. Moreover, in the secondprice auction, the effect of an agent’s bid on her own payoff is less than in a first-price auction as the bid does not affect how much the bidder pays for the auctioned good. These facts about the payoff function in the second-price auction underlie the convergence to asymmetric strategy profiles, where one agent overbids while the remaining agents underbid. This outcome renders agent payoffs very insensitive to agents’ own strategies, thereby dampening learning. Consequently, strategies are very unstable, having 10 or more times the variance of strategies in first-price auctions. This variability leads the second-price auction to clear on occasion at a very high price, especially when there are many bidders and thus many chances for one high outlier bid. However, on average, second-price auction revenue is lower than first-price revenue, with the difference being very large (more than 15%) when the number of bidders is small. The importance of learning externalities for obtaining this result is highlighted by the fact that, when an artificial agent is matched up with a fixed bidding strategy rather than other artificial agents, that agent converges to the dominant strategy Nash solution 100% of the time. This failure of revenue in second-price auctions to reach the predicted level contrasts with first-price auctions under certainty, where revenues but not strategies conform to the Nash prediction. However, even this revenue conformity in first-price auctions fails when the auctioned good’s value becomes uncertain. With uncertainty, overbidding results in expected losses, but on occasion is rewarded, while underbidding may sometimes produce a high payoff. These effects make learning much more difficult in the first-price auction with value uncertainty and greatly increase (more than threefold) the variance of agent strategies. To win the auction, a bidder must only beat the maximum bid of the other bidders. Hence, when the number of bidders is small and bidder strategies are highly volatile, the maximum bid of the other bidders will frequently be substantially below value. When this occurs, underbidding is rewarded and agents learn to underbid. In fact, we see that in the two-bidder case, the average bid is about 20% below the value for the good. As the number of bidders increases, because of the volatility in bidding strategies, the highest bid increases toward the good’s value even as individual bids remain low. However, Nash revenues are again not attained. Once the expected winning bid approaches bid value, the expected payoff to nonmarginal bidders becomes flat at bids below value and stochastic yet with a negative expected slope of one for bids above the expected value the work of Gale, Binmore, and Samuelson (1995) on dominance and evolutionary stable strategies (ESS), our results are very different from those an ESS model such as Gale, Binmore, and Samuelson would produce. For example, in contrast to the ESS solution, which by definition has symmetric players playing symmetric strategies, our adaptive agent simulations lead to highly asymmetric strategies being played by different artificial agents.
3
of the auctioned good. Among nonmarginal bidders, this result produces strategies that drift around in the region below value; Bids that reach above value are usually but not always punished by losses. Because of the possibility that bids above auction value will not be punished, there is a small probability that a bidder will learn to bid more than the value of the auctioned good. Consequently, when the number of bidders is very large, say 50, because of the small chance that any one bid is above value, there is a large chance that the highest bid, which wins the auction, is above value. Thus, we find that in such cases, the first-price auction clears on average at about 10% more than the value of the auctioned good. Hence, in first-price auctions under uncertainty, auction revenue is substantially lower than predicted by theory when the number of bidders is small and substantially higher than predicted by theory when the number of bidders is very large. Given that even under certainty, second-price auctions do not conform with the dominant strategy solution, it is not surprising that conformity is also not observed under payoff uncertainty. In fact, for second-price auctions, the introduction of payoff uncertainty has little effect on strategies or revenues: Agents underbid but less than in the case of first-price auctions, and bid variance remains very high. In summary, our analysis shows that even absent private information, private valuations, or risk aversion, learning dynamics combined with the highly nonlinear structure of auction payoffs can produce sharp predictions regarding auction outcomes. For example, • Bidders will on average shade, bidding below the value of the auctioned good in both first- and second-price auctions. • In the absence of payoff uncertainty, auction revenues will be higher and less risky in first-price auctions. • Bidder payoff uncertainty leads to lower and more variable bids in first-price auctions but has little effect on second-price auctions. • With payoff uncertainty, second-price auctions are on average both riskier and more profitable for sellers. These results have a number of implications for experimental, theoretical, and simulation research on auction design. First, note that we show that without private information regarding the common value of the auctioned good, artificial agents underbid. Such underbidding without private information is inconsistent with the Nash solution. However, it is similar to the underbidding behavior documented by Andreoni and Miller (1995) who parameterize auctions using the Kagel and Levin (1986) framework and study bidding by artificial agents who receive private signals regarding the common value of an auctioned good. Andreoni and Miller find 4
that adaptive agents converge to the approximate Kagel and Levin solution of bidding at the lower end point of the support of the conditional distribution of the good’s value and conclude that this result demonstrates that agents learn Nash equilibrium bidding strategies in common value first-price auctions. Our results, however, provide an alternative explanation for their results—the general tendency of artificial agents to underbid in auctions. A second prediction of our model is that the behavior of human subjects bidding in auctions where learning is important and the underlying value distribution for the auctioned good is poorly understood, for example when agents are bidding in an auction for new technology regarding which there is Knightian uncertainty, should exhibit specific cross-sectional differences based on the auction mechanism. For example, in second-price auctions with a large number of bidders most subjects should have great difficulty fixing on a single bidding strategy. This should result in significant uncertainty regarding auction revenue. Our analysis also shows that when uncertainty is low or the auctioneer is risk averse and the number of bidders is not too large, differences in the distributions of revenue from first- and second-price auctions might lead auctioneers to prefer first-price auctions. The first-price design “trains” bidders better and this frequently helps the auctioneer. Hence, our result may explain why most sealed bid auctions are first-price auctions, despite the lower equilibrium revenue from first-price auctions in many situations. Moreover, our results support, albeit from a very different perspective, the conclusions of Klemperer (2003) that designing auctions to maximize participation is more important than maximizing extractions from a fixed set of auction participants. In a very general sense our approach to analyzing auction behavior is consistent with Freidman and Sunder (2004), which argues that non-linearity and learning dynamics can parsimoniously account for many phenomena that economists have typically analyzed through complex models of risk preferences. Having a simple theory allows for sharp implication based directly on measurable auction characteristics such as auction rules, the unconditional distribution of payoffs from the auctioned good, and the number of auction participants. For example, in first-price auctions, we predict that a high degree of uncertainty coupled with low bidder participation leads to auction prices substantially below true value, while with high participation auction prices exceed value. This suggests that, for high volatility assets in thin markets, auction prices will be low and thus the realized returns from holding the auctioned good will be supernormal. When markets are thick, prices will be high and returns will be subnormal. Perhaps this insight partially explains the underperformance of newly issued stocks, which tend to be bid on by many market participants and feature highly uncertain payoffs. As well as having value for predicting human agent behavior, our artificial agent model looks forward to a future where markets are populated by artificial agents. This future may be closer than we think. For example, the DEAL logistics consortium has developed artificial bid5
ding agents that bid for cargo in online auctions.3 Dave Cliff, former head of Hewlett Packard’s Digital Media Systems Laboratory, predicts that in the near future“some or possibly all major financial markets will be implemented as e-marketplaces populated by autonomous softwareagent traders” (Cliff 2003). Thus, as well as being an attempt to explain some familiar, and well-studied features of human auction behavior, this paper also represents an exploratory examination of the behavior of the human agents’ new rival, the artificial agent. The remainder of the paper is organized as follows: The next section describes the structure of our experiment. Section II describes our central results. In Section III, we present results from an experiment employing an alternative specification of our genetic algorithm. The paper concludes with a discussion in Section IV. I
Implementation of the Auctions via a Genetic Algorithm
Consider an agent attempting to sell one unit of a good at an auction. Every bidder participating in the auction knows that the good will generate a payoff, x, ˜ after the auction is complete. In simulations where there is no uncertainty regarding the value of the auctioned good x˜ = 0.5. Otherwise x˜ ∈ [0, 1]. The seller can choose to employ either a first-price or a second-price auction. Regardless of the type of auction selected, all bids are sealed and every bidder can enter only one bid. The good is awarded to the bidder submitting the highest bid, with ties being broken by splitting both the payment for the good and the good’s payoffs among the highest bidders. Let P1 and P2 be the highest and the next highest bid in the auction, respectively. If the seller employs a first-price auction, the bidder placing the bid P1 receives the good and pays the seller P1 . This results in a realized payoff of π1B = x − P1 to the winning bidder and a revenue of π1S = P1 to the seller. All other bidders receive a payoff of 0. If the second-price auction is employed to sell the good, the highest bidder is once again selected as the winner. However, in this case, he pays the seller P2 . Thus the winning bidder receives a realized payoff of π2B = x − P2 and the seller receives revenue π2S = P2 . We simulate the bidder behavior by playing a game between virtual bidders. Agent actions are determined by a genetic algorithm. Each run of the genetic-algorithm simulation consists of a number of rounds. In each round, the virtual agents select strategies and receive payoffs. The auction design is fixed by the seller at the beginning of every run. Conditional on the auction design, bidders submit a bid for the good being auctioned. Each bidder’s strategy is drawn at random from a strategy pool associated with that auction design. For each auction design, each bidder has a strategy pool consisting of 80 chromosomes. Each chromosome is encoded with a strategy. The strategies are binary strings of 0s and 1s that decode to a ten-place, base-two 3 See
http://www.ercim.org/publication/Ercim News/enw56/t hoen.html.
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representation of a number, z, from zero to 1023. For example, the string 0000000010 decodes to the number 2. If the strategy encoded by the drawn chromosome decodes to the number z, z the bid by the bidder is 1023 . Initially, the 80 chromosomes in each pool are randomly encoded, using a uniform distribution over strategies.4 After each round, chromosome pools are updated. This involves selection, crossover, mutation, and election.5 Selection is performed first and involves estimation of the
profitability of each strategy. Selection takes the form of a lottery where chromosomes are selected at random to form a new pool of chromosomes. The probability that a chromosome will be selected into the new pool is proportional to the profit from employing the strategy encoded by the chromosome. Bidders’ profits are transformed by adding 1 to the profit for each strategy. Because bidder profits are bounded from below by −1, the transformed profits, πnaB , are always positive. Specifically, after each round of an auction of design a, the probability that chromosome n is selected into the new pool is proportional to the transformed profitability of chromosome n, πnaB . This probability equals πnaB . ∑m πm aB Selection continues until the new pool consists of 80 chromosomes. A new chromosome pool is produced from the post-selection pool as follows. First, two chromosomes are randomly chosen from the pool, and, with a probability of 0.6, they are
crossed. A crossing of two chromosomes involves splitting each chromosome into two parts at a randomly selected position and then swapping the parts between the pair of chromosomes. For example, suppose that two chosen chromosomes are encoded with 0000010101 and 0000111000, respectively. If the randomly selected position for splitting the chromosomes is seven, two new chromosomes are generated such that the chromosome encoded with 0000010101 is recoded with 0000011000, and the chromosome encoded with 0000111000 is recoded with 0000110101. The two new chromosomes are then subject to mutation with probability 0.003.6 Mutation involves replacing the chromosome with a new chromosome that is randomly encoded using a uniform probability measure over the 1024 permitted strategies. The new chromosomes generated through crossover and mutation, together with the two original chromosomes, are then subjected to election; that is, only the two chromosomes encoding the 4 In
simulations not reported in this paper, we constructed the initial bidder chromosome pools using alternate procedures. The simulations generated results that are qualitatively similar to the ones reported in this paper. 5 The process for selection, mutation, crossover, and election employed here is similar to the process used in Arifovic (1996). 6 Results from simulations not reported in this paper indicate that, qualitatively, our results are not sensitive to material changes in mutation rates employed in the reported simulations.
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most profitable strategies are added to the new chromosome pool. Each time this process of crossover, mutation, and election is repeated, two new chromosomes are added to the new pool. The process continues until the new pool contains 80 chromosomes. This completion of the new chromosome pool marks the end of a round. A run of the genetic algorithm starts with the initial encoding of chromosome pools for the N bidders. A run ends when 1,000 rounds have elapsed.7 Each experiment consists of 1,000 runs. II
Results In this section, we characterize the results of simulations implementing the algorithm de-
scribed above. First, we present results from simulations in which the value of the auctioned good is certain and fixed at 0.5. Next, we present results from simulations where the auctioned good’s value is uniformly distributed over the interval [0, 1]. A
Fixed value auctioned good
In Table 1, we present the outcomes of simulations where the value of the auctioned good was fixed at 0.5 and bidder behavior was permitted to evolve. The first set of columns presents results from runs where the good was auctioned using a first-price auction while the second set of columns presents results from runs where the good was auctioned using a second-price auction. Each row presents the outcomes of auctions with a given number of bidders. Results are reported for auctions with 2, 4, 8, 20, and 50 bidders. In simulations using first-price auctions, average seller revenue was always close to 0.5. Only in auctions with two and four bidders did revenues diverge from 0.5, and only in auctions with two bidders did the standard deviation of auction revenue exceed 1% of the value of the auctioned good. This result contrasts strongly with the results from simulations employing second-price auctions where average revenue was perceptibly lower than the value of the auctioned good. As can be seen in Figure 1, average revenue in second-price auctions increased with the number of bidders because of a shift in the probability mass from the lower tail of the revenue distribution to its upper tail, with the modal revenue remaining close to 0.5. For example, average revenue was 36% lower than the value of the auctioned good when only two bidders participated, and 1% lower when 50 bidders participated. The second-price auction revenues also displayed higher standard deviations, which we term “revenue volatility.” With only two bidders, revenue volatility was 30% of the value of the auctioned good. However, 7 Results
from simulations not reported in this paper indicate that 1,000 rounds provide enough time for the bidders to learn. Our results are not sensitive to material changes in the number of rounds per run when we extend all runs to 10,000 rounds.
8
revenue volatility dropped as the number of bidders increased, falling to 4% of the value of the auctioned good when the number of bidders increased to 50. The difference between first-price and second-price auctions was also reflected in agents’ strategy pools. We characterize these strategy pools statistically as follows: For each agent we can compute an average bid and a bid variance using the bids in that agent’s strategy pool weighted by their representation in the pool. We call the average across all agents of their average bids, the “mean bid,” and the average of the standard deviations of the agents’ bids “bid volatility.” We also compute the extent of “bidder heterogeneity” as captured by the standard deviation of the average bids. As we can see from Figure 2, average bids in first-price auctions never exceeded 0.5, the value of the good. Average bids that were less than 0.5 were approximately uniformly distributed between zero and 0.5. In marked contrast, when only two bidders participated, average bids in second-price auctions varied across the entire range of feasible bids, [0, 1]. As the number of bidders increased, the incidence of average bids in second-price auctions above 0.5 fell dramatically. However, even with 50 bidders, there was a non-trivial fraction of bidders whose average bid exceeded 0.5. These patterns are captured by bidder pool statistics presented in Table 1. In first-price auctions, the mean bid was 2% lower than good value when only two bidders participated. The mean bid decreased with the number of bidders and was approximately 19% lower than the value of the auctioned good for auctions with 50 bidders. Mean bids in the second-price auctions displayed a steeper decline, falling from approximately 0.5 when only two bidders participated to approximately 0.25 when 50 bidders participated. In first-price auctions, bid volatility also tended to increase with the number of bidders, from 0.003 in first-price auctions with two bidders to 0.023 when 50 bidders participated. Because of this increase, bid volatility exceeded revenue volatility when four or more bidders participated in first-price auctions. The pattern of bid volatility in second-price auctions was quite different. Bid volatility in second-price auctions was much higher, ranging from 30 times higher in auctions with two bidders to 3 times higher in auctions with 50 bidders. Further, in contrast to the trend in first-price auctions, volatility declined with the number of bidders, dropping from approximately 0.085 when only two bidders participated to 0.067 with 50 bidders. However, the even faster decline in revenue volatility led to higher bid volatility than revenue volatility with eight or more bidders. The statistics in Table 1 also suggest that in both first- and second-price auctions, changes in the number of participants had contrasting effects on bidder heterogeneity. In first-price auctions, bidder heterogeneity increased from approximately 0.02 when only two bidders participated to 0.15 with 50 participating bidders. In contrast, in second-price auctions bidder heterogeneity dropped from approximately 0.22 when only two bidders participated in a second-price 9
auction to 0.13 when 50 bidders participated. To provide further insights into the evolution of bidder strategies, we now present information on bidder strategy pools that had converged to a pure strategy, that is, bidder strategy pools where a single strategy accounted for all bids in the pool. Statistics describing these pools are presented in Table 2. In excess of 90% of bidder pools converged in first-price auctions when only two bidders participated, and the frequency with which bidder pools converged declined as the number of bidders rose. Further, in first-price auctions, the vast majority of bidder pools tended to converge to bids close to the true value of the auction good. In auctions with more than eight bidders, a small fraction of bidder pools tended to converge to bids distant from 0.5, and the frequency of such behavior declined as the number of bidders increased. This contrasts strongly with the evolution of bidder pools in the second-price auctions. Between one and two percent of bidder pools converged, and these pools never converged to a value close to 0.5. Table 3 presents summary statistics for bidder pools that did not converge to a pure strategy. In the first-price auction designs with 8, 20, and 50 bidders, where a large fraction of bidders failed to converge to pure strategies, the average bid in the non-converged strategy pools was approximately 0.23, and the average value of the standard deviation of bids was approximately 0.012. In the corresponding second-price auctions, the behavior of non-converged bidders exhibited more dependence on the auction parameters. The average non-converged bid and the standard deviation of these bids both declined as the number of bidders rose. The auction behavior documented above has a relatively simple explanation based on the nonlinear payoff structure of auctions and learning dynamics. In first-price auctions, overbids lead to sure losses to at least one of the overbidders. Such overbidding strategies are quickly weeded out. Moreover, topping a winning bid that is substantially below the value of the auctioned good generates a large reward. Thus, some of the bidders quickly converge to bidding close to the value of the auctioned good. Once a few bidders are bidding the value of the auctioned good, the payoffs to other bidders become flat below the good’s value and fall steeply with bids above value. Thus, the remaining bidders no longer receive enough feedback to fix on a specific bidding strategy, and so become a group of nonmarginal underbidders whose bids wander aimlessly over the underbidding range. Thus, in first-price auctions, expected revenue is not affected by the auction design (i.e., number of bidders) and equals the value of the auctioned good. However, as the number of bidders increases, so does the likelihood that a given bidder becomes an untrained underbidder, boosting bid volatility and lowering the mean bid. The feedback mechanism in second-price auctions is quite different. The key feature of such auctions from the perspective of our simulation is that a lone overbidder will not be penalized with losses and may lose by lowering his bid. Moreover, such an agent’s bid will cause 10
other agents to realize either losses or zero profits if they overbid, and realize the same zero payoff for all underbids. Thus, a lone overbidder will condition other bidders to underbid, but will not condition them to pick any particular underbidding strategy. In turn, the underbidders will condition a lone overbidder to continue to overbid, but will not condition any particular high bid. Under this pattern of behavior, agents are unlikely to converge to any pure strategy and thus bid volatility will be high. Moreover, as the number of bidders increases, the fraction of bidders who become underbidders increases. Thus, more and more bids are crowded into the underbidding range of 0.00 to 0.50. This reduces both the bid mean and bid volatility. Revenue, which equals the second highest bid, will tend to equal the maximum bid of the underbidders. Thus, as the number of auction participants increases, despite a decline in the average bid, this maximum bid from the underbidders, and thus auction revenue, increases. The pattern of bidding in second-price auctions contrasts with the dominant strategy solution, which calls for all bidders to bid the value of the auctioned good. The problem with the dominant strategy solution is that, once it is played by more than one agent, the payoffs to all other agents are independent of their strategies so long as they do not top the dominant strategy bid. This lack of feedback blocks learning. Thus, learning externalities are key to the failure of the dominant strategy solution to emerge in our artificial agent simulation. To verify this assertion, we simulated a two-bidder auction in which the bids from one of the agents are drawn from a uniform distribution over [0, 1]. In this simulation, the incentives for the other bidder are identical to those produced by the Becker, DeGroot, and Marschak (1964) procedure for incentive compatible elicitation of valuations. In this simulation, the bidders’ strategies converged in every simulation to the dominant strategy solution of bidding value, 0.50. Thus, individually the artificial agents learn to play the dominant strategy solution; collectively, however, they converge to a very different pattern of behavior. Our result in second-price auctions is consistent with experimental behavior in which a core group of subjects overbids, and this demonstrates that such behavior can result even in the absence of perceptual biases to which the literature frequently appeals (see, e.g., Kagel 1995). B
Uncertain value for the auctioned good
In Table 4, we present the outcomes of simulations where the value of the auctioned good was uniformly distributed over the interval [0, 1] and bidder behavior was permitted to evolve. The first set of columns presents results from runs where the good was auctioned using a first-price auction; the second set of columns presents results from runs where the good was auctioned using a second-price auction. Each row presents the outcomes of auctions with a given number of bidders. Results are reported for auctions with 2, 4, 8, 20, and 50 bidders. In the first-price auction simulations the introduction of uncertainty lowers revenue substan11
tially when only a small number of bidders participates and raises revenue when 20 or more bidders participate. Further, revenue volatility increases dramatically, with the increase most marked in auctions with few bidders. Revenue in second-price auctions was not affected as dramatically. However, the introduction of uncertainty raised revenue, especially when many bidders participated. The variance of revenue also increased, especially when there were many bidders. This increase, however, was not as marked as the increase in revenue volatility in first-price auctions. These effects on revenue of introducing uncertainty are also illustrated in Figure 3. The introduction of uncertainty also affected bidding strategies in first-price auctions more dramatically than bidding strategies in second-price auctions. In first-price auctions, as can also be seen in Figure 4, uncertainty resulted in lower average bids which tended to vary between 0.225 and 0.300. The introduction of uncertainty also boosted bid volatility; however, the bid volatility tended to be lower than the volatility of revenue. Bidder heterogeneity also increased with the introduction of uncertainty, with the largest increase occuring for auctions with two bidders. In second-price auctions, the introduction of uncertainty increased bid averages, especially when the number of bidders was from 4 and 20. The volatility of bids was lower than the volatility of auction revenues. Further, the introduction of uncertainty resulted in a marked decline in bid volatility for auctions with a large number of participants. Bidder heterogeneity also tended to increase, most significantly in auctions with 20 or more bidders. Table 5 presents information on the strategy pools of bidders who have converged to a pure strategy, and Table 6 presents this information for the remaining bidders. Once again, the effect of the introduction of uncertainty is very dramatic in the case of first-price auctions. There is no longer any convergence to bids close to 0.5. Further, there is a dramatic decline in the incidence of convergence, with less than 30% of participants converging to any pure strategy in auctions with more than two bidders. In addition, in first-price auctions with more than 20 participants, the average bid of participants who converged to a pure strategy are similar to the average bids of the remaining participants. In the second-price auctions, the rate of convergence remained low with the exception of auctions with more than 20 participants. In these auctions, the average bids of participants who converged to a pure strategy was similar to the mean value of the bids of the remaining participants. From the above data we see that value uncertainty has a profound effect on the outcomes of auctions, especially for first-price auctions. The key driver for this effect is that with uncertainty, overbids do not always result in losses; thus, overbidding is not subject to such intense selection pressure and consequently remains in some agents’ strategy pools. At the same time, simply bidding the expected value of the auctioned good and winning the auction can generate losses; thus, bids below value are sometimes selected into the pool. These two effects greatly 12
increase the volatility of the marginal bid. This increased volatility also blocks learning, further increasing the randomness of bidding strategies. This increased randomness, in turn, raises the gain from placing low bids. Consequently, the distribution of mean bids shifts left and becomes more left skewed. Nevertheless, because of the greater bid volatility, auction revenue still increases because each agent has some probability of overbidding. In the case of second-price auctions, convergence to pure strategies was almost nonexistent even in the absence of uncertainty. Thus, the introduction of uncertainty, which further impedes learning, has less effect on bidder strategies. However, under certainty, overbidding by more than one bidder generates sure losses while under uncertainty it does not. Thus, overbidding by the second highest bidder is frequent and hence auction revenue increases, especially when there are many bidders. III
Conclusion
Auction-like mechanisms have historically been employed to allocate high value assets such as oil leases, antiques, equity in corporations, government contracts, and so on. Recently, with advances in communication technology which have reduced the costs of organizing and participating in auctions, auctions are emerging as an important means of retailing a wider range of goods and services to both businesses and consumers. Consequently, more and more auctioneers face bidders who are both unsophisticated and likely participants in the auctioneer’s future auctions. For both these reasons auctioneers need to consider the influence of the auction mechanism on bidder learning. This paper provides, in a symmetric information setting, guidance to these auctioneers. First we show that the behavior of unsophisticated bidders is unlikely to conform to the predictions of auction theory and thus rational expectations theoretical models may not provide a useful guide to the modern day auctioneer. Further, we identify the learning externalities embedded in two common auction designs—first-price and second-price auctions. Our results demonstrate that second-price auctions generate greater negative learning externalities, resulting in more uncertain auction revenue. Unless the number of bidders is large and uncertainty about the auctioned good’s value is high, this increased randomness lowers auction revenue. These results may explain why goods are almost never sold using sealed-bid second-price auctions and why, in cases where the good’s value is highly uncertain, winning bidders overbid even when private information regarding the common value of the good is not salient.8 Given that our effort is one of the first to examine the potential of agent learning to explain auction behavior, it has featured very simple specifications of auction design, the pref8 For
example, many winning bidders in the 1994 U.S. Federal Communication Commission spectrum auction were bankrupted by their aggressive bids despite the fact that they had similar valuations for bandwidth and believed that common value information was not important in determining their bidding strategies (see, e.g., Cramton 1995).
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erences of bidders, and the learning mechanism. However, even this very simple framework has yielded some fairly powerful insights. We believe that future research projects within this research program, by enriching our specification with more complex designs and agent preference structures, could yield even greater insights into the tradeoffs that underlie real-world auction design. For example, while theory identifes an isomorphism between second-price sealed bid auctions and open outcry English auctions, the sequential nature of English auctions would offer a very different training environment for agents. Thus, we would not be surprised if these two designs yielded very different outcomes. Further, it would appear that multistage auctions, by weeding out poorly trained bidders who submit low bids, may improve the learning of the remaining bidders. Finally, in the learning framework, heterogenous valuation for the auctioned good would have an additional effect not present in a rational expectations setting. High valuation bidders would submit marginal bids much more often thus such bidders would become better trained. Low valuation bidders, receiving little feedback from their mostly intramarginal bids, would be poorly trained and thus have high variance bidding strategies. This competition from “noise bidders” could force up the bids of high valuation bidders, perhaps explaining, without relying on the hypotheses of risk aversion, the observed overbidding in private value auctions.
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Table 1: Outcomes of Simulations When the Value of the Good is Fixed at 0.5 This table presents outcomes when the seller fixes on either first-price auction or second-price auction with the number of bidders (N). It presents the mean (RAVE) and standard deviation of the seller’s revenue (RSTD) in the last round of 1000 runs. The mean values of bid averages (BAVE), the average value of standard deviations of bidders’ chromosome pools (BSTD), and the standard deviation of bid averages across all bidders (STAVE) in the last round across all 1,000 runs are reported in the table as well. The results were generated by simulations employing crossover and mutation rates for bidders of 0.6 and 0.003, respectively. Bidders employ lottery-based selection. Each run lasts 1000 rounds.
N 2 4 8 20 50
RAVE 0.495 0.499 0.500 0.500 0.500
First-Price Auctions Second-Price Auctions RSTD BAVE BSTD STAVE RAVE RSTD BAVE BSTD 0.011 0.492 0.003 0.023 0.318 0.149 0.499 0.085 0.001 0.467 0.008 0.095 0.396 0.102 0.358 0.080 0.000 0.443 0.013 0.122 0.445 0.065 0.293 0.074 0.000 0.420 0.019 0.138 0.478 0.034 0.259 0.070 0.000 0.404 0.023 0.146 0.494 0.021 0.251 0.067
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STAVE 0.219 0.205 0.169 0.140 0.128
Table 2: Distribution of Converged Bidder Pools When the Value of the Good is Fixed at 0.5 This table presents the distribution of converged bidder pools when the seller fixes on either first-price auction or second-price auction with varying number of bidders (N). It presents the percentages of bidder pools that have converged to the highest or the second highest tick below 0.5 (CNV12), and any other tick (CNVE). The mean values of bid averages (BAVE) and the standard deviations of bid averages (STAVE) in the last round across all bidder pools that have converged to a tick other than the highest and the second highest below 0.5 are reported in the table as well. The results were generated by simulations employing crossover and mutation rates for bidders of 0.6 and 0.003, respectively. Bidders employ lottery-based selection. Each run lasts 1000 rounds. First-Price Auctions Second-Price Auctions N CNV12 CNVE BAVE STAVE CNV12 CNVE BAVE STAVE 2 45.10 47.30 0.493 0.012 0.05 1.15 0.406 0.200 4 85.63 1.93 0.454 0.111 0.00 0.98 0.335 0.200 8 79.03 0.66 0.228 0.152 0.03 1.25 0.283 0.165 20 70.25 0.64 0.233 0.143 0.00 1.50 0.269 0.167 50 63.83 0.82 0.242 0.140 0.01 1.76 0.253 0.149
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Table 3: Distribution of Non-Converged Bidder Pools When the Value of the Good is Fixed at 0.5 This table presents the distribution of non-converged bidder pools when the seller fixes on either first-price auction or second-price auction with varying number of bidders (N). It presents the percentages of bidder pools that are not converged (NOCNV). The mean values of bid averages (BAVE), the average value of standard deviations of bidders’ chromosome pools (BSTD), and the standard deviation of bid averages (STAVE) in the last round across all non-converged bidder pools are reported in the table as well. The results were generated by simulations employing crossover and mutation rates for bidders of 0.6 and 0.003, respectively. Bidders employ lottery-based selection. Each run lasts 1000 rounds. First-Price Auctions Second-Price Auctions N NOCNV BAVE BSTD STAVE NOCNV BAVE BSTD STAVE 2 7.60 0.450 0.064 0.044 98.80 0.500 0.219 0.086 4 12.45 0.250 0.127 0.063 99.03 0.358 0.205 0.080 8 20.31 0.231 0.120 0.064 98.73 0.293 0.169 0.075 20 29.12 0.233 0.119 0.066 98.50 0.258 0.139 0.071 50 35.36 0.236 0.120 0.065 98.24 0.251 0.128 0.068
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Table 4: Outcomes of Simulations When the Value of the Good is Uniformly Distributed Over the Interval [0, 1]. This table presents outcomes when the seller fixes on either first-price auction or second-price auction with varying number of bidders (N). It presents the mean (RAVE) and standard deviation of the seller’s revenue (RSTD) in the last round of 1000 runs. The mean values of bid averages (BAVE), the average value of standard deviations of bidders’ chromosome pools (BSTD), and the standard deviation of bid averages across all bidders (STAVE) in the last round across all 1,000 runs are reported in the table as well. The results were generated by simulations employing crossover and mutation rates for bidders of 0.6 and 0.003, respectively. Bidders employ lottery-based selection. Each run lasts 1000 rounds.
N 2 4 8 20 50
RAVE 0.300 0.409 0.455 0.502 0.546
First-Price Auctions Second-Price Auctions RSTD BAVE BSTD STAVE RAVE RSTD BAVE BSTD 0.077 0.279 0.019 0.074 0.318 0.160 0.499 0.095 0.066 0.305 0.043 0.118 0.433 0.129 0.374 0.091 0.053 0.279 0.056 0.137 0.500 0.102 0.314 0.093 0.050 0.247 0.031 0.149 0.552 0.073 0.268 0.048 0.052 0.233 0.035 0.149 0.596 0.067 0.255 0.050
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STAVE 0.218 0.201 0.176 0.173 0.164
Table 5: Distribution of Converged Bidder Pools When the Value of the Good is Uniformly Distributed Over the Interval [0, 1]. This table presents the distribution of converged bidder pools When the seller fixes on either first-price auction or second-price auction with varying number of bidders (N). It presents the percentages of bidder pools that have converged to the highest or the second highest tick below 0.5 (CNV12), and any other tick (CNVE). The mean values of bid averages (BAVE) and the standard deviations of bid averages (STAVE) in the last round across all bidder pools that have converged to a tick other than the highest and the second highest below 0.5 are reported in the table as well. The results were generated by simulations employing crossover and mutation rates for bidders of 0.6 and 0.003, respectively. Bidders employ lottery-based selection. Each run lasts 1000 rounds. First-Price Auctions Second-Price Auctions N CNV12 CNVE BAVE STAVE CNV12 CNVE BAVE STAVE 2 0.00 44.90 0.287 0.069 0.00 0.55 0.443 0.186 4 0.00 20.58 0.385 0.069 0.00 1.03 0.384 0.199 8 0.00 10.18 0.417 0.093 0.00 0.83 0.360 0.214 20 0.00 28.80 0.273 0.169 0.00 17.85 0.272 0.188 50 0.00 24.77 0.245 0.169 0.00 17.09 0.258 0.182
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Table 6: Distribution of Non-Converged Bidder Pools When the Value of the Good is Uniformly Distributed Over the Interval [0, 1]. This table presents the distribution of non-converged bidder pools when the seller fixes on either first-price auction or second-price auction with varying number of bidders (N). It presents the percentages of bidder pools that are not converged (NOCNV). The mean values of bid averages (BAVE), the average value of standard deviations of bidders’ chromosome pools (BSTD), and the standard deviation of bid averages (STAVE) in the last round across all non-converged bidder pools are reported in the table as well. The results were generated by simulations employing crossover and mutation rates for bidders of 0.6 and 0.003, respectively. Bidders employ lottery-based selection. Each run lasts 1000 rounds. First-Price Auctions Second-Price Auctions N NOCNV BAVE BSTD STAVE NOCNV BAVE BSTD STAVE 2 55.10 0.272 0.077 0.035 99.45 0.500 0.218 0.095 4 79.43 0.284 0.119 0.054 98.98 0.374 0.201 0.092 8 89.81 0.264 0.132 0.062 99.18 0.313 0.176 0.094 20 71.16 0.237 0.139 0.043 82.11 0.267 0.170 0.058 50 75.17 0.229 0.141 0.047 82.87 0.254 0.160 0.060
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(a) 2 bidders
(b) 4 bidders
(c) 8 bidders
(d) 20 bidders
(e) 50 bidders
Figure 1: Distribution of auction revenues when the value of the good is fixed at 0.5. Blank bars are for first-price auctions and dashed bars are for second-price auctions.
(a) 2 bidders
(b) 4 bidders
(c) 8 bidders
(d) 20 bidders
(e) 50 bidders
Figure 2: Distribution of bid pool averages when the value of the good is fixed at 0.5. Blank bars are for first-price auctions and dashed bars are for second-price auctions.
(a) 2 bidders
(b) 4 bidders
(c) 8 bidders
(d) 20 bidders
(e) 50 bidders
Figure 3: Distribution of auction revenues when the value of the good is random between 0 and 1. Blank bars are for first-price auctions and dashed bars are for second-price auctions.
(a) 2 bidders
(b) 4 bidders
(c) 8 bidders
(d) 20 bidders
(e) 50 bidders
Figure 4: Distribution of bid pool averages when the value of the good is random between 0 and 1. Blank bars are for first-price auctions and dashed bars are for second-price auctions.
