The winner's curse: Conditional reasoning and belief formation

Abstract

In explaining the winner's curse, recent approaches have focused on one of two cognitive processes: conditional reasoning and belief formation. We provide the first joint experimental analysis of the role of these two obstacles. First, we observe that overbidding decreases significantly between a simple common-value auction and a transformed version of this auction that does not require conditional reasoning. Second, assistance in belief formation leads to comparable behavioral changes in both games. The two effects are of similar magnitude and amplify each other when jointly present. We conclude that the combination and the interaction of the two cognitive processes in auctions lead to relatively low strategic sophistication compared to other domains. The study's focus on games' objective cognitive challenges is potentially useful for improving predictions across games and complements the common focus on behavioral models and their explanatory power.

Introduction

The “winner's curse” (WC) in common-value auctions (CVA) refers to the systematic overbidding relative to the Bayesian Nash equilibrium (BNE) that leads to losses for winners in field settings and laboratory experiments.¹ This phenomenon is one of the most important and robust findings in empirical auction analysis and has generated ample theoretical work.

Two main departures from the BNE have been modeled. Both maintain the assumption that players best respond to their beliefs but relax the requirement of consistency of beliefs. First, in equilibrium models such as cursed equilibrium (CE, Eyster and Rabin, 2005), behavioral equilibrium (Esponda, 2008), and the application of analogy-based expectation equilibrium to auctions (Jehiel, 2005; Jehiel and Koessler, 2008), beliefs do not fully take into account what bids tell about underlying signals, capturing that agents do not optimally adjust for the information revealed by winning. Second, the level-k model assumes non-equilibrium beliefs that result from iterated best responses (Nagel, 1995; Stahl and Wilson, 1995). It has been applied to private information games such as auctions and zero-sum betting (Crawford and Iriberri, 2007; Brocas et al., 2014). When one assumes beliefs of uninformed play, this approach can implicitly capture that agents do not fully account for revealed information.

Doubts have been cast on the sufficiency of these belief-based models to explain auction behavior. With an innovative semi-computerized version of the maximal game, Ivanov et al. (2010, ILN) experimentally study whether these models can explain the WC and claim that they cannot. Along these lines, Charness and Levin (2009, CL) use computerized sellers in an acquiring-a-company game and document that subjects have a more general problem with conditional reasoning – drawing appropriate conclusions from hypothetical events – that seems not to be fully captured by the relaxation of beliefs.

In turn, however, Costa-Gomes and Shimoji (2015) criticize ILN's use of game theoretical concepts when the interaction with a known computer program is a single-person decision problem. They argue that belief-based models are indeed compatible with some observations from ILN's experiment. Moreover, Camerer et al. (2016) suggest on the basis of the Quantal Response Equilibrium (QRE, McKelvey and Palfrey, 1995) that imprecise best responses combined with non-equilibrium beliefs could explain observed behavior.

This discussion shows that no consensus has been reached on how to explain the WC. In this study, we do not test concrete models of reasoning, but take a step back and focus on two objective game complexities whose relative importance in causing the WC is – as shown above – disputed in the literature: the needs for conditional reasoning and for belief formation. Both activities are indispensable to reach a best response. In any strategic situation, subjects have to form beliefs about their opponents' behavior in order to know what to best respond to. In CVAs, best responding further requires conditioning on hypothetical situations induced by the game's structure. For example, one's bid is only relevant when winning, which implies that all others have bid less. Crucially, which of the two complexities poses a more substantial challenge for bidders in CVAs remains an open empirical question. By providing the first joint experimental analysis that disentangles the impact of these two cognitive processes in a CVA setting, we are able to determine – as our paper's main contribution – whether the WC is predominantly driven by conditional reasoning or belief formation. Studying strategic behavior with a focus on objective game complexities enables us to establish how two of these complexities – which can be found in a variety of important game – generally affect behavior. Notably, this analysis is not constrained by a more specific structure on how people think about these problems. Physics can predict the bending of a horizontal steel bar due to vertical forces without a detailed model of the tensions inside the bar. Similarly, we propose to relate deviations from equilibrium play to objective game complexities such as the need for conditional reasoning or for elements of belief formation. This approach has potential for improving predictions across very different games, an area of study so far put in second place. Our approach allows for improvements in predictions in the sense of managing expectations as to how close subjects could be to a model prediction such as equilibrium. Or in terms of heterogeneous types, how many observations can reasonably be expected to feature sophisticated play.

Our starting point is a simple first-price CVA adapted from Kagel and Levin (1986). At the core of our investigation is a transformation of this game that maintains the strategic nature of the original auction game in terms of best response functions and equilibria but removes the need to engage in conditional reasoning. This allows us to cleanly identify the effect that this cognitive activity has on bidding behavior and the WC. Independently of this variation, we further change the need to form beliefs in two ways. First, we fully remove the need to form beliefs by letting subjects play against naïve computer opponents that follow a known simple strategy. Second, we partially remove crucial parts of belief formation but maintain the strategic uncertainty associated with human opponents when we let subjects play against human opponents after they played against the computer. The preceding encounter with the naïve computer confronts subjects with one particular action of the opponent, providing an occasion to think about a reaction to such a basic belief.

Following our focus on objective cognitive complexities, we provide a simple formalization to measure the complexities' behavioral impact in a flexible and general way. Defining a measure μ on the action space, we normalize the distance to 1 between equilibrium play, ${\mu }^e=0$, and uninformed random play, ${\mu }^u=1$. We judge a cognitive complexity by the sign and magnitude of the change Δμ caused in the direction away from optimal behavior.

In the modified auction setting that requires neither conditional reasoning nor belief formation for optimal behavior, we observe bids that are close to equilibrium play with μ of 0.29. From there, we obtain three main results. First, introducing the need to condition – without requiring any belief formation – increases bids significantly and moves them further away from optimal play with $\mathrm{\Delta }\mu =0.18$. In addition, it increases the incidence of the WC by 15 percentage points. Second, requiring partial or full belief formation – in the absence of conditional reasoning – leads to remarkably similar increases in bids, Δμ of 0.15 and 0.20, and raises the number of subjects falling prey to the WC by 8 to 11 percentage points, respectively. Interestingly, the partial belief manipulation suggests that the mere need to form a first basic belief, at a given level of strategic uncertainty, already proves challenging for subjects. Third, no generally significant differences emerge when comparing the magnitude of the effects of conditional reasoning and belief formation. Although both effects individually worsen game play, the fraction of plausible bids still remains non-negligible. Combining conditional reasoning and full belief formation results in behavior fairly far away from equilibrium, $\mu =0.81$, as usually observed in CVA settings. Interestingly, the combination of both effects, $\mathrm{\Delta }\mu =0.52$, leads further away from equilibrium than expected by the sum of the two individual effects, $\mathrm{\Delta }\mu =0.38$, implying that the two strengthen each other and exhibit what we call cognitive diseconomies.

The two cognitive complexities jointly produce an extreme case of game-dependent sophistication that is not fully captured by belief-based models. This explains why CL and ILN do not find support for those models when at the same time such support is abundant in other domains in which conditional reasoning is not required (see Crawford et al., 2013).

A number of further papers are closely related to our work. Levin et al. (2016) analyze the conditioning problem in the WC in more detail by separating the involved Bayesian updating from non-probabilistic reasoning. In particular, the authors compare results from a first-price auction with a strategically equivalent Dutch-CVA that makes the conditioning problem more salient. Relatedly, but in non-auction settings, Esponda and Vespa (2014), Louis (2015), and Ngangoue and Weizsäcker (2015) have analyzed conditioning in more depth by separating two involved steps – hypothetical thinking per se and conditioning on hypothetical events – and comparing behavior in simultaneous and sequential games. With our transformation, we propose a complementary way of studying conditional reasoning in auction settings. Crucially, we do not provide a differentiated analysis of conditional reasoning itself but relate the impact of the overall conditioning effect in causing the WC to the impact of belief formation. Moreover, Charness et al. (2014) observe the WC in a generalized information environment in which bidders hold identical and public information. Their innovative design allows them to disentangle the influence of heterogeneity in estimating the common value from non-optimal bidding behavior. They show that both are relevant for the WC. Complementing their results, our study only focuses on the bidding behavior but additionally disentangles the role of conditional reasoning and belief formation. Finally, Levin and Reiss (2012) construct a behavioral auction design in which the payment rule incorporates the adverse selection problem that is at the origin of the WC. They observe that the WC is still present in their data. The authors adjust the payment rule but do not transform the auction game as we do.

Due to our method of transformation, our paper also relates to the broad set of studies that investigate behavior using strategically very similar games. The largest fraction of those studies considers framing effects that influence subjects' behavior but do not result from the strategic nature of the situation (for example Tversky and Kahneman, 1986; Osborne and Rubinstein, 1994; Chou et al., 2009). Another methodologically interesting instance of strategic equivalence is the experimental, so-called “strategy method” in which participants make contingent decisions for all decision nodes that they will possibly encounter in a game (Brandts and Charness, 2011). In a different manner, strategically equivalent versions of a game can facilitate the investigation of particular aspects of behavior. For example, Nagel and Tang (1998) use a repeated, normal-form centipede game to investigate learning behavior without aspects of sequential reciprocity.

In our study, we craft two similar games that differ in the cognitive process under investigation: conditional reasoning. To the best of our knowledge, our experiment is the first that uses such a transformation as a means to investigate the impact of this particular cognitive activity in strategic reasoning. By this virtue, our approach opens further avenues for investigation in settings with similar cognitive processes. For example, conditioning on being pivotal in a jury decision is part of strategic voting (Feddersen and Pesendorfer, 1998) and conditioning on message selection is part of being optimally persuaded (Glazer and Rubinstein, 2004).