理论论文

In a recent paper, Lin & Palfrey (2024) developed a theory of cognitive hierarchies (CH) in sequential games and observed that this solution concept is not reduced-normal-form invariant. In this paper I qualify and explain this observation. I show that the CH model is normal-form invariant, and that the differences arising from the application of the CH model to the reduced normal form depend only on how randomization by level-0 types is modeled. Indeed, while the uniform behavior strategy in the extensive form yields the uniform mixed strategy in the normal form, the latter does not correspond to the uniform randomization in the reduced normal form, because different reduced strategies may correspond to sets of equivalent strategies with different cardinalities. I also comment on (i) the invariance of the CH model to some transformations of the sequential game, and (ii) the independence of conditional beliefs about co-players' level-types.

We study optimal service allocation through queues when agents have homogeneous preferences and must be incentivized to enter and remain in the queue. The designer controls entry, exit, service priority, and information about queue priorities. Under a mild regularity condition, the optimal mechanism admits agents only up to a cutoff queue length, never removes admitted agents, serves them according to a first-come, first-served (FCFS) rule, and provides no information beyond recommendations. FCFS is also necessary for optimality in a rich domain. The result identifies a new role for queue disciplines in regulating agents’ beliefs and dynamic incentives, and uncovers a hitherto unrecognized virtue of FCFS in this regard.

We analyze optimal procurement contracts when suppliers learn about project costs over time and retain an ex-post right to exit. In the first period, the firm observes a private signal on the likelihood of high costs; in the second, it learns the realized cost type and decides whether to continue or exercise its outside option. We show that when exit opportunities are sufficiently attractive, the optimal mechanism pools all first-period signals and reduces to a cost-plus contract, with payments tied only to realized, verifiable costs. Attempts to elicit cost forecasts ex-ante backfire: firms strategically under-report expectations, triggering exit in high-cost states and generating overruns. This provides a mechanism-based interpretation of cost overruns as failures of incentive design. Finally, we demonstrate that more generous outside options can raise procurement costs and lower firm profits, identifying conditions where limiting exit rights is ex-ante Pareto improving. (JEL D82, D86, H57)

Quantal response equilibrium (QRE), a statistical generalization of Nash equilibrium, is a standard benchmark in the analysis of experimental data. Despite its influence, nonparametric characterizations and tests of QRE are unavailable beyond the case of finite games. We address this gap by completely characterizing the set of QRE in a class of binary-action games with a continuum of types. Our characterization provides sharp predictions in settings such as global games, volunteer's dilemma, and the compromise game. Further, we leverage our results to develop nonparametric tests of QRE. As an empirical application, we revisit the experimental data from Carrillo and Palfrey (2009) on the compromise game.

The effect of group size and effort cost on rent dissipation in the standard Tullock rent seeking model is examined using a laboratory experiment. Data support qualitative comparative static Nash equilibrium predictions as reductions in both group size and effort cost increase effort levels. However, effort and dissipation rates are above Nash predictions for all treatments. In addition, while cutting per-unit effort costs in half increases individual effort levels, they do not fully double as predicted and therefore lower rent dissipation, creating a policy-relevant “streamlining effect”. A quantal response equilibrium, which adds stochastic elements into the game structure, predicts the observed higher effort levels and dissipation but is proven to be unable to explain the streamlining effect. A modification of quantal response that incorporates implementation error noise and loss aversion outperforms other estimated models and tracks the observed all data patterns more closely, including the streamlining effect.

This paper aims to connect epistemic and behavioral game theory by examining the epistemic foundations of quantal response equilibrium (QRE) in static games. We focus on how much information agents possess about the probability distributions of idiosyncratic payoff shocks, in addition to the standard assumptions of rationality and common belief in rationality. When these distributions are transparent, we obtain a solution concept called Δ p -rationalizability, which includes action distributions derived from QRE; we also give a condition under which this relationship holds true in reverse. When agents only have common belief in the monotonicity of these distributions (for example, extreme value distributions), we obtain another solution concept called Δ M -rationalizability, which includes action distributions derived from rank-dependent choice equilibrium, a parameter-free variant of QRE. Our solution concepts also provide insights for interpreting experimental and empirical data.

We consider contests with a large set (continuum) of participants and axiomatize contest success functions that arise when performance is composed of both effort and a random element, and when winners are those whose performance exceeds a cutoff determined by the designer’s budget constraint. Any such Random Performance Function (RPF) satisfies a Co-monotonicity property, which implies that opponents’ effort distributions can be unambiguously ranked based on how competitive they are. We prove that Co-monotonicity, along with standard continuity and monotonicity properties, in fact characterize the class of RPFs. We also describe necessary and sufficient conditions for the noise term to have an additive structure. “The idea of a continuum of traders may seem outlandish to the reader. Actually, it is no stranger than a continuum of prices or of strategies or a continuum of ‘particles’ in fluid mechanics... It should be emphasized that our consideration of a continuum of traders is not merely a mathematical exercise; it is the expression of an economic idea.” In: Markets with a continuum of traders, Robert J. Aumann, Econometrica 1964

We report on the topological structure of logit quantal response equilibria for asymmetric, two-player, two-choice games in normal form, via catastrophe theory. We present three main outcomes: first, a novel, smooth potential function for the underlying dynamics, relative to which logit equilibria arise as stationary points comprising a parameterised hyper-surface; secondly, a proof that all catastrophes within this manifold are of order at most three, i.e. “folds and cusps”; thirdly, discovery of a new topological phenomenon, the “pleated loop”, corresponding to a pair of opposed pitchfork bifurcations. This we exhibit for games, such as Prisoner's Dilemma, that have a unique Nash equilibrium, with multiple logit equilibria for finite precision parameters. These results extend work at the intersection of stochastic decision theory and catastrophe theory with economic games. Their application to the problem of bifurcations in the tracing procedure for logit solutions, proposed by McKelvey and Palfrey, is discussed throughout.

This paper introduces a refinement of Nash equilibrium (NE) to determine the relevance of weakly dominated NE in games. We adapt Harsanyi (1973)'s concept of “approachability,” originally based on games with randomly disturbed payoffs, and extend it to a broader family that encompasses the most widely used equilibrium models of imperfectly rational behavior. We identify a singular NE refinement, approximate payoff monotone equilibrium (APME), that coincides with the set of NE that are approachable by regular quantal response equilibria as well as control cost models. We advocate for APME based on its 1) foundation in models that rationalize empirically observed behavior; 2) ability to discriminate between equilibria without automatically discarding weakly dominated behavior; and 3) computational simplicity in applications where the structure of NE is known. We illustrate the value of APME in three applications in which empirical studies suggest that weakly dominated NE are relevant.

We propose an extensive-form solution concept, with players who neglect information from hypothetical events but make inferences from observed events. Our concept modifies cursed equilibrium (Eyster and Rabin 2005) and allows that players can be cursed about endogenous information.

We consider a group of voters that needs to decide between two candidates. We propose a novel family of neutral and strategy-proof rules, which we call sequential unanimity rules. By demonstrating their formal equivalence to the M-winning coalition rules of Moulin (1983), we show that sequential unanimity rules are characterized by neutrality and strategy-proofness. We establish our results by developing algorithms that transform a given M-winning coalition rule into an equivalent sequential unanimity rule and vice versa. The analysis can be extended to accommodate the full preference domain in which voters may be indifferent between candidates.

We study the optimal auction design problem when bidders are ambiguity averse and follow the max-min expected utility model. Each bidder’s set of priors consists of beliefs that are close to the seller’s belief, where “closeness” is defined by a divergence. For a given allocation rule, we show that optimal transfers belong to a specific class of transfers, termed win-lose dependent transfers, in which bidders’ transfers upon winning and losing depend only on their own types but not on their opponents’ type reports. This result effectively reduces the infinite-dimensional problem of identifying an optimal transfer function into a two-dimensional problem of determining two constants-one for winning and another for losing. Solving this reduced problem, we show that among efficient mechanisms without transfers to losing bidders, the first-price auction is optimal, thereby outperforming other auction formats such as the second-price auction. We also discuss how the structure of the set of priors is related to the revenue ranking between the first- and second-price auctions.

We consider a setting where each player of a normal-form game privately designs an information structure before playing the game. One of these designs is chosen at random to determine the distribution of the private messages that players receive. These messages allow players to correlate their actions; however, private information design implies a push from correlated to Nash equilibria. Indeed, the sequential equilibrium payoffs of the private information design extensive-form game are correlated equilibrium payoffs of the underlying normal-form game, but not all correlated equilibrium payoffs are sequential equilibrium payoffs. In generic 2-player games, the latter are specific convex combinations of two Nash equilibrium payoffs. In any 2-player game, imposing optimal beliefs from private information design allows us to obtain sufficient epistemic conditions for Nash equilibrium that require only knowledge of players' rationality.

We study priority-based matching markets with public and private endowments. We show that efficiency-adjusted deferred acceptance (EADA) is justified-envy minimal in the class of efficient mechanisms, while Top trading cycles (TTC) and other popular mechanisms are not. Our findings highlight EADA as an interesting alternative to TTC in the context of transplantation-organ markets. Restricting attention to strategyproof mechanisms, we show that TTC is justified-envy minimal, strengthening the result of Abdulkadiroglu et al. (2020).

In a game in extensive form, we say that a sequence of trembles destabilizes an outcome if there is no corresponding sequence of payoff perturbations and Nash outcomes converging to the outcome. We use equilibrium destabilization to characterize sequential equilibria and sequentially stable outcomes, and to reinterpret standard selection criteria in signaling games as a means of ruling out implausible equilibria by identifying destabilizing tremble sequences. We show that the existence of a maximally destabilizing sequence of trembles (destabilizing all outcomes but one) significantly simplifies studying strategically stable behavior in complex games such as finitely repeated games.

By showing that a one-shot deviation property holds for almost all large games with i.i.d. randomly drawn payoffs in which a convention (in the sense of Young, 1993a) exists, long-run outcomes under behavioral rules are characterized by social choice rules.

In the context of one-to-one matching markets, we study myopic-farsighted stable sets, which are internally and externally stable when myopic agents consider immediate payoffs from their deviations, while farsighted agents anticipate counter-deviations and consider final payoffs. We constructively prove the existence of a (rational expectations) myopic-farsighted stable set, in which farsighted agents receive a single payoff while myopic agents may receive multiple payoffs. Our existence result extends to settings with enforcing coalitions of arbitrary size, yielding coalitional myopic-farsighted stable sets, and to settings where not all members of an enforcing coalition must strictly gain, yielding myopic-farsighted weakly stable sets. When all farsighted agents have unit demand, our results also extend to many-to-one matching markets. As a key corollary, we provide a foundation for the efficiency-adjusted deferred acceptance algorithm by showing that its outcome constitutes a singleton myopic-farsighted stable set when one side is farsighted and the other is myopic.

In school choice problems, the pursuit of student welfare (efficiency) is constrained by the requirement to respect schools’ priorities (fairness). Among fair matchings, even the welfare maximizing one, the Student-Optimal Stable Matching (SOSM), is inefficient. Moreover, any mechanism that yields welfare gains over the SOSM is manipulable by students. This paper investigates the fairness and incentive properties of efficient mechanisms. We compare matchings using the notion that one matching is less unfair than another if it generates a smaller set of students whose priorities are violated, and we define minimal unfairness accordingly. We show that the Efficiency-Adjusted Deferred Acceptance (EADA) mechanism is minimally unfair within the class of Pareto efficient mechanisms satisfying a simple incentive requirement, top-manipulation-proofness. Moreover, the EADA satisfies a stronger property, upper-manipulation-proofness. Upper-manipulation-proofness is a broad incentive property one can maintain while improving upon the SOSM. Together, these results highlight a sharp frontier in school choice: when the objective is efficiency and improving student welfare over the SOSM, the EADA emerges as a compelling mechanism in both fairness and incentive terms.

This paper analyzes the extent to which well-known results on the relationship between the bargaining set, the core, and the kernel in one-to-one assignment games generalize to many-to-one assignment markets, and by extension, many-to-many markets. Using a minimal counterexample, we show that the bargaining set does not necessarily coincide with the core and that the kernel may not be contained within the core. We would like to highlight that the failure of the coincidence between the core and the bargaining set, as observed in the many-to-one assignment game, is quite notable. This is especially true when compared to various other highly structured games, many of which emerge from combinatorial optimization problems, such as the one underlying many-to-one assignment games.

Economists often rely on people’ s choices to infer their preferences. However, inferring preferences from choices becomes problematic when people face unobserved constraints. In this paper, I study how to (cautiously) infer preferences from choices when the choices of the members of a social group are subject to a constraint that we only imperfectly know. When this happens, heterogeneity in the group’ s choices, along with more information about the constraint, helps recover the preferences of group members, whereas homogeneity in choices hinders it. I further argue that standard approaches to choice-based welfare analysis can lead to misleading inferences about welfare when there are unobserved social constraints.