A Structure Theorem for Rationalizability with Application to Robust Predictions of Refinements

Rationalizability is a central solution concept of game theory. Economic models often have many rationalizable outcomes, motivating economists to use refinements of rationalizability, including equilibrium refinements. In this paper we try to achieve a general understanding of when this multiplicity occurs and how one should deal with it. Assuming that the set of possible payoff functions and belief structures is sufficiently rich, we establish a revealing structure of the correspondence of beliefs to sets of rationalizable outcomes. We show that, for any rationalizable action a of any type, we can perturb the beliefs of the type in such a way that a is uniquely rationalizable for the new type. This unique outcome will be robust to further small changes. When multiplicity occurs, then we are in a “knife‐edge” case, where the unique rationalizable outcome changes, sandwiched between open sets of types where each of the rationalizable actions is uniquely rationalizable. As an immediate application of this result, we characterize, for any refinement of rationalizability, the predictions that are robust to small misspecifications of interim beliefs. These are only those predictions that are true for all rationalizable strategies, that is, the predictions that could have been made without the refinement.     

An Efficient Dynamic Mechanism

This paper constructs an efficient, budget‐balanced, Bayesian incentive‐compatible mechanism for a general dynamic environment with quasilinear payoffs in which agents observe private information and decisions are made over countably many periods. First, under the assumption of “private values” (other agents' private information does not directly affect an agent's payoffs), we construct an efficient, ex post incentive‐compatible mechanism, which is not budget‐balanced. Second, under the assumption of “independent types” (the distribution of each agent's private information is not directly affected by other agents' private information), we show how the budget can be balanced without compromising agents' incentives. Finally, we show that the mechanism can be made self‐enforcing when agents are sufficiently patient and the induced stochastic process over types is an ergodic finite Markov chain.     

Robust Predictions for Bilateral Contracting with Externalities

The paper studies bilateral contracting between one principal and N agents when each agent's utility depends on the principal's unobservable contracts with other agents. We show that allowing deviations to menu contracts from which the principal chooses bounds equilibrium outcomes in a wide class of bilateral contracting games without imposing ad hoc restrictions on the agents' beliefs. This bound yields, for example, competitive convergence as N →∞ in environments in which an appropriately‐defined notion of competitive equilibrium exists. We also examine the additional restrictions arising in two common bilateral contracting games: the “offer game” in which the principal makes simultaneous offers to the agents, and the “bidding game” in which the agents make simultaneous offers to the principal.     

Isotone Equilibrium in Games of Incomplete Information

An isotone pure strategy equilibrium exists in any game of incomplete information in which each player's action set is a finite sublattice of multidimensional Euclidean space, types are multidimensional and atomless, and each player's interim expected payoff function satisfies two “nonprimitive conditions” whenever others adopt isotone pure strategies: (i) single‐crossing in own action and type and (ii) quasi‐supermodularity in own action. Conditions (i), (ii) are satisfied in supermodular and log‐supermodular games given affiliated types,and in games with independent types in which each player's ex post payoff satisfies supermodularity in own action and nondecreasing differences in own action and type. This result is applied to provide the first proof of pure strategy equilibrium existence in the uniform price auction when bidders have multi‐unit demand, nonprivate values, and independent types.     

Information Independence and Common Knowledge

In Bayesian environments with private information, as described by the types of Harsanyi, how can types of agents be (statistically) disassociated from each other and how are such disassociations reflected in the agents' knowledge structure? Conditions studied are (i) subjective independence (the opponents' types are independent conditional on one's own) and (ii) type disassociation under common knowledge (the agents' types are independent, conditional on some common‐knowledge variable). Subjective independence is motivated by its implications in Bayesian games and in studies of equilibrium concepts. We find that a variable that disassociates types is more informative than any common‐knowledge variable. With three or more agents, conditions (i) and (ii) are equivalent. They also imply that any variable which is common knowledge to two agents is common knowledge to all, and imply the existence of a unique common‐knowledge variable that disassociates types, which is the one defined by Aumann.     

Private Information and Insurance Rejections

Across a wide set of nongroup insurance markets, applicants are rejected based on observable, often high‐risk, characteristics. This paper argues that private information, held by the potential applicant pool, explains rejections. I formulate this argument by developing and testing a model in which agents may have private information about their risk. I first derive a new no‐trade result that theoretically explains how private information could cause rejections. I then develop a new empirical methodology to test whether this no‐trade condition can explain rejections. The methodology uses subjective probability elicitations as noisy measures of agents' beliefs. I apply this approach to three nongroup markets: long‐term care, disability, and life insurance. Consistent with the predictions of the theory, in all three settings I find significant amounts of private information held by those who would be rejected; I find generally more private information for those who would be rejected relative to those who can purchase insurance, and I show it is enough private information to explain a complete absence of trade for those who would be rejected. The results suggest that private information prevents the existence of large segments of these three major insurance markets.     

Level‐k Auctions: Can a Nonequilibrium Model of Strategic Thinking Explain the Winner's Curse and Overbidding in Private‐Value Auctions?

This paper proposes a structural nonequilibrium model of initial responses to incomplete‐information games based on “level‐k” thinking, which describes behavior in many experiments with complete‐information games. We derive the model's implications in first‐ and second‐price auctions with general information structures, compare them to equilibrium and Eyster and Rabin's (2005) “cursed equilibrium,” and evaluate the model's potential to explain nonequilibrium bidding in auction experiments. The level‐k model generalizes many insights from equilibrium auction theory. It allows a unified explanation of the winner's curse in common‐value auctions and overbidding in those independent‐private‐value auctions without the uniform value distributions used in most experiments.     

Co‐opetition and Investment for Supply‐Chain Resilience

This paper considers the problem of disruption risk management in global supply chains. We consider a supply chain with two participants, who face interdependent losses resulting from supply chain disruptions such as terrorist strikes and natural hazards. The Harsanyi–Selten–Nash bargaining framework is used to model the supply chain participants' choice of risk mitigation investments. The bargaining approach allows a framing of both joint financing of mitigation activities before the fact and loss‐sharing net of insurance payouts after the fact. The disagreement outcome in the bargaining game is assumed to be the result of the corresponding non‐cooperative game. We describe an incentive‐compatible contract that leads to First Best investment and equal “gain” for all players, when the solution is “interior” (as it almost certainly is in practice). A supplier that has superior security practices (i.e., is inherently safer) exploits its informational advantage by extracting an “information rent” in the usual spirit of incomplete information games. We also identify a special case of this contract, which is robust to moral hazard. The role of auditing in reinforcing investment incentives is also examined.     

Randomization Tests Under an Approximate Symmetry Assumption

This paper develops a theory of randomization tests under an approximate symmetry assumption. Randomization tests provide a general means of constructing tests that control size in finite samples whenever the distribution of the observed data exhibits symmetry under the null hypothesis. Here, by exhibits symmetry we mean that the distribution remains invariant under a group of transformations. In this paper, we provide conditions under which the same construction can be used to construct tests that asymptotically control the probability of a false rejection whenever the distribution of the observed data exhibits approximate symmetry in the sense that the limiting distribution of a function of the data exhibits symmetry under the null hypothesis. An important application of this idea is in settings where the data may be grouped into a fixed number of “clusters” with a large number of observations within each cluster. In such settings, we show that the distribution of the observed data satisfies our approximate symmetry requirement under weak assumptions. In particular, our results allow for the clusters to be heterogeneous and also have dependence not only within each cluster, but also across clusters. This approach enjoys several advantages over other approaches in these settings.     

A Dynamic Reinterpretation of Nash Bargaining With Endogenous Threats

This paper concerns the two‐stage game introduced in Nash (1953). It formalizes a suggestion made (but not pursued) by Nash regarding equilibrium selection in that game, and hence offers an arguably more solid foundation for the “Nash bargaining with endogenous threats” solution. Analogous reasoning is then applied to an infinite horizon game to provide equilibrium selection in two‐person repeated games with contracts. In this setting, issues about enforcement of threats are much less problematic than in Nash's static setting. The analysis can be extended to stochastic games with contracts.     

Discounted Stochastic Games With No Stationary Nash Equilibrium: Two Examples

We present two examples of discounted stochastic games, each with a continuum of states, finitely many players, and actions, that possess no stationary equilibria. The first example has deterministic transitions—an assumption undertaken in most of the early applications of dynamics games in economics—and perfect information, and does not possess even stationary approximate equilibria or Markovian equilibria. The second example satisfies, in addition to stronger regularity assumptions, that all transitions are absolutely continuous with respect to a fixed measure—an assumption that has been widely used in more recent economic applications. This assumption has been undertaken in several positive results on the existence of stationary equilibria in special cases, and in particular, guarantees the existence of stationary approximate equilibria.     

On greedy and strategic evaders in sequential interdiction settings with incomplete information

We consider a class of sequential network interdiction problem settings where the interdictor has incomplete initial information about the network while the evader has complete knowledge of the network including its structure and arc costs. In each decision epoch, the interdictor can block (for the duration of the epoch) at most k arcs known to him/her. By observing the evader’s actions, the interdictor learns about the network structure and costs and thus, can adjust his/her actions in subsequent decision epochs. It is known from the literature that if the evader is greedy (i.e., the shortest available path is used in each decision epochs), then under some assumptions the greedy interdiction policies that block k-most vital arcs in each epoch are efficient and have a finite regret. In this paper, we consider the evader’s perspective and explore deterministic “strategic” evasion policies under the assumption that the interdictor is greedy. We first study the theoretical computational complexity of the evader’s problem. Then we derive basic constructive properties of optimal evasion policies for two decision epochs when the interdictor has no initial information about the network structure. These properties are then exploited for the design of a heuristic algorithm for a strategic evader in a general setting with an arbitrary time horizon and any initial information available to the interdictor. Our computational experiments demonstrate that the proposed heuristic outperforms the greedy evasion policy on several classes of synthetic network instances under either perfect or noisy information feedback. Finally, some interesting insights from our theoretical and computational results conclude the paper.     

Robust Predictions in Games With Incomplete Information

We analyze games of incomplete information and offer equilibrium predictions that are valid for, and in this sense robust to, all possible private information structures that the agents may have. The set of outcomes that can arise in equilibrium for some information structure is equal to the set of Bayes correlated equilibria. We completely characterize the set of Bayes correlated equilibria in a class of games with quadratic payoffs and normally distributed uncertainty in terms of restrictions on the first and second moments of the equilibrium action–state distribution. We derive exact bounds on how prior knowledge about the private information refines the set of equilibrium predictions. We consider information sharing among firms under demand uncertainty and find new optimal information policies via the Bayes correlated equilibria. We also reverse the perspective and investigate the identification problem under concerns for robustness to private information. The presence of private information leads to set rather than point identification of the structural parameters of the game.     

Revealed Altruism1

This paper develops a nonparametric theory of preferences over one's own and others' monetary payoffs. We introduce “more altruistic than” (MAT), a partial ordering over such preferences, and interpret it with known parametric models. We also introduce and illustrate “more generous than” (MGT), a partial ordering over opportunity sets. Several recent studies focus on two‐player extensive form games of complete information in which the first mover (FM) chooses a more or less generous opportunity set for the second mover (SM). Here reciprocity can be formalized as the assertion that an MGT choice by the FM will elicit MAT preferences in the SM. A further assertion is that the effect on preferences is stronger for acts of commission by FM than for acts of omission. We state and prove propositions on the observable consequences of these assertions. Finally, empirical support for the propositions is found in existing data from investment and dictator games, the carrot and stick game, and the Stackelberg duopoly game and in new data from Stackelberg mini‐games.     

Cursed Equilibrium

There is evidence that people do not fully take into account how other people's actions depend on these other people's information. This paper defines and applies a new equilibrium concept in games with private information, cursed equilibrium, which assumes that each player correctly predicts the distribution of other players' actions, but underestimates the degree to which these actions are correlated with other players' information. We apply the concept to common‐values auctions, where cursed equilibrium captures the widely observed phenomenon of the winner's curse, and to bilateral trade, where cursedness predicts trade in adverse‐selections settings for which conventional analysis predicts no trade. We also apply cursed equilibrium to voting and signalling models. We test a single‐parameter variant of our model that embeds Bayesian Nash equilibrium as a special case and find that parameter values that correspond to cursedness fit a broad range of experimental datasets better than the parameter value that corresponds to Bayesian Nash equilibrium.     

Stable and Coordinating Contracts for a Supply Chain with Multiple Risk‐Averse Suppliers

We analyze a decentralized supply chain with a single risk‐averse retailer and multiple risk‐averse suppliers under a Conditional Value at Risk objective. We define coordinating contracts and show that the supply chain is coordinated only when the least risk‐averse agent bears the entire risk and the lowest‐cost supplier handles all production. However, due to competition, not all coordinating contracts are stable. Thus, we introduce the notion of contract core, which reflects the agents' “bargaining power” and restricts the set of coordinating contracts to a subset which is “credible.” We also study the concept of contract equilibrium, which helps to characterize contracts that are immune to opportunistic renegotiation. We show that, the concept of contract core imposes conditions on the share of profit among different agents, while the concept of contract equilibrium provide conditions on how the payment changes with the order quantity.     

Moral Hazard Contracting and Private Credit Markets

This paper studies the impact of credit markets on optimal contracting, when the agent's “interim preference” over upcoming contracts is private information because personal financial decisions affect it via the wealth effect. The main result is a severe loss of incentive provision: equilibrium contracts invariably cause the agent to shirk (i.e., exert minimal effort) if the agent's private financial decision precedes moral hazard contracting. The basic intuition is that committing on another private variable, other than the effort level, exposes the parties to further exploitation of efficient risk‐sharing by relaxing the incentive constraint that was binding ex ante, unless the risk‐sharing was fully efficient to begin with.     

A Theory of Disagreement in Repeated Games With Bargaining

This paper proposes a new approach to equilibrium selection in repeated games with transfers, supposing that in each period the players bargain over how to play. Although the bargaining phase is cheap talk (following a generalized alternating‐offer protocol), sharp predictions arise from three axioms. Two axioms allow the players to meaningfully discuss whether to deviate from their plan; the third embodies a “theory of disagreement”—that play under disagreement should not vary with the manner in which bargaining broke down. Equilibria that satisfy these axioms exist for all discount factors and are simple to construct; all equilibria generate the same welfare. Optimal play under agreement generally requires suboptimal play under disagreement. Whether patient players attain efficiency depends on both the stage game and the bargaining protocol. The theory extends naturally to games with imperfect public monitoring and heterogeneous discount factors, and yields new insights into classic relational contracting questions.     

Causal inference with observational data: A tutorial on propensity score analysis

When treatment cannot be manipulated, propensity score analysis provides a useful way to making causal claims under the assumption of no unobserved confounders. However, it is still rarely utilised in leadership and applied psychology research. The purpose of this paper is threefold. First, it explains and discusses the application and key assumptions of the method with a particular focus on propensity score weighting. This approach is readily implementable since a weighted regression is available in most statistical software. Moreover, the approach can offer a “double robust” protection against misspecification of either the propensity score or the outcome model by including confounding variables in both models. A second aim is to discuss how propensity score analysis (and propensity score weighting, specifically) has been conducted in recent management studies and examine future challenges. Finally, we present an advanced application of the approach to illustrate how it can be employed to estimate the causal impact of leadership succession on performance using data from Italian football. The case also exemplifies how to extend the standard single treatment analysis to estimate the separate impact of different managerial characteristic changes between the old and the new manager.     

Truthful Equilibria in Dynamic Bayesian Games

This paper characterizes an equilibrium payoff subset for dynamic Bayesian games as discounting vanishes. Monitoring is imperfect, transitions may depend on actions, types may be correlated, and values may be interdependent. The focus is on equilibria in which players report truthfully. The characterization generalizes that for repeated games, reducing the analysis to static Bayesian games with transfers. With independent private values, the restriction to truthful equilibria is without loss, except for the punishment level: if players withhold their information during punishment‐like phases, a folk theorem obtains.