Games played in a contracting environment

We analyze normal form games where a player has to pay a price to a supplier in order to play a specific action. Our focus is on supplier competition, arising from the fact that distinct suppliers supply different players, and possibly different actions of the same player. With private contracts, where a player only observes the prices quoted by his own suppliers, the set of equilibrium distributions over player actions coincides with the set of equilibrium distributions when all actions are supplied competitively, at cost. With public contracts, the two distributions differ dramatically even in simple games.     

Commitment and observability in a contracting environment

This paper examines leader-follower games where a leader must purchase an essential input from a price-setting supplier in order to take an action. We show that equilibrium outcomes when the followers perfectly observe the leaders' actions cannot be approximated by mixed equilibrium outcomes of the game where followers imperfectly observe the leaders' actions, i.e. they are not accessible. Accessibility fails since in a pure strategy equilibrium, a supplier makes positive profits; however in an equilibrium where a leader randomizes, supplier profits must be zero. Our result follows from a generalized indifference principle that mixed strategies must satisfy in economic environments. While supplier profits cannot be approximated, player action profiles are accessible. Our results also apply to games with costly observation.     

Representing equilibrium aggregates in aggregate games with applications to common agency

An aggregate game is a normal-form game with the property that each playerʼs payoff is a function of only his own strategy and an aggregate of the strategy profile of all players. Such games possess properties that can often yield simple characterizations of equilibrium aggregates without requiring that one solves for the equilibrium strategy profile. When payoffs have a quasi-linear structure and a degree of symmetry, we construct a self-generating maximization program over the space of aggregates with the property that the solution set corresponds to the set of equilibrium aggregates of the original n-player game. We illustrate the value of this approach in common-agency games where the playersʼ strategy space is an infinite-dimensional space of nonlinear contracts. We derive equilibrium existence and characterization theorems for both the adverse selection and moral hazard versions of these games.     

INCOMPLETE INFORMATION GAMES WITH MULTIPLE PRIORS*

We present a model of incomplete information games, where each player is endowed with a set of priors. Upon arrival of private information, it is assumed that each player “updates” his set of priors to a set of posterior beliefs, and then evaluates his actions by the most pessimistic posterior beliefs. So each player's preferences may exhibit aversion to ambiguity or uncertainty. We define a couple of equilibrium concepts, establish existence results for them, and demonstrate by examples how players’ views on uncertainty about the environment affect the strategic outcomes.     

Decompositions and potentials for normal form games

A basic model of commitment is to convert a two-player game in strategic form to a “leadership game” with the same payoffs, where one player, the leader, commits to a strategy, to which the second player always chooses a best reply. This paper studies such leadership games for games with convex strategy sets. We apply them to mixed extensions of finite games, which we analyze completely, including nongeneric games. The main result is that leadership is advantageous in the sense that, as a set, the leader's payoffs in equilibrium are at least as high as his Nash and correlated equilibrium payoffs in the simultaneous game. We also consider leadership games with three or more players, where most conclusions no longer hold.     

Timing and Virtual Observability in Ultimatum Bargaining and “Weak Link” Coordination Games

Previous studies have shown that simply knowing one player moves first can affect behavior in games, even when the first-mover's moves are known to be unobservable. This observation violates the game-theoretic principle that timing of unobserved moves is irrelevant, but is consistent with virtual observability, a theory of how timing can matter without the ability to observe actions. However, this previous research only shows that timing matters in games where knowledge that one player moved first can help select that player's preferred equilibrium, presenting an alternative explanation to virtual observability. We extend this work by varying timing of unobservable moves in ultimatum bargaining games and “weak link” coordination games. In the latter, the equilibrium selection explanation does not predict any change in behavior due to timing differences. We find that timing without observability affects behavior in both games, but not substantially.     

Dynamics in near-potential games

We consider discrete-time learning dynamics in finite strategic form games, and show that games that are close to a potential game inherit many of the dynamical properties of potential games. We first study the evolution of the sequence of pure strategy profiles under better/best response dynamics. We show that this sequence converges to a (pure) approximate equilibrium set whose size is a function of the “distance” to a given nearby potential game. We then focus on logit response dynamics, and provide a characterization of the limiting outcome in terms of the distance of the game to a given potential game and the corresponding potential function. Finally, we turn attention to fictitious play, and establish that in near-potential games the sequence of empirical frequencies of player actions converges to a neighborhood of (mixed) equilibria, where the size of the neighborhood increases according to the distance to the set of potential games.     

Notes on optimal transportation

We study equilibrium in hedonic markets, when consumers and suppliers have reservation utilities, and the utility functions are separable with respect to price. There is one indivisible good, which comes in different qualities; each consumer buys 0 or 1 unit, and each supplier sells 0 or 1 unit. Consumer types, supplier types and qualities can be either discrete of continuous, in which case they are allowed to be multidimensional. Prices play a double role: they keep some agents out of the market, and they match the remaining ones pairwise. We define equilibrium prices and equilibrium distributions, and we prove that equilibria exist, we investigate to what extend equilibrium prices and distributions are unique, and we prove that equilibria are efficient. In the particular case when there is a continuum of types, and a generalized Spence–Mirrlees condition is satisfied, we prove the existence of a pure equilibrium, where demand distributions are in fact demand functions, and we show to what extent it is unique. The proofs rely on convex analysis, and care has been given to illustrate the theory with examples.     

Existence, uniqueness and efficiency of equilibrium in hedonic markets with multidimensional types

We study equilibrium in hedonic markets, when consumers and suppliers have reservation utilities, and the utility functions are separable with respect to price. There is one indivisible good, which comes in different qualities; each consumer buys 0 or 1 unit, and each supplier sells 0 or 1 unit. Consumer types, supplier types and qualities can be either discrete of continuous, in which case they are allowed to be multidimensional. Prices play a double role: they keep some agents out of the market, and they match the remaining ones pairwise. We define equilibrium prices and equilibrium distributions, and we prove that equilibria exist, we investigate to what extend equilibrium prices and distributions are unique, and we prove that equilibria are efficient. In the particular case when there is a continuum of types, and a generalized Spence–Mirrlees condition is satisfied, we prove the existence of a pure equilibrium, where demand distributions are in fact demand functions, and we show to what extent it is unique. The proofs rely on convex analysis, and care has been given to illustrate the theory with examples.     

Rock-paper-scissors and cycle-based games

The present work characterizes the unique Nash equilibrium for games that are based on a cyclic preference relation. In the Nash equilibrium of these games, each player randomizes between three specific actions. In particular, an alternative way of deriving the unique Nash equilibrium of the rock-paper-scissors game is proposed.     

Repeated signaling games

I analyze a class of repeated signaling games in which the informed player's type is persistent and the history of actions is perfectly observable. In this context, a large class of possibly complex sequences of signals can be supported as the separating equilibrium actions of the “strong type” of the informed player. I characterize the set of such sequences. I also characterize the sequences of signals in least cost separating equilibria (LCSE) of these games. In doing this, I introduce a state variable that can be interpreted as a measure of reputation. This gives the optimization problem characterizing the LCSE a recursive structure. I show that, in general, the equilibrium path sequences of signals have a simple structure. The shapes of the optimal sequences depend critically on the relative concavities of the payoff functions of different types, which measure the relative preferences towards payoff smoothing.     

Unobservable Contracts as Precommitments

It is well known that signing publicly observable contracts with third parties is a means of credibly committing to certain actions and hence may yield strategic advantages. Previous work on the commitment value of unobservable contracts has been limited to normal form games and extensive form games in which only one party has the option to sign a contract. In this paper, we extend the analysis to extensive form games in which both players can sign contracts, and characterize the set of sequential equilibria. We show that any Nash equilibrium outcome of the original game in which both players receive more than their individually rational payoffs can be supported as a sequential equilibrium outcome. Therefore, delegation acts not only as a commitment device to gain advantage over the opponent, but also as a cooperative device to attain Pareto improvements over the subgame perfect equilibrium outcome. I would like to thank Atila Abdulkadiroglu, Jean-Pierre Benoit, Alberto Bisin, Boyan Jovanovic, Ehud Kalai, Giuseppe Lopomo, George Mailath, Efe Ok, Ariel Rubinstein, Andy Schotter, seminar participants at various universities and conferences, and anonymous referees for helpful comments and suggestions. Support from the Alfred P. Sloan Foundation and Program for Economic Research at Columbia University is also gratefully acknowledged.     

Strategic Delegation By Unobservable Incentive Contracts

Many strategic interactions in the real world take place among delegates empowered to act on behalf of others. Although there may be a multitude of reasons why delegation arises in reality, one intriguing possibility is that it yields a strategic advantage to the delegating party. In the case where only one party has the option to delegate, we analyse the possibility that strategic delegation arises as an equilibrium outcome under completely unobservable incentive contracts within the class of two-person extensive form games. We show that delegation may arise solely due to strategic reasons in quite general economic environments even under unobservable contracts. Furthermore, under some reasonable restrictions on out-of-equilibrium beliefs and actions of the outside party, strategic delegation is shown to be the only equilibrium outcome.     

Learning with bounded memory in games

We study learning with bounded memory in zero-sum repeated games with one-sided incomplete information. The uninformed player has only a fixed number of memory states available. His strategy is to choose a transition rule from state to state, and an action rule, which is a map from each memory state to the set of actions. We show that the equilibrium transition rule involves randomization only in the intermediate memory states. Such randomization, or less frequent updating, is interpreted as a way of testing the opponent, which generates inertia in the player's behavior and is the main short-run bias in information processing exhibited by the bounded memory player.     

On Sustaining Cooperation without Public Observations

This paper examines a dynamic game in which each player only observes a private and imperfect signal on the actions played. Our main result is that in a repeated prisoner's dilemma where defections are irreversible (at least for a long enough period of time), patient enough players may achieve almost efficient outcomes. Dealing with models of imperfect private monitoring is difficult because (i) continuation games are games of incomplete information, hence they do not have the same structure as the original game. In particular, continuation equilibria are correlated equilibria. (ii) Players are typically uncertain about their opponents' past observations and actions, and they use their entire own private history to learn about these actions. As a result equilibrium strategies are in general nontrivial and increasingly complex functions of past observations. We bypass these difficulties by looking at correlated equilibria of the original game and find correlated equilibria in which the decision problem faced by each player remains the same over time. Journal of Economic Literature Classification Numbers: C72.     

On the impossibility of achieving no regrets in repeated games

Regret-minimizing strategies for repeated games have been receiving increasing attention in the literature. These are simple adaptive behavior rules that lead to no regrets and, if followed by all players, exhibit nice convergence properties: the average play converges to correlated equilibrium, or even to Nash equilibrium in certain classes of games. However, the no-regret property relies on a strong assumption that each player treats her opponents as unresponsive and fully ignores the opponents’ possible reactions to her actions. We show that if at least one player is slightly responsive, it is impossible to achieve no regrets, and convergence results for regret minimization with responsive opponents are unknown.     

Uniqueness of Stationary Equilibrium Payoffs in the Baron-Ferejohn Model

We consider a multilateral sequential bargaining model in which the players may differ in their probability of being selected as the proposer and the rate at which they discount future payoffs. For games in which agreement requires less than unanimous consent, we characterize the set of stationary subgame perfect equilibrium payoffs. With this characterization, we establish the uniqueness of the equilibrium payoffs. For the case where the players have the same discount factor, we show that the payoff to a player is nondecreasing in his probability of being selected as the proposer. For the case where the players have the same probability of being selected as the proposer, we show that the payoff to a player is nondecreasing in his discount factor. Journal of Economic Literature Classification numbers: C72, C78, D70.     

政府补贴下区分制造商类型的合约设计

考虑政府补贴下由一个供应商和一个制造商构成的绿色供应链.供应商通过对原材料和产品生产过程的控制来提供绿色产品,也因此能够获得政府补贴.政府依据供应链的最终产品数量向供应商提供补贴,供应商选择制造商以完成最终产品的生产过程,并除采购价格以外按一定比例与制造商分享政府补贴,制造商有生产成本等私人信息.在分析供应商和制造商成本、收益的基础上,建立了一个逆向选择模型,其中供应商提供包括供应数量和补贴分享比例的合约菜单,制造商从中选择一个合约或者拒绝全部合约,依据成本制造商可分为两种类型:较高机会成本和较低机会成本,通过优化合约设计,供应商对不同类型制造商提供不同的供应数量和补贴比例,以实现供应商收益最大化.     

Communication in Repeated Games with Private Monitoring

The paper considers repeated games where each player can be observed by only a subset of the other players, and where players can make public announcements about the behavior of the players they observed. We address the following question: What is the minimal level of observability that is required to obtain efficient outcomes? The main result is that the limit set of sequential equilibrium payoffs, when the discount factor tends to one, contains the set of individual rational payoffs whenever each player is observed by at leasttwoother players.Journal of Economic LiteratureClassification Number: C73.     

Learning from private information in noisy repeated games

We study the perfect type-contingently public ex-post equilibrium (PTXE) of repeated games where players observe imperfect public signals of the actions played, and both the payoff functions and the map from actions to signal distributions depend on an unknown state. The PTXE payoffs when players are patient are determined by the solutions to a family of linear programming problems. Using this characterization, we develop conditions under which play can be as if the players have learned the state. We provide a sufficient condition for the folk theorem, and a characterization of the PTXE payoffs in games with a known monitoring structure.