Monotone Games with Positive Spillovers

A monotone game comprises the infinitely repeated play of an n-person stage game, subject to the constraint that players' actions be monotonically nondecreasing over time. These games represent a variety of strategic situations in which players are able to make (partial) commitments. If the stage games have positive spillovers and satisfy certain other conditions, then the limit points of the subgame perfect equilibria are precisely the approachable action profiles. This characterization is applied to voluntary contribution games, market games, and coordination games. Journal of Economic Literature Classification Number: C7.     

Stochastic Cooperative Games: Superadditivity, Convexity, and Certainty Equivalents

This paper extends the notions of superadditivity and convexity to stochastic cooperative games. It is shown that convex games are superadditive and have nonempty cores, and that these results also hold in the context of NTU games. Furthermore, a subclass of stochastic cooperative games to which one can associate a deterministic cooperative game is considered. It is shown that such a stochastic cooperative game satisfies properties like nonemptiness of the core, superadditivity, and convexity if and only if the corresponding deterministic game satisfies these properties.Journal of Economic LiteratureClassification Number: C71.     

Existence of Sparsely Supported Correlated Equilibria

We show that every N-player K ₁ × ... × K _N game possesses a correlated equilibrium with at least zero entries. In particular, the largest N-player K × ... × K games with unique fully supported correlated equilibrium are two-player games. We thank an anonymous referee for most useful comments. The first author acknowledges financial support from Spanish Ministry of Science and Technology, grant SEJ2004-03619, and in form of a Ramón y Cajal fellowship. The second author acknowledges support by the PASCAL Network of Excellence under EC grant no.506778, as well as from Spanish Ministry of Science and Technology and FEDER, grant BMF2003-03324. Both authors also acknowledge financial support from BBVA grant “Aprender a jugar.”     

Statistical Equilibrium Wealth Distributions in an Exchange Economy with Stochastic Preferences

We describe an exchange market consisting of many agents with stochastic preferences for two goods. When individuals are indifferent between goods, statistical mechanics predicts that goods and wealth will have steady-state gamma distributions. Simulation studies show that gamma distributions arise for a broader class of preference distributions. We demonstrate this mathematically in the limit of large numbers of individual agents. These studies illustrate the potential power of a statistical mechanical approach to stochastic models in economics and suggest that gamma distributions will describe steady-state wealths for a class of stochastic models with periodic redistribution of conserved quantities. Journal of Economic Literature Classification Numbers: C15, C62, C73, D3, D5.     

Reciprocity and Cooperation in Repeated Coordination Games: The Principled-Player Approach

Two long-lived players play a repeated coordination game. I show the restrictions on players' beliefs which imply that cooperation is optimal play. Journal of Economic Literature Classification Number: D83.     

Correlated equilibria, incomplete information and coalitional deviations

This paper proposes new concepts of strong and coalition-proof correlated equilibria where agents form coalitions at the interim stage and share information about their recommendations in a credible way. When players deviate at the interim stage, coalition-proof correlated equilibria may fail to exist for two-player games. However, coalition-proof correlated equilibria always exist in dominance-solvable games and in games with positive externalities and binary actions.     

Public Information, Private Information, and the Multiplicity of Equilibria in Coordination Games

I study coordination games with incomplete public and private information and relate equilibrium convergence to convergence of higher-order beliefs. As the players' signals become more and more precise, the equilibrium manifold converges to the correspondence of common knowledge equilibria, whenever the variance of the public signal converges to 0 at a rate faster than one half the rate of convergence of the variance of private signals. The same condition also determines the convergence of common p-belief to common knowledge, which leads to a simple intuition for its origin and an immediate generalization of the former results about equilibrium convergence. Journal of Economic Literature Classification Numbers: C72, D82.     

Implementation in Economic Environments with Incomplete Information: The Use of Multi-Stage Games

This paper shows that, in economic environments with incomplete information, incentive compatibility and a preference reversal condition are sufficient for implementation in sequential equilibrium.Journal of Economic LiteratureClassification Numbers: C72, D71, D82.     

Noisy Directional Learning and the Logit Equilibrium

We specify a dynamic model in which agents adjust their decisions toward higher payoffs, subject to normal error. This process generates a probability distribution of players’ decisions that evolves over time according to the Fokker–Planck equation. The dynamic process is stable for all potential games, a class of payoff structures that includes several widely studied games. In equilibrium, the distributions that determine expected payoffs correspond to the distributions that arise from the logit function applied to those expected payoffs. This “logit equilibrium” forms a stochastic generalization of the Nash equilibrium and provides a possible explanation of anomalous laboratory data.     

Self-Fulfilling Mechanisms in Bayesian Games

Amechanismfor a Bayesian gameGis a mapping μ from the set of states of nature to the set of players' actions. μ isself-fulfillingif players are truthful at the communication stage and, given the information revealed by μ, no player can gain in unilaterally deviating from the action prescribed by the mechanism. We investigate the properties of self-fulfilling mechanisms and we show in particular that they correspond to inert solutions of the infinitely repeated game generated byG. We also discuss applications to market games, regulation, and R&D games.Journal of Economic LiteratureClassification Numbers: D82, C72.     

Market Games in Large Economies with a Finite Number of Types

We study market games derived from an exchange economy with a continuum of agents, each having one of finitely many possible types. The type of agent determines his initial endowment and utility function. It is shown that, unlike the well-known Shapley–Shubik theorem on market games (Shapley and Shubik in J Econ Theory 1:9–25, 1969), there might be a (fuzzy) game in which each of its sub-games has a non-empty core and, nevertheless, it is not a market game. It turns out that, in order to be a market game, a game needs also to be homogeneous. We also study investment games – which are fuzzy games obtained from an economy with a finite number of agents cooperating in one or more joint projects. It is argued that the usual definition of the core is inappropriate for such a model. We therefore introduce and analyze the new notion of comprehensive core. This solution concept seems to be more suitable for such a scenario. We finally refer to the notion of feasibility of an allocation in games with a large number of players. Some of the results in this paper appear in a previous draft distributed by the name “Cooperative investment games or Population games”. An anonymous referee of Economic Theory is acknowledged for his/her comments     

Strong Equilibrium in Congestion Games

[6]introduced the class of congestion games and proved that they always possess a Nash equilibrium in pure strategies. Here we obtain conditions for the existence of a strong equilibrium in this class of games, as well as for the equivalence of Nash and strong equilibria. We also give conditions for uniqueness and for Pareto optimality of the Nash equilibrium. Except for a natural monotonicity assumption on the utilities, the conditions are expressed only in terms of the underlying congestion game form. It turns out that avoiding a certain type of bad configuration in the strategy spaces is essential to positive results.Journal of Economic LiteratureClassification Numbers: C71, C72, D62.     

Resolving Indeterminacy in Coordination Games: A New Approach Applied to a Pay-as-you-go Pension Scheme

This paper shows that when players ignore what outcome will emerge because of the presence of multiple equilibria, they can coordinate their expectations by forming an initial belief based on the principle of indifference followed by a process of reasoning that updates this belief. Since this procedure describes a natural way to form beliefs under indeterminacy, it is reasonable for every agent to conjecture that all the others form their beliefs according to the same logic. Exactly the fact that agents are aware that they form their beliefs following the same procedure allows them to successfully coordinate their expectations.      

Correlated equilibrium and sunspot equilibrium

Summary We show by an example that the sunspot equilibria of a competitive economy are not equivalent to the correlated equilibria if sunspots generate transfers between (extrinsic) states of nature (through a contingent commodities market). Nevertheless, we prove that the sunspot equilibrium allocations of a standard overlapping generations economy coincide with the (strategic form) correlated equilibrium allocations of a natural market game mimicking the economy.     

Cournot and Bertrand Games

The author describes a series of matrix choice games illustrating monopoly, shared monopoly, Cournot, Bertrand, and Stackelberg behavior given either perfect complements or perfect substitutes. The games are created by using a spreadsheet to fill out a profit table given the choices of two players. One player selects the column, the other the row, and the table gives the profit of the row chooser. Because each player has a table, each thinks of him- or herself as the row chooser and the other as the column chooser. The games may be applied to international trade through the traditional Boeing v. Airbus story or, more currently, through foreign sales corporations. Addition of Bertrand competition allows discussion of price wars, and addition of perfect complements allows discussion of the proposed Microsoft breakup.     

Equilibrium in a Market Game

This paper studies the equilibrium of an exchange economy with the same number of agents and commodities. It is shown that under rather mild conditions on demand the market have a price equilibrium—independently of the divisibility of the commodities. The result extends Gale’s equilibrium theorem for indivisible commodities to the case where some commodities are perfectly divisible. The proof is based on a topological lemma contained in previous paper of one of the authors.     

Socially Subjective Equilibrium in Strategic Form Games

This study provides a new framework and a new equilibrium concept, which are able to describe the situation where people have various images of the society and have various solution concepts for social outcomes, and where people accept the social outcomes. In socially subjective equilibrium, people have a coherence of their own norms in two senses. One is the consistency of the norm itself. Imagined outcomes should satisfy a certain (subjective) solution concept. The other is the consistency between the imagined outcomes and realized one. These are the main features of our equilibrium concept. This paper forms a part of my doctoral thesis, which is titled “On socially subjective equilibrium”. The first person I would like to thank is my direct supervisor Professor Ken Urai (Osaka University). I obtained a basic idea of the main concept of my doctoral thesis, that is, the solution concept scheme, from Professor Urai. I am grateful to Professor Hiroaki Nagatani (Osaka University) and Professor Ken-Ichi Shimomura (Kobe University), who monitored my works and took efforts in providing me with valuable comments on earlier versions of my thesis. I also obtained a fruitful advice from Professor Kenichi Amaya (Kobe University). Lastly, I specially thank to Kozo Shiraishi (Osaka University).     

Correlated equilibria in continuous games: Characterization and computation

We present several new characterizations of correlated equilibria in games with continuous utility functions. These have the advantage of being more computationally and analytically tractable than the standard definition in terms of departure functions. We use these characterizations to construct effective algorithms for approximating a single correlated equilibrium or the entire set of correlated equilibria of a game with polynomial utility functions.