Takashi Ui

A Shapley Value Representation of Potential Games

Abstract In potential games, as considered by Monderer and Shapley (1996a, Games Econom. Behav.14, 124–143), each players gain from a deviation is equal to the gain in a potential function. We prove that a game has a potential function if and only if its payoff functions coincide with the Shapley value of a particular class of cooperative games indexed by the set of strategy profiles. Also a potential function of a noncooperative potential game coincides with the potentials (cf. Hart and Mas-Colell, 1989, Econometrica57, 589–614) of cooperative games indexed by the strategy set. Journal of Economic Literature Classification Numbers: C71, C72.

Generalized potentials and robust sets of equilibria

This paper introduces generalized potential functions of complete information games and studies the robustness of sets of equilibria to incomplete information. A set of equilibria of a complete information game is robust if every incomplete information game where payoffs are almost always given by the complete information game has an equilibrium which generates behavior close to some equilibrium in the set. This paper provides sufficient conditions for the robustness of sets of equilibria in terms of argmax sets of generalized potential functions and shows that the sufficient conditions generalize the existing sufficient conditions for the robustness of equilibria.

Robust Equilibria of Potential Games

Potential games are games with potential functions. Technically, the potential function defines a refinement concept. We provide justification for this refinement concept using the notion of robustness of equilibria. A Nash equilibrium of a complete information game is said to be robust if every incomplete information game where payoffs are almost always given by the complete information game has an equilibrium which generates behavior close to the Nash equilibrium. We show that Nash equilibria that maximize potential functions are generically robust.

Correlated equilibrium and concave games

This paper shows that if a game satisfies the sufficient condition for the existence and uniqueness of a pure-strategy Nash equilibrium provided by Rosen (Econometrica 33:520, 1965), then the game has a unique correlated equilibrium, which places probability one on the unique pure-strategy Nash equilibrium. In addition, it shows that a weaker condition suffices for the uniqueness of a correlated equilibrium. The condition generalizes the sufficient condition for the uniqueness of a correlated equilibrium provided by Neyman (Int J Game Theory 26:223, 1997) for a potential game with a strictly concave potential function.

Interim efficient allocations under uncertainty

This paper considers an exchange economy under uncertainty with asymmetric information. Uncertainty is represented by multiple priors and posteriors of agents who have either Bewleys incomplete preferences or Gilboa-Schmeidlers maximin expected utility preferences. The main results characterize interim efficient allocations under uncertainty; that is, they provide conditions on the sets of posteriors, thus implicitly on the way how agents update the sets of priors, for non-existence of a trade which makes all agents better off at any realization of private information. For agents with the incomplete preferences, the condition is necessary and sufficient, but for agents with the maximin expected utility preferences, the condition is sufficient only. A couple of necessary conditions for the latter case are provided.

Discrete Concavity for Potential Games

This paper proposes a discrete analogue of concavity appropriate for potential games with discrete strategy sets. It guarantees that every Nash equilibrium maximizes a potential function.

Agreeable bets with multiple priors

This paper considers a two agent model of trade with multiple priors. First, we characterize the existence of an agreeable bet on some event in terms of the set of priors. It is then shown that the existence of an agreeable bet on some event is a strictly stronger condition than the existence of an agreeable trade, whereas the two conditions are equivalent in the standard Bayesian framework. Secondly, we show that the two conditions are equivalent when the set of priors is the core of a convex capacity.

A Note on Discrete Convexity and Local Optimality

One of the most important properties of a convex function is that a local optimum is also a global optimum. This paper explores the discrete analogue of this property. We consider arbitrary locality in a discrete space and the corresponding local optimum of a function over the discrete space. We introduce the corresponding notion of discrete convexity and show that the local optimum of a function satisfying the discrete convexity is also a global optimum. The special cases include discretely-convex, integrally-convex, M-convex, M♮-convex, L-convex, and L♮-convex functions.

Correlated quantal responses and equilibrium selection

Abstract This paper considers incomplete information games with payoffs subject to correlated random disturbances. It explains the connection between the uniqueness of quantal response equilibria, where large noise is required, and the uniqueness of equilibria in global games, where small noise is required.

The Myerson value for complete coalition structures

In order to describe partial cooperation structures, this paper introduces complete coalition structures as sets of feasible coalitions. A complete coalition structure has a property that, for any coalition, if each pair of players in the coalition belongs to some feasible coalition contained in the coalition then the coalition itself is also feasible. The union stable structures, which constitute the domain of the Myerson value, are a special class of the complete coalition structures. As an allocation rule on complete coalition structures, this paper proposes an extension of the Myerson value for complete coalition structures and provides an axiomatization.