The Lipschitz constant of a finite normal-form game is the maximal change in some player's payoff when a single opponent changes his strategy. We prove that games with small Lipschitz constant admit pure {\epsilon}-equilibria, and pinpoint the maximal Lipschitz constant that is sufficient to imply existence of pure {\epsilon}-equilibrium as a function of the number of players in the game and the number of strategies of each player. Our proofs use the probabilistic method.Comment: minor changes, forthcoming in Mathematics of Operations Researc

We consider the design of decision rules in an environment with two alternatives, independent private values and no monetary transfers. The utilitarian rule subject to incentive compatibility constraints is a weighted majority rule, where agents' weights correspond to expected gains given that their favorite alternative is chosen. It is shown that a rule is interim incentive efficient if and only if it is a weighted majority rule, and we characterize those weighted majority rules that are ex ante incentive efficient. We also discuss efficiency in the class of anonymous mechanisms and the stability of weighted majority rules.

A (TU) cooperative game is extendable (Kikuta and Shapley, 1986) if every core allocation of each sub-game can be extended to a core allocation of the game. It is strongly extendable if any minimal vector in the upper core of any of its sub-games can be extended to a core allocation. We prove that strong extendability is equivalent to largeness of the core (Sharkey, 1982). Further, we characterize extendability in terms of an extension of the balanced cover of the game. It is also shown how this extension can unify the analysis of many families of games under one roof. JEL classification: C71

Abstract. We consider games with incomplete informationà la Harsanyi, where the payoff of a player depends on an unknown state of nature as well as on the profile of chosen actions. As opposed to the standard model, the players in the game are not necessarily expected utility maximizers. Rather, their preferences over state-contingent utility vectors are represented by arbitrary functionals. Our first contribution is to provide simple and applicable definitions of both ex-ante and interim equilibria in this generalized setting. Second, we characterize equilibrium existence in terms of the preferences of the participating players. It turns out that, given some standard properties of the functionals, equilibrium exists in every game if and only if all players are averse to uncertainty. Finally, for a sub-class of preferences' representing functionals we show that there exists a symmetric equilibrium in every symmetric game if and only if all players share preferences.

Abstract. We prove existence of envy-free allocations in markets with heterogenous indivisible goods and money, when a given quantity is supplied from each of the goods and agents have unit demands. We depart from most of the previous literature by allowing agents' preferences over the goods to depend on the entire vector of prices. Our proof uses Shapley's K-K-M-S theorem and Hall's marriage lemma. We then show how our theorem may be applied in two related problems: Existence of envy-free allocations in a version of the cake-cutting problem, and existence of equilibrium in an exchange economy with indivisible goods and money.