112,341 research outputs found

    Is Having a Unique Equilibrium Robust?

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    We investigate whether having a unique equilibrium (or a given number of equilibria) is robust to perturbation of the payoffs, both for Nash equilibrium and correlated equilibrium. We show that the set of n-player finite games with a unique correlated equilibrium is open, while this is not true of Nash equilibrium for n>2. The crucial lemma is that a unique correlated equilibrium is a quasi-strict Nash equilibrium. Related results are studied. For instance, we show that generic two-person zero-sum games have a unique correlated equilibrium and that, while the set of symmetric bimatrix games with a unique symmetric Nash equilibrium is not open, the set of symmetric bimatrix games with a unique and quasi-strict symmetric Nash equilibrium is

    Correlated equilibria, good and bad : an experimental study

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    We report results from an experiment that explores the empirical validity of correlated equilibrium, an important generalization of the Nash equilibrium concept. Specifically, we seek to understand the conditions under which subjects playing the game of Chicken will condition their behavior on private, third–party recommendations drawn from known distributions. In a “good–recommendations” treatment, the distribution we use is a correlated equilibrium with payoffs better than any symmetric payoff in the convex hull of Nash equilibrium payoff vectors. In a “bad–recommendations” treatment, the distribution is a correlated equilibrium with payoffs worse than any Nash equilibrium payoff vector. In a “Nash–recommendations” treatment, the distribution is a convex combination of Nash equilibrium outcomes (which is also a correlated equilibrium), and in a fourth “very–good–recommendations” treatment, the distribution yields high payoffs, but is not a correlated equilibrium. We compare behavior in all of these treatments to the case where subjects do not receive recommendations. We find that when recommendations are not given to subjects, behavior is very close to mixed–strategy Nash equilibrium play. When recommendations are given, behavior does differ from mixed–strategy Nash equilibrium, with the nature of the differ- ences varying according to the treatment. Our main finding is that subjects will follow third–party recommendations only if those recommendations derive from a correlated equilibrium, and further,if that correlated equilibrium is payoff–enhancing relative to the available Nash equilibria

    Strategic Behavior in Non-Atomic Games

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    Typically, economic situations featuring a large number of agents are not modelled with a finite normal form game, rather by a non-atomic game. Consequently, the possibility of strategic interaction may be completely ignored. In order to restore strategic interaction among agents we propose a refinement of Nash equilibrium, strategic equilibrium, for non-atomic games with a continuum of agents, each of whose payoÂź depends on what he chooses and a societal choice. Given a non-atomic game, we consider a perturbed game in which every player believes that he alone has a small, but positive, impact on the societal choice. A strategy profile is a strategic equilibrium if it is a limit point of a sequence of Nash equilibria of games in which each player's belief about his impact on the societal choice goes to zero. After proving the existence of strategic equilibria, we show that every strategic equilibrium must be a Nash equilibrium of the original non-atomic game, thus, our concept of strategic equilibrium is indeed a refinement of Nash equilibrium. Next, we show that the concept of strategic equilibrium is the natural extension of Nash equilibrium infinite normal form games, to non-atomic games: That is, given any finite normal form game, we consider its non- atomic version, and prove that a strategy profile, in the non-atomic version of the given finite normal form game, is a strategic equilibrium if and only if the associated strategy profile in the finite form game is a Nash equilibrium. Finally, applications of strategic equilibrium is presented examples in which the set of strategic equilibria, in contrast with the set of Nash equilibria, does not contain any implausible Nash equilibrium strategy profiles. These examples are: a game of proportional voting, a game of allocation of public resources, and finally non-atomic Cournot oligopoly

    Implementation in mixed Nash equilibrium

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    A mechanism implements a social choice correspondence f in mixed Nash equilibrium if at any preference profile, the set of all pure and mixed Nash equilibrium outcomes coincides with the set of f-optimal alternatives at that preference profile. This definition generalizes Maskin’s definition of Nash implementation in that it does not require each optimal alternative to be the outcome of a pure Nash equilibrium. We show that the condition of weak set-monotonicity, a weakening of Maskin’s monotonicity, is necessary for implementation. We provide sufficient conditions for implementation and show that important social choice correspondences that are not Maskin monotonic can be implemented in mixed Nash equilibrium
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