A Bound of the Proportion of Pure Strategy Equilibria in Generic Games

In a generic finite normal form game with 2(α) + 1 Nash equilibria, at least alpha of the equilibria are nondegenerate mixed strategy equilibria (that is, they involve randomization by some players)

A Bound of the Proportion of Pure Strategy Equilibria in Generic Games

In a generic finite normal form game with 2(alpha) + 1 Nash equilibria, at least alpha of the equilibria are nondegenerate mixed strategy equilibria (that is, they involve randomization by some players).Normal form, mixed strategy, game theory

On Sustainable Equilibria

Following the ideas laid out in Myerson (1996), Hofbauer (2000) defined an equilibrium of a game as sustainable if it can be made the unique equilibrium of a game obtained by deleting a subset of the strategies that are inferior replies to it, and then adding others. Hofbauer also formalized Myerson's conjecture about the relationship between the sustainability of an equilibrium and its index: for a generic class of games, an equilibrium is sustainable iff its index is +1. Von Schemde and von Stengel (2008) proved this conjecture for bimatrix games. This paper shows that the conjecture is true for all finite games. More precisely, we prove that an isolated equilibrium of a given game has index +1 if and only if it can be made unique in a larger game obtained by adding finitely many inferior reply strategies

A Continuation Method for Nash Equilibria in Structured Games

Structured game representations have recently attracted interest as models for multi-agent artificial intelligence scenarios, with rational behavior most commonly characterized by Nash equilibria. This paper presents efficient, exact algorithms for computing Nash equilibria in structured game representations, including both graphical games and multi-agent influence diagrams (MAIDs). The algorithms are derived from a continuation method for normal-form and extensive-form games due to Govindan and Wilson; they follow a trajectory through a space of perturbed games and their equilibria, exploiting game structure through fast computation of the Jacobian of the payoff function. They are theoretically guaranteed to find at least one equilibrium of the game, and may find more. Our approach provides the first efficient algorithm for computing exact equilibria in graphical games with arbitrary topology, and the first algorithm to exploit fine-grained structural properties of MAIDs. Experimental results are presented demonstrating the effectiveness of the algorithms and comparing them to predecessors. The running time of the graphical game algorithm is similar to, and often better than, the running time of previous approximate algorithms. The algorithm for MAIDs can effectively solve games that are much larger than those solvable by previous methods

Strategic Equilibrium

An outcome in a noncooperative game is said to be self-enforcing, or a strategic equilibrium, if, whenever it is recommended to the players, no player has an incentive to deviate from it.This paper gives an overview of the concepts that have been proposed as formalizations of this requirement and of the properties and the applications of these concepts.In particular the paper discusses Nash equilibrium, together with its main coarsenings (correlated equilibrium, rationalizibility) and its main refinements (sequential, perfect, proper, persistent and stable equilibria).There is also an extensive discussion on equilibrium selection.noncooperative games;equilibrium analysis

Essays on conventions in games and anticipation-dependent preferences

This thesis consists of three chapters. Two chapters fall into the field of game theory and one into the field of decision theory. In the first chapter, I study strategic interaction when people are familiar with the setting they interact in. In such situations, social conventions often emerge and tend to dictate how people behave. Conventions in which people disregard alternatives outside of the convention not only help people coordinate their interactions but also simplify their decision-making. Motivated by this, I develop a novel game-theoretic concept that captures outcomes that are consistent with the existence of such self-enforcing conventions. The resulting solution concept is operational and allows for decomposing games into smaller self-contained games that can be studied in isolation. In the second chapter, I ask whether behavior consistent with the just-described conventions can be given evolutionary interpretations. In such interpretations, the convention is the resulting pattern of behavior in a large population of individuals after they have interacted for some time, with their behavior adjusting over time in response to the payoffs that their actions have given in the past. These interpretations differ from the standard justification of solution concepts based on the assumption of rational individuals that have correct expectations about others’ behavior. I find that indeed these conventions admit such interpretations, and, moreover, standard notions of evolutionarily stable behavior are often consistent with the adherence to such conventions. In the last chapter, I develop a model of a decision-maker who evaluates outcomes as gains and losses relative to her recent expectations. The decision-maker forms her expectations of an uncertain future outcome by trading off the joy from anticipating a higher outcome with the risk of being disappointed by the outcome. These expectations are then taken as given when the outcome nears. Moreover, the decision-maker is loss averse in the sense that losses relative to these expectations are felt worse than same-sized gains are felt good. The main result is a complete description of the observable choices that are consistent with this behavior. More specifically, I provide necessary and sufficient conditions on choices in the form of axioms such that it is as-if the decision-maker acts as described by the model.Cette thèse se compose de trois chapitres. Deux chapitres relèvent du domaine de la théorie des jeux et un autre du domaine de la théorie de la décision. Dans le premier chapitre, j’étudie des interactions stratégiques dans un contexte où les agents connaissent le cadre dans lequel ils interagissent. Souvent dans de telles situations, des conventions sociales émergent et tendent à dicter comment les agents se comportent. Les conventions dans lesquelles les gens ne tiennent pas compte des alternatives hors de convention aident les agents non seulement à coordonner leurs interactions mais aussi à simplifier leur prise de décision. Motivé par cela, je développe un nouveau concept en théorie des jeux qui capture des résultats compatibles avec l’existence de conventions qui s’enforcent par eux-mêmes. Le concept de solution qui en résulte est opérationnel et permet de décomposer des jeux en jeux autonomes plus petits qui peuvent être étudiés isolément. Dans le deuxième chapitre, je me demande si un comportement conforme aux conventions qui viennent d’être décrites peut recevoir des interprétations évolutionnaires. Dans une telle interprétation, la convention est le modèle qui résulte du comportement d‘individus dans une grande population, après qu’ils ont interagi pendant un certain temps, leur comportement s’ajustant au fil du temps en réponse au paiement que leurs actions ont généré dans le passé. Ces interprétations diffèrent de la justification standard des concepts de solution basée sur l’hypothèse d’individus rationnels qui ont des attentes correctes sur le comportement des autres. Je prouve dans ce chapitre qu’en effet ces conventions admettent de telles interprétations, et les notions standard de comportement stable sur le plan de l’évolution sont souvent compatibles avec l’adhésion à de telles conventions. Dans le dernier chapitre, je développe un modèle de décideur qui évalue les résultats comme des gains et des pertes par rapport à ses anticipations récentes. Le décideur forme ses anticipations d’un résultat futur et incertain, en échangeant la joie d’anticiper un résultat plus élevé avec le risque d’être déçu par le résultat. Ces anticipations sont alors considérées comme données lorsque la réalisation du résultat approche. De plus, le décideur est averse aux pertes dans le sens où les pertes par rapport à ces attentes sont ressenties pire que les gains de même taille. Le résultat principal est une description complète des choix observables qui sont cohérents avec ce comportement. Plus précisément, je fournis des conditions nécessaires et suffisantes sur les choix sous forme d’axiomes, comme si le décideur agissait comme décrit par le modèle