This textbook brings together the main developments in non-cooperative game theory from the 1950s to the present. After opening with a number of lively examples, Ritzberger starts by considering the âŠ
This textbook brings together the main developments in non-cooperative game theory from the 1950s to the present. After opening with a number of lively examples, Ritzberger starts by considering the theory of decisions under uncertainty. He then turns to representations of games, first introducing extensive forms and then normal forms. The remainder of the text is devoted to solution theory, going from basic solution concepts like rationalizable strategies, Nash equilibrium, and correlated equilibrium to refinements of Nash equilibrium. Foundations of Non-Cooperative Game Theory covers all material relevant for a first graduate course in game theory, plus some issues only touched on by other texts. In particular, this book contains an in-depth discussion of perfect recall and related concepts, including a proof of Kuhn's theorem. It provides an introduction to the Thompson transformations for extensive forms, and a section on the reflection of extensive form structures in normal form games. In addition to the standard material on basic solution concepts, strategy subsets closed under rational behavior are covered, as well as fixed sets under the best-reply correspondence. Refinements of Nash equilibrium driven by backwards induction in the extensive form or by strategy trembles in the normal form are presented, and strategic stability, the geometry of the Nash equilibrium correspondence, and index theory for Nash equilibrium components are discussed in depth. Ritzberger provides numerous examples and exercises to aid the reader's understanding, most of which are motivated by applications of game theory in economics. While advanced mathematical machinery is used on occasions, an effort has been made to include as many explanations for formal concepts as possible, making this text an invaluable tool for teachers, students, and researchers of microeconomics and game theory.
John Nash has contributed to game theory and economics two solution concepts for nonconstant sum games. One, the non-cooperative solution [9], is a generalization of the minimax theorem for two âŠ
John Nash has contributed to game theory and economics two solution concepts for nonconstant sum games. One, the non-cooperative solution [9], is a generalization of the minimax theorem for two person zero sum games and of the Cournot solution; and the other, the cooperative solution [10], is completely new. It is the purpose of this paper to present a non-cooperative equilibrium concept, applicable to supergames, which fits the Nash (non-cooperative) definition and also has some features resembling the Nash cooperative solution. âSupergameâ describes the playing of an infinite sequence of âordinary gamesâ over time. Oligopoly may profitably be viewed as a supergame. In each time period the players are in a game, and they know they will be in similar games with the same other players in future periods.
Cooperative game theory is a branch of (micro-)economics that studies the behavior of self-interested agents in strategic settings where binding agreements among agents are possible. Our aim in this b
Cooperative game theory is a branch of (micro-)economics that studies the behavior of self-interested agents in strategic settings where binding agreements among agents are possible. Our aim in this b
Cooperative game theory is a branch of (micro-)economics that studies the behavior of self-interested agents in strategic settings where binding agreements among agents are possible. Our aim in this b
Cooperative game theory is a branch of (micro-)economics that studies the behavior of self-interested agents in strategic settings where binding agreements among agents are possible. Our aim in this b
In many multiagent scenarios, agents distribute resources, such as time or energy, among several tasks. Having completed their tasks and generated profits, task payoffs must be divided among the agents âŠ
In many multiagent scenarios, agents distribute resources, such as time or energy, among several tasks. Having completed their tasks and generated profits, task payoffs must be divided among the agents in some reasonable manner. Cooperative games with overlapping coalitions (OCF games) are a recent framework proposed by Chalkiadakis et al. (2010), generalizing classic cooperative games to the case where agents may belong to more than one coalition. Having formed overlapping coalitions and divided profits, some agents may feel dissatisfied with their share of the profits, and would like to deviate from the given outcome. However, deviation in OCF games is a complicated matter: agents may decide to withdraw only some of their weight from some of the coalitions they belong to; that is, even after deviation, it is possible that agents will still be involved in tasks with non-deviators. This means that the desirability of a deviation, and the stability of formed coalitions, is to a great extent determined by the reaction of non-deviators. In this work, we explore algorithmic aspects of OCF games, focusing on the core in OCF games. We study the problem of deciding if the core of an OCF game is not empty, and whether a core payoff division can be found in polynomial time; moreover, we identify conditions that ensure that the problem admits polynomial time algorithms. Finally, we introduce and study a natural class of OCF games, Linear Bottleneck Games. Interestingly, we show that such games always have a non-empty core, even assuming a highly lenient reaction to deviations.
The core of an M-person game, though used already by von Neumann and Morgenstern [15], was first explicitly defined by Gillies [5].Gillies's definition is restricted to cooperative games with side âŠ
The core of an M-person game, though used already by von Neumann and Morgenstern [15], was first explicitly defined by Gillies [5].Gillies's definition is restricted to cooperative games with side payments and unrestrictedly transferable utilities(2), but the basic idea is very simple and natural, and appears in many approaches to game theory.We consider a certain set of "outcomes" to a game, and define a relation of "dominance"(usually not transitive) on this set.The core is then defined to be the subset of outcomes maximal with respect to the dominance relation; in other words, the subset of outcomes from which there is no tendency to move away-the equilibrium states.To turn this intuitive description of the core notion into a mathematical definition, we need precise characterizations of (a) the kind of game-theoretic situation to which we are referring (cooperative game, noncooperative game, etc.); (b) what we mean by "outcome"; and (c) what we mean by "dominance."Different ways of interpreting these three elements yield different applications of the generalized "core" notion, many of them well-known in game theory.Gillies's core, Luce's ^-stability [lO], Nash's equilibrium points [12], Nash's solution to the bargaining problem [l3](3), and the idea of Pareto optimality-to mention only some of the applications-can all be obtained in this way.Here we shall be concerned exclusively with cooperative games without side payments(4).Our procedure will be to generalize von Neumann's fundamental notion of characteristic function to this case, and on the basis of this generalization to define the core in a way that generalizes and parallels the core in the classical theory-i.e., Gillies's core.The generalization of the characteristic function is of interest for its own sake also; for example, a theory of "solutions" has been developed that generalizes and parallels the classical theory of solutions and is based on the characteristic function [3; 16].
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We introduce a new family of Parrondo's games of alternating losing strategies in order to get a winning result. In our version of the games we consider an ensemble of âŠ
We introduce a new family of Parrondo's games of alternating losing strategies in order to get a winning result. In our version of the games we consider an ensemble of players and use "social" rules in which the probabilities of the games are defined in terms of the actual state of the neighbors of a given player.
We introduce a new family of Parrondo's games of alternating losing strategies in order to get a winning result. In our version of the games we consider an ensemble of âŠ
We introduce a new family of Parrondo's games of alternating losing strategies in order to get a winning result. In our version of the games we consider an ensemble of players and use "social" rules in which the probabilities of the games are defined in terms of the actual state of the neighbors of a given player.
Cores of cooperative games are ubiquitous in information theory and arise most frequently in the characterization of fundamental limits in various scenarios involving multiple users. Examples include classical settings in âŠ
Cores of cooperative games are ubiquitous in information theory and arise most frequently in the characterization of fundamental limits in various scenarios involving multiple users. Examples include classical settings in network information theory such as Slepian-Wolf source coding and multiple access channels, classical settings in statistics such as robust hypothesis testing, and new settings at the intersection of networking and statistics such as distributed estimation problems for sensor networks. Cooperative game theory allows one to understand aspects of all these problems from a fresh and unifying perspective that treats users as players in a game, sometimes leading to new insights. At the heart of these analyses are fundamental dualities that have been long studied in the context of cooperative games; for information theoretic purposes, these are dualities between information inequalities on the one hand and properties of rate, capacity, or other resource allocation regions on the other.
Abstract : A model of a pure exchange economy is investigated without the usual assumption of convex preference sets for the participating traders. The concept of core, taken from the âŠ
Abstract : A model of a pure exchange economy is investigated without the usual assumption of convex preference sets for the participating traders. The concept of core, taken from the theory of games, is applied to show that if there are sufficiently many participants, the economy as a whole will possess a solution that is sociologically stable--i.e., that cannot profitably be upset by any coalition of traders.
Abstract This survey paper presents the basic concepts of cooperative game theory, at an elementary level. Five examples, including three insurance applications, are progressively developed throughout the paper. The characteristic âŠ
Abstract This survey paper presents the basic concepts of cooperative game theory, at an elementary level. Five examples, including three insurance applications, are progressively developed throughout the paper. The characteristic function, the core, the stable sets, the Shapley value, the Nash and Kalai-Smorodinsky solutions are defined and computed for the different examples.
The kernel of a cooperative game is a subset of the bargaining set ^d(i) .It is sensitive to symmetry relations and their generalizations, which may exist in the characteristic function.The âŠ
The kernel of a cooperative game is a subset of the bargaining set ^d(i) .It is sensitive to symmetry relations and their generalizations, which may exist in the characteristic function.The present paper offers an interesting representation formula for the kernel.This formula is applied to deriving properties of the kernel as well as practical methods for its computation.In particular, we provide an algebraic proof to the theorem stating that for each coalition structure in a cooperative game there exists a payoff in the kernel (and therefore also in the bargaining set ^#Î (i) ).(AH other known proofs of this theorem are based on the Brouwer fixed-point theorem.)We also prove that the maximal dimension of the kernel of an ?? -person game is n -[logz(n -J)] -2, and this bound is sharp.Two players in a game are called symmetric, if the game remains invariant when these players exchange roles.One generalizes this concept by defining a player k to be more desirable than a player I, if player k always contributes not less than player I by joining coalitions which contain none of these players.It turns out that the payoffs in the kernel always preserve the order determined by the desirability relations.This fact may simplify the representation formula significantly.
Abstract The kernel of a cooperative nâperson game is defined. It is a subset of the bargaining set đ (i) . Its existence and some of its properties are studied. âŠ
Abstract The kernel of a cooperative nâperson game is defined. It is a subset of the bargaining set đ (i) . Its existence and some of its properties are studied. We apply it to the 3âperson games, to the 4âperson constantâsum games, to the symmetric and nâquota games and to games in which only the n and the (nâ1)âperson coalitions are allowed to be nonâflat. In order to illustrate its merits and demerits as a predictor of an actual outcome in a realâlife situation, we exhibit an example in which the kernel prediction seems frustrating. The opinions of other authors are quoted in order to throw some light on this interesting example.
One of the properties characterizing cooperative game solutions is consistency connecting solution vectors of a cooperative game with finite set of players and its reduced game defined by removing one âŠ
One of the properties characterizing cooperative game solutions is consistency connecting solution vectors of a cooperative game with finite set of players and its reduced game defined by removing one or more players and by assigning them the payoffs according to some specific principle (e.g., a proposed payoff vector). Consistency of a solution means that any part (defined by a coalition of the original game) of a solution payoff vector belongs to the solution set of the corresponding reduced game. In the paper the proportional solutions for TU-games are defined as those depending only on the proportional excess vectors in the same manner as translation covariant solutions depend on the usual DavisâMaschler excess vectors. The general form of the reduced games defining consistent proportional solutions is given. The efficient, anonymous, proportional TU cooperative game solutions meeting the consistency property with respect to any reduced game are described.