John von Neumann's Contribution to Modern Game Theory

2004 ◽  
Vol 54 (1) ◽  
pp. 73-84 ◽  
Author(s):  
Ferenc Forgó
Keyword(s):  
Game Theory ◽  
Cooperative Games ◽  
Minimax Theorem ◽  
Stable Set ◽  
Zero Sum Games ◽  
Side Payments ◽  
Basic Concepts ◽  
Definition Of ◽  
Zero Sum ◽  
Abstract Games

The paper gives a brief account of von Neumann's contribution to the foundation of game theory: definition of abstract games, the minimax theorem for two-person zero-sum games and the stable set solution for cooperative games with side payments. The presentation is self-contained, uses very little mathematical formalism and caters to the nonspecialist. Basic concepts and their implications are in focus. It is also indicated how von Neumann's groundbreaking work initiated further research, and a few unsolved problems are also mentioned.

MONTE CARLO METHODS IN FUZZY GAME THEORY

2007 ◽  
Vol 03 (02) ◽  
pp. 259-269 ◽  
Author(s):  
AREEG ABDALLA ◽  
JAMES BUCKLEY
Keyword(s):  
Game Theory ◽  
Monte Carlo ◽  
Monte Carlo Method ◽  
Monte Carlo Methods ◽  
Mixed Strategy ◽  
Minimax Theorem ◽  
Approximate Solutions ◽  
Mixed Strategies ◽  
Fuzzy Game ◽  
Zero Sum

In this paper, we consider a two-person zero-sum game with fuzzy payoffs and fuzzy mixed strategies for both players. We define the fuzzy value of the game for both players [Formula: see text] and also define an optimal fuzzy mixed strategy for both players. We then employ our fuzzy Monte Carlo method to produce approximate solutions, to an example fuzzy game, for the fuzzy values [Formula: see text] for Player I and [Formula: see text] for Player II; and also approximate solutions for the optimal fuzzy mixed strategies for both players. We then look at [Formula: see text] and [Formula: see text] to see if there is a Minimax theorem [Formula: see text] for this fuzzy game.

scholarly journals A computational model of outguessing in two-player non-cooperative games

2014 ◽  
Vol 6 (1) ◽  
pp. 71-88
Author(s):  
Tamás László Balogh ◽  
János Kormos
Keyword(s):  
Numerical Simulation ◽  
Computational Model ◽  
Cooperative Games ◽  
Behavioral Game Theory ◽  
Equilibrium Concept ◽  
Zero Sum Games ◽  
Simulation Based ◽  
Behavioral Game ◽  
Zero Sum ◽  
Non Cooperative Games

Abstract Several behavioral game theory models aim at explaining why “smarter“ people win more frequently in simultaneous zero-sum games, a phanomenon, which is not explained by the Nash equilibrium concept. We use a computational model and a numerical simulation based on Markov chains to describe player behavior and predict payoffs.

Game theory and its economic applications

2018 ◽  
Author(s):  
Анатолий Сигал ◽  
Anatoliy Sigal
Keyword(s):  
Game Theory ◽  
Matrix Games ◽  
Managerial Decisions ◽  
Combined Application ◽  
Finite Games ◽  
Basic Concepts ◽  
Zero Sum ◽  
Risk Managers ◽  
Large Groups ◽  
Antagonistic Games

The manual describes the main sections of game theory, the basic concepts of the theory of economic risks and the conceptual framework for modeling the process of making managerial decisions in the economy based on the combined application of statistical and antagonistic games. Antagonistic games (AG) are the finite games of two persons with zero sum, i.e. matrix games, with classical antagonistic games called AG, given by completely known matrices, and neoclassical antagonistic games – AG, given by partially known matrices. The manual is intended primarily for masters who study in the direction of training "business information". However, it will be useful for students and postgraduates studying in large groups of areas of training "Economics and Management", "Mathematics and Mechanics", "Management in Technical Systems", as well as scientific and pedagogical workers specializing in the field of the theory of games, risk managers , managers and economists-practitioners.

Learning Nash Equilibria in Non-Cooperative Games

2011 ◽  
pp. 1018-1023 ◽  
Author(s):  
Alfredo Garro
Keyword(s):  
Game Theory ◽  
Nash Equilibrium ◽  
Cooperative Games ◽  
Nash Equilibria ◽  
Applied Mathematics ◽  
Solution Concept ◽  
The Other ◽  
Political Sciences ◽  
Definition Of ◽  
Non Cooperative Games

Game Theory (Von Neumann & Morgenstern, 1944) is a branch of applied mathematics and economics that studies situations (games) where self-interested interacting players act for maximizing their returns; therefore, the return of each player depends on his behaviour and on the behaviours of the other players. Game Theory, which plays an important role in the social and political sciences, has recently drawn attention in new academic fields which go from algorithmic mechanism design to cybernetics. However, a fundamental problem to solve for effectively applying Game Theory in real word applications is the definition of well-founded solution concepts of a game and the design of efficient algorithms for their computation. A widely accepted solution concept of a game in which any cooperation among the players must be selfenforcing (non-cooperative game) is represented by the Nash Equilibrium. In particular, a Nash Equilibrium is a set of strategies, one for each player of the game, such that no player can benefit by changing his strategy unilaterally, i.e. while the other players keep their strategies unchanged (Nash, 1951). The problem of computing Nash Equilibria in non-cooperative games is considered one of the most important open problem in Complexity Theory (Papadimitriou, 2001). Daskalakis, Goldbergy, and Papadimitriou (2005), showed that the problem of computing a Nash equilibrium in a game with four or more players is complete for the complexity class PPAD-Polynomial Parity Argument Directed version (Papadimitriou, 1991), moreover, Chen and Deng extended this result for 2-player games (Chen & Deng, 2005). However, even in the two players case, the best algorithm known has an exponential worst-case running time (Savani & von Stengel, 2004); furthermore, if the computation of equilibria with simple additional properties is required, the problem immediately becomes NP-hard (Bonifaci, Di Iorio, & Laura, 2005) (Conitzer & Sandholm, 2003) (Gilboa & Zemel, 1989) (Gottlob, Greco, & Scarcello, 2003). Motivated by these results, recent studies have dealt with the problem of efficiently computing Nash Equilibria by exploiting approaches based on the concepts of learning and evolution (Fudenberg & Levine, 1998) (Maynard Smith, 1982). In these approaches the Nash Equilibria of a game are not statically computed but are the result of the evolution of a system composed by agents playing the game. In particular, each agent after different rounds will learn to play a strategy that, under the hypothesis of agent’s rationality, will be one of the Nash equilibria of the game (Benaim & Hirsch, 1999) (Carmel & Markovitch, 1996). This article presents SALENE, a Multi-Agent System (MAS) for learning Nash Equilibria in noncooperative games, which is based on the above mentioned concepts.

scholarly journals A robust von Neumann minimax theorem for zero-sum games under bounded payoff uncertainty

2011 ◽  
Vol 39 (2) ◽  
pp. 109-114 ◽  
Author(s):  
V. Jeyakumar ◽  
G.Y. Li ◽  
G.M. Lee
Keyword(s):  
Minimax Theorem ◽  
Von Neumann ◽  
Zero Sum Games ◽  
Zero Sum ◽  
Payoff Uncertainty

Solving Two-Person Zero-Sum Game by Matlab

2011 ◽  
Vol 50-51 ◽  
pp. 262-265 ◽  
Author(s):  
Yan Mei Yang ◽  
Yan Guo ◽  
Li Chao Feng ◽  
Jian Yong Di
Keyword(s):  
Game Theory ◽  
Linear Programming ◽  
Special Class ◽  
The Other ◽  
Matrix Game ◽  
Matlab Software ◽  
Zero Sum Games ◽  
Programming Techniques ◽  
Zero Sum

In this article we present an overview on two-person zero-sum games, which play a central role in the development of the theory of games. Two-person zero-sum games is a special class of game theory in which one player wins what the other player loses with only two players. It is difficult to solve 2-person matrix game with the order m×n(m≥3,n≥3). The aim of the article is to determine the method on how to solve a 2-person matrix game by linear programming function linprog() in matlab. With linear programming techniques in the Matlab software, we present effective method for solving large zero-sum games problems.

scholarly journals Justifying optimal play via consistency

2019 ◽  
Vol 14 (4) ◽  
pp. 1185-1201
Author(s):  
Florian Brandl ◽  
Felix Brandt
Keyword(s):  
Nash Equilibrium ◽  
Minimax Theorem ◽  
Bimatrix Games ◽  
Zero Sum Games ◽  
Zero Sum ◽  
Normative Foundations ◽  
Coherent Behavior

Developing normative foundations for optimal play in two‐player zero‐sum games has turned out to be surprisingly difficult, despite the powerful strategic implications of the minimax theorem. We characterize maximin strategies by postulating coherent behavior in varying games. The first axiom, called consequentialism, states that how probability is distributed among completely indistinguishable actions is irrelevant. The second axiom, consistency, demands that strategies that are optimal in two different games should still be optimal when there is uncertainty regarding which of the two games will actually be played. Finally, we impose a very mild rationality assumption, which merely requires that strictly dominated actions will not be played. Our characterization shows that a rational and consistent consequentialist who ascribes the same properties to his opponent has to play maximin strategies. This result can be extended to characterize Nash equilibrium in bimatrix games whenever the set of equilibria is interchangeable.

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