Games of fixed rank: a hierarchy of bimatrix games

We propose and investigate a hierarchy of bimatrix games (A, B), whose (entry-wise) sum of the pay-off matrices of the two players is of rank k, where k is a constant. We will say the rank of such a game is k. For every fixed k, the class of rank k-games strictly generalizes the class of zero-sum ga...

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Bibliographic Details
Published in:Economic theory Vol. 42; no. 1; pp. 157 - 173
Main Authors: Kannan, Ravi, Theobald, Thorsten
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Berlin/Heidelberg : Springer-Verlag 01.01.2010
Springer
Springer-Verlag
Springer Nature B.V
Subjects:
Algorithms
Approximation
Approximation algorithms
Bimarix game
Computer science
Economic behaviour
Economic theory
Economic Theory/Quantitative Economics/Mathematical Methods
Economics
Economics and Finance
Equilibrium
Fixed rank
Game Theory
Games
Hierarchies
Hierarchy
Linear programming
Mathematics
Matrices
Matrix algebra
Microeconomics
Mixed strategy
Nash equilibrium
Pay-off
Polynomials
Public Finance
Social and Behav. Sciences
Studies
Symposium
Symposium on Computation of Nash Equilibria in Finite Games
Vertices
Zero sum games
C63
Nash equilibrium
Approximation
Fixed rank
Bimarix game
C72
ISSN:0938-2259, 1432-0479
Online Access:Get full text
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Summary:We propose and investigate a hierarchy of bimatrix games (A, B), whose (entry-wise) sum of the pay-off matrices of the two players is of rank k, where k is a constant. We will say the rank of such a game is k. For every fixed k, the class of rank k-games strictly generalizes the class of zero-sum games, but is a very special case of general bimatrix games. We study both the expressive power and the algorithmic behavior of these games. Specifically, we show that even for k = 1 the set of Nash equilibria of these games can consist of an arbitrarily large number of connected components. While the question of exact polynomial time algorithms to find a Nash equilibrium remains open for games of fixed rank, we present polynomial time algorithms for finding an ε-approximation.
Bibliography:http://dx.doi.org/10.1007/s00199-009-0436-2
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ISSN:0938-2259
1432-0479
DOI:10.1007/s00199-009-0436-2