2,680 research outputs found
Finding All Nash Equilibria of a Finite Game Using Polynomial Algebra
The set of Nash equilibria of a finite game is the set of nonnegative
solutions to a system of polynomial equations. In this survey article we
describe how to construct certain special games and explain how to find all the
complex roots of the corresponding polynomial systems, including all the Nash
equilibria. We then explain how to find all the complex roots of the polynomial
systems for arbitrary generic games, by polyhedral homotopy continuation
starting from the solutions to the specially constructed games. We describe the
use of Groebner bases to solve these polynomial systems and to learn geometric
information about how the solution set varies with the payoff functions.
Finally, we review the use of the Gambit software package to find all Nash
equilibria of a finite game.Comment: Invited contribution to Journal of Economic Theory; includes color
figure
Note on maximally entangled Eisert-Lewenstein-Wilkens quantum games
Maximally entangled Eisert-Lewenstein-Wilkens games are analyzed. For a
general class of gate operators defined in the previous papers of the first
author the general conditions are derived which allow to determine the form of
gate operators leading to maximally entangled games. The construction becomes
particularly simple provided one does distinguish between games differing by
relabelling of strategies. Some examples are presented.Comment: 20 pages, no figures, appendix added, references added, concluding
remarks extende
Computing Equilibria of Semi-algebraic Economies Using Triangular Decomposition and Real Solution Classification
In this paper, we are concerned with the problem of determining the existence
of multiple equilibria in economic models. We propose a general and complete
approach for identifying multiplicities of equilibria in semi-algebraic
economies, which may be expressed as semi-algebraic systems. The approach is
based on triangular decomposition and real solution classification, two
powerful tools of algebraic computation. Its effectiveness is illustrated by
two examples of application.Comment: 24 pages, 5 figure
Approximate well-supported Nash equilibria in symmetric bimatrix games
The -well-supported Nash equilibrium is a strong notion of
approximation of a Nash equilibrium, where no player has an incentive greater
than to deviate from any of the pure strategies that she uses in
her mixed strategy. The smallest constant currently known for
which there is a polynomial-time algorithm that computes an
-well-supported Nash equilibrium in bimatrix games is slightly
below . In this paper we study this problem for symmetric bimatrix games
and we provide a polynomial-time algorithm that gives a
-well-supported Nash equilibrium, for an arbitrarily small
positive constant
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