Issue
EPL
Volume 78, Number 2, April 2007
Article Number 20005
Number of page(s) 6
Section General
DOI http://dx.doi.org/10.1209/0295-5075/78/20005
Published online 03 April 2007
EPL, 78 (2007) 20005
DOI: 10.1209/0295-5075/78/20005

Two-population replicator dynamics and number of Nash equilibria in matrix games

T. Galla

The Abdus Salam International Centre for Theoretical Physics - Strada Costiera 11, 34014 Trieste, Italy and CNR-INFM, Trieste-SISSA Unit - V. Beirut 2-4, 34014 Trieste, Italy


received 9 September 2006; accepted in final form 7 March 2007; published April 2007
published online 3 April 2007

Abstract
We study the connection between the evolutionary replicator dynamics and the number of Nash equilibria in large random bi-matrix games. Using techniques of disordered systems theory we compute the statistical properties of both, the fixed points of the dynamics and the Nash equilibria. Except for the special case of zero-sum games, one finds a transition as a function of the so-called co-operation pressure between a phase in which there is a unique stable fixed point of the dynamics coinciding with a unique Nash equilibrium, and an unstable phase in which there are exponentially many Nash equilibria with statistical properties different from the stationary state of the replicator equations. Our analytical results are confirmed by numerical simulations of the replicator dynamics, and by explicit enumeration of Nash equilibria.

PACS
02.50.Le - Decision theory and game theory.
75.10.Nr - Spin-glass and other random models.
87.23.Kg - Dynamics of evolution.

© Europhysics Letters Association 2007