Europhys. Lett, 48 (2), pp. 129-135 (1999)
Entropy and typical properties of Nash equilibria in two-player games
J. Berg 1 and M. Weigt 2
1 Institut für Theoretische Physik,
PF 4120, 39106 Magdeburg, Germany
2 Laboratoire de Physique Théorique, Ecole Normale Supérieure
24 rue Lhomond, 75231 Paris cedex 05, France
(received 17 May 1999; accepted in final form 30 August 1999)
PACS. 05.20 - Classical statistical mechanics.
PACS. 02.50Le - Decision theory and game theory.
PACS. 64.60Cn - Order-disorder transformations; statistical mechanics of model systems.
We use techniques from the statistical mechanics of disordered systems to analyse the properties of Nash equilibria of bimatrix games with large random payoff matrices. By means of an annealed bound, we calculate their number and analyse the properties of typical Nash equilibria, which are exponentially dominant in number. We find that a randomly chosen equilibrium realizes almost always equal payoffs to either player. This value and the fraction of strategies played at an equilibrium point are calculated as a function of the correlation between the two payoff matrices. The picture is complemented by the calculation of the properties of Nash equilibria in pure strategies.
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