# Étude du fictitious play dans le cas d'un jeu à fonctions d'utilité identiques

### From LRDE

- Auteurs
- Jean-Philippe Garcia Ballester
- Type
- techreport
- Année
- 2008

## Résumé

Fictitious play, in game theory, is a learning rule in which each player presumes that his opponents are playing a stationary strategy —potentially mixed, i.e. a probability distribution over a set of strategies. At each round, each player thus best responds to his opponents' strategy, computed empirically using their previous moves. Convergence of such strategies is not always assured, yet we know that if there is convergence, then the strategies used will correspond statistically to a Nash-equilibrium. It is thus very interesting to know when fictitious play converges. We will address for this presentation the case where utility functions —a player's payoff with respect to the played strategies— of each player are identical. We will first study results about convergence in this special case. In order to reduce the computationnal complexity we will see a modified version of this algorithm that allows errors in players' best replies. We will finally introduce an example of use of fictitious play to solve a problem not a priori connected to game theory: an optimization problem, i.e. computing the maximum of the values taken by a function.