# Alternate Fictitious Play study and implementation

## Abstract

Nash equilibria computation in finite games is a problem which is known to be PPAD-complete, which means it currently seems impossible to find an efficient solution ; worst case complexity of well known algorithms is in ${\displaystyle 2^{0(n)}}$ for any game of size ${\displaystyle n}$. For this reasonresearch in this domain currently focuses on ${\displaystyle \varepsilon }$-equilibria, situations which approximately satisfy the conditions of a Nash equilibrium. The emphFictitious Play algorithm fits in this approach. At each iteration of this algorithm, each of the players “strengthens” the strategy that has the highest utility in the current context. For some specific game classes this algorithm converges to a Nash equilibrium, therefore providing an efficient approximation method. However, convergence can only be proved for a small amount of game classes. It is therefore useful to study other algorithms (based on Fictitious Play) in order to find other convergence cases. In this study, we will focus on the alternate Fictitious Play algorithm, in which only one player at a time strengthens one of his strategies : the player which is the “further” from his maximum payoff.