## Abstract

We consider reinforcement learning algorithms in normal form games. Using two-time-scales stochastic approximation, we introduce a model-free algorithm which is asymptotically equivalent to the smooth fictitious play algorithm, in that both result in asymptotic pseudotrajectories to the flow defined by the smooth best response dynamics. Both of these algorithms are shown to converge almost surely to Nash distribution in two-player zero-sum games and N-player partnership games. However, there are simple games for which these, and most other adaptive processes, fail to converge â€” in particular, we consider the N-player matching pennies game and Shapleyâ€™s variant of the rockâ€“scissorsâ€“paper game. By extending stochastic approximation results to multiple time scales we can allow each player to learn at a different rate. We show that this extension will converge for two-player zero-sum games and two-player partnership games, as well as for the two special cases we consider.

Translated title of the contribution | Convergent multiple-timescales reinforcement learning algorithms in normal form games |
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Original language | English |

Pages (from-to) | 1231 - 1251 |

Number of pages | 21 |

Journal | Annals of Applied Probability |

Volume | 13 (4) |

DOIs | |

Publication status | Published - Nov 2003 |