Complexity and Optimality of the Best Response Algorithm in Random Potential Games

Abstract

In this paper we compute the worst-case and average execution time of the Best Response Algorithm (BRA) to compute a pure Nash equilibrium in finite potential games. Our approach is based on a Markov chain model of BRA and a coupling technique that transform the average execution time of this discrete algorithm into the solution of an ordinary differential equation. In a potential game with N players and A strategies per player, we show that the worst case complexity of BRA (number of moves) is exactly $N A^{N-1}$, while its average complexity over random potential games is equal to $e^\gamma N + O(N)$, where $\gamma$ is the Euler constant. We also show that the effective number of states visited by BRA is equal to $\log N + c + O(1/N)$ (with $c \leqslant e^\gamma$), on average. Finally, we show that BRA computes a pure Nash Equilibrium faster (in the strong stochastic order sense) than any local search algorithm over random potential games.