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 γ N + O(N ), where γ 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 e γ ), 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.
It has been shown in REF that in random potential games with n players in which every player has at most a strategies, the worst case convergence time is n · a n−1 .
Complexity and Optimality of the Best Response Algorithm in Random Potential Games
SAGT
Mathematics Computer Science