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.
The average case complexity of a random potential game was shown to be polynomial in the number of players and strategies by Durand and Gaujal REF .
Complexity and Optimality of the Best Response Algorithm in Random Potential Games
SAGT
Mathematics Computer Science