Efficiency of Best Response Dynamics with High Playing Rates in Potential Games

Abstract

In this paper we design and analyze distributed best response dynamics to compute Nash equilibria in potential games. This algorithm uses local Poisson clocks for each player and does not rely on the usual but unrealistic assumption that players take no time to compute their best response. If this time (denotedδ) is taken into account, distributed best response dynamics (BRD) may suffer from overlaps: one player starts to play while another player has not changed its strategy yet. An overlap may lead to a decrease of the potential but we can show that they do not jeopardize eventual convergence to a Nash equilibrium. Our main result is to use a Markovian approach to show that the average execution time of the algorithm E[TBRD] can be bounded: 2δn log n/ log log n + O(n) ≤E[TBRD] ≤ 4e δn log n/ log log n+O(n), where ' is the Euler constant, n is the number of players and δ is the time taken by one player to compute its best response. These bounds are obtained by using an asymptotically optimal playing rate λ. Our analytic bound shows that 2δλ = log log n - log C, where C is a constant. This induces a large probability of overlap (p = 1 - C/ log1/2 n). In practice, numerical simulations also show that using high playing rates is efficient, with an optimal probability of overlap popt ≈ 0.78, for n up to 250. This shows that best response dynamics are unexpectedly efficient to compute Nash equilibria, even in a distributed setting.