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 (denoted δ) is taken into account, distributed best response dynamics may suffer from overlaps: one player starts to play while another player has not changed its strategy yet. Overlaps may lead to drops 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 can be bounded from above by e γ δn log n log log n (1 + o(1)) and from below by δn log n log log n (1 + o(1)), where γ 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 bounds show that this λ is high: ˆ λ = log log n−log log log n δ. This induces a large probability of overlap (ˆ p = 1 − log log n/ log n). In practice, numerical simulations also show that using high playing rates is efficient, with an optimal probability of overlap p opt ≈ 0.78 up to n = 250. This implies that best response dynamics are unexpectedly efficient to compute Nash equilibria, even in a distributed setting.