Asynchronous Best-Reply Dynamics Noam Nisan 1 , Michael Schapira 2 , and Aviv Zohar 2 1 Google Tel-Aviv and The School of Computer Science and Engineering, The Hebrew University of Jerusalem, Israel. 2 The School of Computer Science and Engineering, The Hebrew University of Jerusalem, Israel. {noam,mikesch,avivz}@cs.huji.ac.il Abstract. In many real-world settings (e.g., interdomain routing in the Internet) strategic agents are instructed to follow best-reply dynamics in asynchronous environments. In such settings players learn of each other’s actions via update messages that can be delayed or even lost. In particular, several players might update their actions simultaneously, or make choices based on outdated information. In this paper we analyze the convergence of best- (and better-)reply dynamics in asynchronous environments. We provide sufficient conditions, and necessary conditions for convergence in such settings, and also study the convergence-rate of these natural dynamics. 1 Introduction Many real-life protocols can be regarded as executions of best-reply dynamics, i.e, players (computational nodes) are instructed to repeatedly best-reply to the actions of other players. In many cases, like Internet settings, this occurs in asynchronous environments: Think of the players as residing in a computer network, where their best-replies are transmitted to other players and serve as the basis for the other players’ best-replies. These update messages that players send to each other may be delayed or even lost, and so players may update their actions simultaneously, and do so based on outdated information. Perhaps the most notable example for this is the Border Gateway Protocol (BGP) that handles interdomain routing in the Internet. As observed in [1], BGP can indeed be seen as an execution of best-reply dynamics in asynchronous environments. Asynchronous best-reply dynamics. The most fundamental question re- garding best-reply dynamics in asynchronous settings is “When are such dy- namics guaranteed to converge?”. This will certainly not happen if a pure Nash equilibrium does not exist, but is not guaranteed even in very simple and well- structured games that have a pure Nash. We present a formal framework for the analysis of best-reply dynamics in asynchronous environments. We then exhibit a simple class of games for which convergence to a unique pure Nash equilib- rium is guaranteed. We term this class, which contains all strictly-dominance- solvable games (games where iterated elimination of strictly dominated strategies leaves a single strategy profile [2]), “max-solvable-games ”. We also discuss the