The Curse of Simultaneity

Typical models of strategic interactions in computer science use simultaneous move games. However, in applications simultaneity is often hard or impossible to achieve. In this paper, we study the robustness of the Nash Equilibrium when the assumption of simultaneity is dropped. In particular we propose studying the sequential price of anarchy: the quality of outcomes of sequential versions of games whose simultaneous counterparts are prototypical in algorithmic game theory. We study different classes of games with high price of anarchy, and show that the subgame perfect equilibrium of their sequential version is a much more natural prediction, ruling out unreasonable equilibria, and leading to much better quality solutions. We consider three examples of such games: Cost Sharing Games, Unrelated Machine Scheduling Games and Consensus Games. For Machine Cost Sharing Games, the sequential price of anarchy is at most O(log(n)), an exponential improvement of the O(n) price of anarchy of their simultaneous counterparts. Further, the subgame perfect equilibrium can be computed by a polynomial time greedy algorithm, and is independent of the order the players arrive. For Unrelated Machine Scheduling Games we show that the sequential price of anarchy is bounded as a function of the number of jobs n and machines m (by at most O(m2 n )), while in the simultaneous version the price of anarchy is unbounded even for two players and two machines. For Consensus Games we observe that the optimal outcome for generic weights is the unique equilibrium that arises in the sequential game. We also study the related Cut Games, where we show that the sequential price of anarchy is at most 4. In addition we study the complexity of finding the subgame perfect equilibrium outcome in these games