Cooperative Search Games: Symmetric Equilibria, Robustness, and Price of Anarchy
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Assume that a treasure is placed in one of M boxes according to a known distribution and that k searchers are searching for it in parallel during T rounds. We study the question of how to incentivize selfish players so that the success probability, namely, the probability that at least one player finds the treasure, would be maximized. We focus on congestion policies C(s) that specify the reward that a player receives if it is one of s players that (simultaneously) find the treasure for the first time. We show that the exclusive policy, in which C(1)=1 and C(s)=0 for s>1, yields a price of anarchy of (1-(1-1/k)^k)^-1. This is the best possible price among all symmetric reward mechanisms. For this policy, we also have an explicit description of a symmetric equilibrium, which is in some sense unique, and moreover enjoys the best success probability among all symmetric profiles. For general congestion policies, we show how to polynomially find, for any θ>0, a symmetric multiplicative (1+θ)(1+C(k))-equilibrium. Together with a reward policy, a central entity can suggest players to play a particular profile at equilibrium. For such purposes, we advocate the use of symmetric equilibria. Besides being fair, symmetric equilibria can also become highly robust to crashes of players. Indeed, in many cases, despite the fact that some small fraction of players crash, symmetric equilibria remain efficient in terms of their group performances and, at the same time, serve as approximate equilibria. We show that this principle holds for a class of games, which we call monotonously scalable games. This applies in particular to our search game, assuming the natural sharing policy, in which C(s)=1/s. For the exclusive policy, this general result does not hold, but we show that the symmetric equilibrium is nevertheless robust under mild assumptions.
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