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Assume that a treasure is placed in one of $M$ boxes according to a knowndistribution 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 thesuccess probability, namely, the probability that at least one player finds thetreasure, would be maximized. We focus on congestion policies $C(s)$ thatspecify the reward that a player receives if it is one of $s$ players that(simultaneously) find the treasure for the first time. Our main technicalcontribution is proving that the exclusive policy, in which $C(1)=1$ and$C(s)=0$ for $s>1$, yields a price of anarchy of $(1-(1-{1}/{k})^{k})^{-1}$,and that this is the best possible price among all symmetric reward mechanisms.For this policy we also have an explicit description of a symmetricequilibrium, which is in some sense unique, and moreover enjoys the bestsuccess probability among all symmetric profiles. For general congestionpolicies, we show how to polynomially find, for any $e>0$, a symmetricmultiplicative $(1+e)(1+C(k))$-equilibrium. Together with an appropriate rewardpolicy, a central entity can suggest players to play a particular profile atequilibrium. As our main conceptual contribution, we advocate the use ofsymmetric equilibria for such purposes. Besides being fair, we argue that inmany 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 principleholds for a class of games, which we call monotonously scalable games. Thisapplies in particular to our search game, assuming the sharing policy, in which$C(s)=1/s$. For the exclusive policy, this general result does not hold, but weshow that the symmetric equilibrium is nevertheless robust under mildassumptions.

Multi-Round Cooperative Search Games with Multiple Players