Multi-round cooperative search games with multiple players

A treasure is placed in one of M boxes according to a known distribution and k searchers are searching for it in parallel during T rounds. How can one incentivize selfish players so that the probability that at least one player finds the treasure is maximized? We focus on congestion policies C(ℓ) specifying the reward a player receives being one of the ℓ players that (simultaneously) find the treasure first. We prove that the exclusive policy, in which C(1)=1 and C(ℓ)=0 for ℓ>1, yields a price of anarchy of (1−(1−1/k)^k)⁻¹, which is the best among all symmetric reward policies. We advocate the use of symmetric equilibria, and show that besides being fair, they are highly robust to crashes of players. Indeed, in many cases, if some small fraction of players crash, symmetric equilibria remain efficient in terms of their group performance while also serving as approximate equilibria.