  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. Our main technical
contribution 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 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 $e>0$, a symmetric
multiplicative $(1+e)(1+C(k))$-equilibrium. Together with an appropriate reward
policy, a central entity can suggest players to play a particular profile at
equilibrium. As our main conceptual contribution, we advocate the use of
symmetric equilibria for such purposes. Besides being fair, we argue that 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 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.
