Intense Competition can Drive Selfish Explorers to Optimize Coverage

We consider a game-theoretic setting in which selfish individuals compete over resources of varying quality. The motivating example is a group of animals that disperse over patches of food of different abundances. In such scenarios, individuals are biased towards selecting the higher quality patches, while, at the same time, aiming to avoid costly collisions or overlaps. Our goal is to investigate the impact of collision costs on the parallel coverage of resources by the whole group. Consider M sites, where a site x has value f(x). We think of f(x) as the reward associated with site x, and assume that if a single individual visits x exclusively, it receives this exact reward. Typically, we assume that if l>1 individuals visit x then each receives at most f(x)/l. In particular, when competition costs are high, each individual might receive an amount strictly less than f(x)/l, which could even be negative. Conversely, modeling cooperation at a site, we also consider cases where each one gets more than f(x)/l. There are k identical players that compete over the rewards. They independently act in parallel, in a one-shot scenario, each specifying a single site to visit, without knowing which sites are explored by others. The group performance is evaluated by the expected coverage, defined as the sum of f(x) over all sites that are explored by at least one player. Since we assume that players cannot coordinate before choosing their site we focus on symmetric strategies. The main takeaway message of this paper is that the optimal symmetric coverage is expected to emerge when collision costs are relatively high, so that the following "Judgment of Solomon" type of rule holds: If a single player explores a site x then it gains its full reward f(x), but if several players explore it, then neither one receives any reward. Under this policy, it turns out that there exists a unique symmetric Nash Equilibrium strategy, which is, in fact, evolutionary stable. Moreover, this strategy yields the best possible coverage among all symmetric strategies. Viewing the coverage measure as the social welfare, this policy thus enjoys a (Symmetric) Price of Anarchy of precisely 1, whereas, in fact, any other congestion policy has a price strictly greater than 1. Our model falls within the scope of mechanism design, and more precisely in the area of incentivizing exploration. It finds relevance in evolutionary ecology, and further connects to studies on Bayesian parallel search algorithms.