The existence of a pure-strategy Nash equilibrium in a discrete ponds dilemma

In a variety of economic situations discrete agents choose one resource among several available resources and, once admitted to the resource of choice, divide it among fellow agents admitted there. The amount of the resource an agent gets is proportional to her relative ability to acquire this particular resource, what we refer to as an agent's weight at the resource. The relevant applications include students self-selecting into colleges, politicians self-selecting into races, and athletes self-selecting into teams. We find that this game has a pure-strategy Nash equilibrium in at least three special cases: 1) when agents have the same weight at each resource, 2) when all resources are the same, 3) when there are only two resources. We also show that this game always has an approximate Nash equilibrium when the number of players is large. Existence in the general case remains an open problem.