Effects of competition in a secretary problem

In a novel multiplayer extension of the famous secretary problem, multiple players seek to employ secretaries from a common labour pool. Secretaries do not accept being put on hold, always accept job offers immediately, and leave the labour pool once rejected by a single player. All players have an identical preference for secretaries, and all players seek to optimize the probability of obtaining the best of all n secretaries. We find that in the Nash equilibrium, as the number, N, of players searching the labour pool grows, the optimal strategy converges to a simple function of N. For the two-player case we also compute how much players can gain through cooperation and how the optimal strategy changes under a payoff structure that promotes spite.