A zero-sum competitive multi-player game

A single period, zero-sum, multi-player game is constructed. Each player can either exit the game for a fixed payoff or stay and split the remaining payoff with the other non-exiting players. The emphasis is put on the rivalrous nature of the payoffs, meaning that the sum of all payoffs is fixed, but the exact allocation is based on the players' decisions. The value at which Nash and optimal equilibria are attained is shown to be unique and it is constructed explicitly.