On a Generalized Ruin Problem

Abstract

We consider a natural generalization of the classical ruin problem to more than two parties. Our “ruin” problem, which we will call the (k, I)-game, starts with k players each having I units as its initial capital. At each round of the game, all remaining k′ players pay 1/k′th unit as game fee, play the game, and one of the players wins and receives the combined game fees of 1 unit. A player who cannot pay the next game fee goes bankrupt, and the game terminates when all players but one are bankrupt. We analyze the length of the game, that is, the number of rounds executed until the game terminates, and give upper and lower bounds for the expected game length.