Maximizing the variance of the time to ruin in a multiplayer game with selection

Abstract

We consider a game with K ≥ 2 players, each having an integer-valued fortune. On each round, a pair (i,j) among the players with nonzero fortunes is chosen to play and the winner is decided by flipping a fair coin (independently of the process up to that time). The winner then receives a unit from the loser. All other players' fortunes remain the same. (Once a player's fortune reaches 0, this player is out of the game.) The game continues until only one player wins all. The choices of pairs represent the control present in the problem. While it is known that the expected time to ruin (i.e. expected duration of the game) is independent of the choices of pairs (i,j) (the strategies), our objective is to find a strategy which maximizes the variance of the time to ruin. We show that the maximum variance is uniquely attained by the (optimal) strategy, which always selects a pair of players who have currently the largest fortunes. An explicit formula for the maximum value function is derived. By constructing a simple martingale, we also provide a short proof of a result of Ross (2009) that the expected time to ruin is independent of the strategies. A brief discussion of the (open) problem of minimizing the variance of the time to ruin is given.