Bounds for the expected duration of the monopolist game

Abstract

The monopolist game is a multi-player ruin process that arose in the study of certain learning processes. We prove that when the players begin with equal stakes, the expected duration of the game is, up to constant factors, the square of their collective initial wealth. This proves a conjecture of Amano, Tromp, Vitányi, and Watanabe. More generally, we find that the expected duration is similarly related to a quadratic function that reflects the uniformity of the initial stakes, and calculate the expected duration exactly for three players.