A Random-Player Game and Derangement Numbers

Abstract

Consider the following game between a random player R and a deterministic player D. There is a pile of n elements at the beginning. The rules for playing are as follows: In each turn of R, if the pile contains exactly m elements, R removes k elements from the pile, where k is independently identically distributed from {1, . . . , m}. In each turn of D, D removes only one element. The winner is the player that, at the end of its round, has no elements remaining. R starts first to play. This short paper shows that Dn, which is defined as the probability of D winning the game (when is initialized with n elements), approaches 1/e when n increases; and more specifically, Dn = dn/n!, where dn is the n-th derangement number.