The gambler's ruin problem with \(n\) players and asymmetric play (Q1962166)

The aim of the paper is to investigate the \(n\)-player gamble's ruin problem with initial fortunes \(a_1,\dots,a_n\), respectively. For the \(n\)-player asymmetric game, the authors derive a closed-form formula for the expected time until ruin, when the initial fortune of each player is equal to \(n\), and when each initial fortune is equal to \(n+1\). There is also computed the probability that the player \(i\) is ruined, \(i=1,2,\dots,n\), in the \(n+1\) cases, \(a_1=a_2= \cdots=a_n=r\), \(1\leq r\leq n+1\). Independence of the ruin time which player is ruined with is also proved, and it is conjectured that this independence holds as well for equal initial fortunes greater than \(n+1\).