Variance of the asymmetric n-player gambler's ruin problem with ties allowed

In this article, the variance of the duration of play in the asymmetric n-player gambler's ruin problem is considered, when the players use equal initial fortunes of d dollars, 1 d n + 1, and ties allowed in each round. Some special games are simulated and the simulation results verify the validity of the proposed formulas. It is shown that when we do not have the possibility of tie in the game, the increase in the number of players will change the ruin time from a random variable to a degenerate random variable. Finally, the three-tower problem with one of its different definitions are introduced and their expected times as well as their variances of the duration are considered.