Advances in Applied Probability

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

Ilie Grigorescu and Yi-Ching Yao

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

Adv. in Appl. Probab., Volume 48, Number 2 (2016), 610-630.

First available in Project Euclid: 9 June 2016

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 91A60: Probabilistic games; gambling [See also 60G40] 93E20: Optimal stochastic control 60C05: Combinatorial probability

Gambler's ruin martingale dynamic programming stochastic control


Grigorescu, Ilie; Yao, Yi-Ching. Maximizing the variance of the time to ruin in a multiplayer game with selection. Adv. in Appl. Probab. 48 (2016), no. 2, 610--630.

Export citation


  • Amano, K., Tromp, J., Vitányi, P. M. B. and Watanabe, O. (2001). On a generalized ruin problem. In Approximation, Randomization, and Combinatorial Optimization (Lecture Notes Comput. Sci. 2129), Springer, Berlin, pp. 181–191.
  • Bach, E. (2007). Bounds for the expected duration of the monopolist game. Inform. Process. Lett. 101, 86–92.
  • Blackwell, D. (1970). On stationary policies. J. R. Statist. Soc. Ser. A 133, 33–37.
  • Bruss, F. T., Louchard, G. and Turner, J. W. (2003). On the $N$-tower problem and related problems. Adv. Appl. Prob. 35, 278–294.
  • Bürger, R. and Ewens, W. J. (1995). Fixation probabilities of additive alleles in diploid populations. J. Math. Biol. 33, 557–575.
  • Engel, A. (1993). The computer solves the three tower problem. Amer. Math. Monthly 100, 62–64.
  • Felsenstein, J. (1974). The evolutionary advantage of recombination. Genetics 78, 737–756.
  • Harik, G., Cantú-Paz, E., Goldberg, D. E. and Miller, B. L. (1999). The gambler's ruin problem, genetic algorithms, and the sizing of populations. Evolutionary Computation 7, 231–253.
  • Knuth, D. E. (1998). The Art of Computer Programming, Vol. 2, 3rd edn. Addison-Wesley, Reading, MA.
  • Ross, S. M. (2009). A simple solution to a multiple player gambler's ruin problem. Amer. Math. Monthly 116, 77–81.
  • Ross, S. M. (2011). The multiple-player ante one game. Prob. Eng. Inf. Sci. 25, 343–353.
  • Stirzaker, D. (1994). Tower problems and martingales. Math. Scientist 19, 52–59.
  • Swan, Y. C. and Bruss, F. T. (2006). A matrix-analytic approach to the $N$-player ruin problem. J. Appl. Prob. 43, 755–766.