## Advances in Applied Probability

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

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

Ilie Grigorescu and Yi-Ching Yao

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 9 June 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.aap/1465490765

**Mathematical Reviews number (MathSciNet)**

MR3511778

**Zentralblatt MATH identifier**

1347.91078

**Subjects**

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

**Keywords**

Gambler's ruin martingale dynamic programming stochastic control

#### Citation

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. https://projecteuclid.org/euclid.aap/1465490765