A game with optimal stopping of random walks

V. V. Mazalov   ; È. A. Kochetov

Teoriâ veroâtnostej i ee primeneniâ, Tome 42 (1997) no. 4, pp. 820-826

Résumé

A two-person game $\Gamma$ is considered which is specified by the following random walks. Let $x_n$ and $y_n$ be independent symmetric random walks on the set $E=\{0,1,\ldots,K\}$. Assume they start from the states $a$ and $b$ respectively $(1\le a < b\le K-1)$, are absorbed with probability $0.5$ at points $0$ and $K$, and are reflected to the points $1$ and $K-1$, respectively, with the same probability $0.5$. Players I and II observe the random walks $x_n$ and $y_n$, respectively, and stop them at Markov times $\tau $ and $\sigma$ being strategies of the game. Each player knows the values of $K, a$, and $b$ but has no information about the behavior of the other player. The rules of the game are as follows. If $x_{\tau} > y_{\sigma}$ then player II pays player I, say, \$1; if $x_{\tau} < y_{\sigma}$ then I pays II \$1; and if $x_{\tau}=y_{\sigma}$ then the outcome of the game is said to be a draw. The aim of each player is to maximize the expected value of hisincome. We find the equilibrium situation and the value of the game.

Détail

Keywords: random walk, reflecting barriers, strategy, stopping time, spectrum
Mots-clés : equilibrium situation.