Geometrical methods for game problems with random walk

The following zero-sum two-person game Gamma is considered. Let x(n) and y( n) be independent random walks on the set E = {0, 1,..., k) that start from the states a and b, respectively (1 less than or equal to a less than or e qual to b less than or equal to k - 1), and are absorbed in the state 0 wit h probability p. The stopping moments tau and sigma of the random walks x(n ), and y(n), respectively, serve as the strategies for players I and II. Ea ch player knows the values k, a, b, and p but has no information about the behavior of the opponent. If x(tau) > y(sigma), the winner is the first pla yer, whereas, if x(tau) < y(sigma), the winner is the second person. If x(t au) = y(sigma), a tie is declared. The equilibrium situation and the value of the game are found.