On a Markov chain roulette-type game

A Markov chain on non-negative integers which arises in a roulette-type game is discussed. The transition probabilities are p01 = ρ, pNj = δNj, pi,i+W = q, pi,i-1 = p = 1 - q, 1 ≤ W < N, 0 ≤ ρ ≤ 1, N - W < j ≤ N and i = 1, 2, ..., N - W. Using formulae for the determinant of a partitioned matrix, a closed form expression for the solution of the Markov chain roulette-type game is deduced. The present analysis is supported by two mathematical models from tumor growth and war with bargaining