Finite state stochastic games: Existence theorems and computational procedures

Let
{X_{n}}
be a Markov process with finite state space and transition probabilities
p_{ij}(u_{i}, v_{i})
depending on u
i
and
v_{i}.
State 0 is the capture state (where the game ends;
p_{oi} \equiv \delta_{oi})
;
u = {u_{i}}
and
v = {v_{i}}
are the pursuer and evader strategies, respectively, and are to be chosen so that capture is advanced or delayed and the cost
C_{i^{u,v}} = E[\Sum_{0}^{\infty} k (u(X_{n}), v(X_{n}), X_{n}) | X_{0} = i]
is minimaxed (or maximined), where
k(\alpha, \beta, 0) \equiv 0
. The existence of a saddle point and optimal strategy pair or e-optimal strategy pair is considered under several conditions. Recursive schemes for computing the optimal or ε-optimal pairs are given.