  Semi-Markov model is one of the most general models for stochastic dynamic
systems. This paper deals with a two-person zero-sum game for semi-Markov
processes. We focus on the expected discounted payoff criterion with
state-action-dependent discount factors. The state and action spaces are both
Polish spaces, and the payoff function is $\omega$-bounded. We first construct
a fairly general model of semi-Markov games under a given semi-Markov kernel
and a pair of strategies. Next, based on the standard regularity condition and
the continuity-compactness condition for semi-Markov games, we derive a "drift
condition" on the semi-Markov kernel and suppose that the discount factors have
a positive lower bound, under which the existence of the value function and a
pair of optimal stationary strategies of our semi-Markov game are proved by
using the Shapley equation. Moreover, when the state and action spaces are both
finite, a value iteration-type algorithm for computing the value function and
$\varepsilon$-Nash equilibrium of the game is developed. The convergence of the
algorithm is also proved. Finally, we conduct numerical examples to demonstrate
our main results.
