  We consider a class of two-player zero-sum stochastic games with finite state
and compact control spaces, which we call stochastic shortest path (SSP) games.
They are undiscounted total cost stochastic dynamic games that have a cost-free
termination state. Exploiting the close connection of these games to
single-player SSP problems, we introduce novel model conditions under which we
show that the SSP games have strong optimality properties, including the
existence of a unique solution to the dynamic programming equation, the
existence of optimal stationary policies, and the convergence of value and
policy iteration. We then focus on finite state and control SSP games and the
classical Q-learning algorithm for computing the value function. Q-learning is
a model-free, asynchronous stochastic iterative algorithm. By the theory of
stochastic approximation involving monotone nonexpansive mappings, it is known
to converge when its associated dynamic programming equation has a unique
solution and its iterates are bounded with probability one. For the SSP case,
as the main result of this paper, we prove the boundedness of the Q-learning
iterates under our proposed model conditions, thereby establishing completely
the convergence of Q-learning for a broad class of total cost finite-space
stochastic games.
