Zero-sum turn games using Q-learning: finite computation with security guarantees

This paper addresses zero-sum ``turn'' games, in which only one player can make decisions at each state. We show that pure saddle-point state-feedback policies for turn games can be constructed from dynamic programming fixed-point equations for a single value function or Q-function. These fixed-points can be constructed using a suitable form of Q-learning. For discounted costs, convergence of this form of Q-learning can be established using classical techniques. For undiscounted costs, we provide a convergence result that applies to finite-time deterministic games, which we use to illustrate our results. For complex games, the Q-learning iteration must be terminated before exploring the full-state, which can lead to policies that cannot guarantee the security levels implied by the final Q-function. To mitigate this, we propose an ``opponent-informed'' exploration policy for selecting the Q-learning samples. This form of exploration can guarantee that the final Q-function provides security levels that hold, at least, against a given set of policies. A numerical demonstration for a multi-agent game, Atlatl, indicates the effectiveness of these methods.