Strategy Complexity of Büchi and Transience Objectives in Concurrent Stochastic Games

We study 2-player zero-sum concurrent (i.e., simultaneous move) stochastic Büchi games and Transience games on countable graphs. Two players, Max and Min, seek respectively to maximize and minimize the probability of satisfying the game objective. The Büchi objective is to visit a given set of target states infinitely often. This can be seen as a special case of maximizing the expected $\limsup$ of the daily rewards, where all daily rewards are in $\{0,1\}$. The Transience objective is to visit no state infinitely often, i.e., every finite subset of the states is eventually left forever. Transience can only be met in infinite game graphs. We show that in Büchi games there always exist $\varepsilon$-optimal Max strategies that use just a step counter (discrete clock) plus 1 bit of public memory. This upper bound holds for all countable graphs, but it is a new result even for the special case of finite graphs. The upper bound is tight in the sense that Max strategies that use just a step counter, or just finite memory, are not sufficient even on finite game graphs. This upper bound is a consequence of a slightly stronger new result: $\varepsilon$-optimal Max strategies for the combined Büchi and Transience objective require just 1 bit of public memory (but cannot be memoryless). Our proof techniques also yield a closely related result, that $\varepsilon$-optimal Max strategies for the Transience objective alone can be chosen as memoryless.