Recursive Concurrent Stochastic Games

We study Recursive Concurrent Stochastic Games (RCSGs), extending our recent analysis of recursive simple stochastic games to a concurrent setting where the two players choose moves simultaneously and independently at each state. For multi-exit games, our earlier work already showed undecidability for basic questions like termination, thus we focus on the important case of single-exit RCSGs (1-RCSGs). We first characterize the value of a 1-RCSG termination game as the least fixed point solution of a system of nonlinear minimax functional equations, and use it to show PSPACE decidability for the quantitative termination problem. We then give a strategy improvement technique, which we use to show that player 1 (maximizer) has \epsilon-optimal randomized Stackless & Memoryless (r-SM) strategies for all \epsilon > 0, while player 2 (minimizer) has optimal r-SM strategies. Thus, such games are r-SM-determined. These results mirror and generalize in a strong sense the randomized memoryless determinacy results for finite stochastic games, and extend the classic Hoffman-Karp strategy improvement approach from the finite to an infinite state setting. The proofs in our infinite-state setting are very different however, relying on subtle analytic properties of certain power series that arise from studying 1-RCSGs. We show that our upper bounds, even for qualitative (probability 1) termination, can not be improved, even to NP, without a major breakthrough, by giving two reductions: first a P-time reduction from the long-standing square-root sum problem to the quantitative termination decision problem for finite concurrent stochastic games, and then a P-time reduction from the latter problem to the qualitative termination problem for 1-RCSGs.

💡 Research Summary

The paper introduces and thoroughly investigates Recursive Concurrent Stochastic Games (RCSGs), a class of infinite‑state zero‑sum stochastic games in which two players simultaneously and independently choose actions at every “play” node. The authors focus on the single‑exit variant, called 1‑RCSG, because for multi‑exit games even basic termination questions are undecidable.

First, the authors formalize the model. An RCSG consists of a finite collection of components; each component contains a finite set of nodes, a set of “boxes” that implement recursive calls to other components, and a distinguished entry node and (for 1‑RCSGs) a single exit node. Nodes are either probabilistic (transitions follow a fixed rational distribution) or concurrent (the next transition is determined by the pair of moves selected by the two players from their finite move alphabets Γ₁ and Γ₂). The global state space is the set of pairs (stack of boxes, current node), which is countably infinite.

The central quantitative problem is the termination game: starting from the empty stack and a designated start node, player 1 (the maximizer) tries to maximize the probability of eventually reaching the unique exit, while player 2 (the minimizer) tries to minimize it. The authors show that the value function V(u) for each global state u satisfies a system of nonlinear minimax functional equations:

• If u is an exit, V(u)=1; if u is a dead‑end, V(u)=0.
• If u is probabilistic, V(u)=∑₍v₎ p_{u,v}·V(v).
• If u is concurrent, V(u)= min_{γ₂∈Γ_u²} max_{γ₁∈Γ_u¹} V_{u}^{γ₁,γ₂},

where V_{u}^{γ₁,γ₂} denotes the value after the joint move (γ₁,γ₂) is taken. This operator is monotone and continuous on the complete lattice