The Complexity of Nash Equilibria in Stochastic Multiplayer Games

We analyse the computational complexity of finding Nash equilibria in turn-based stochastic multiplayer games with omega-regular objectives. We show that restricting the search space to equilibria whose payoffs fall into a certain interval may lead to undecidability. In particular, we prove that the following problem is undecidable: Given a game G, does there exist a Nash equilibrium of G where Player 0 wins with probability 1? Moreover, this problem remains undecidable when restricted to pure strategies or (pure) strategies with finite memory. One way to obtain a decidable variant of the problem is to restrict the strategies to be positional or stationary. For the complexity of these two problems, we obtain a common lower bound of NP and upper bounds of NP and PSPACE respectively. Finally, we single out a special case of the general problem that, in many cases, admits an efficient solution. In particular, we prove that deciding the existence of an equilibrium in which each player either wins or loses with probability 1 can be done in polynomial time for games where the objective of each player is given by a parity condition with a bounded number of priorities.

💡 Research Summary

The paper investigates the computational complexity of finding Nash equilibria in turn‑based stochastic multiplayer games (SMGs) with ω‑regular objectives such as reachability, Büchi, co‑Büchi, and parity conditions. The authors formalize the model as a finite directed arena where each vertex is either controlled by a player or is stochastic, with rational transition probabilities. Each player’s winning condition is a Borel set of infinite colour sequences, and a play’s payoff is a binary vector indicating which players win.

The central decision problem, called NE, asks whether a given SMG admits a Nash equilibrium whose expected payoff lies within a prescribed interval (or equals a given binary vector). The problem is studied under six strategy restrictions: (i) positional (memoryless deterministic), (ii) stationary (memoryless possibly randomised), (iii) pure finite‑state, (iv) randomised finite‑state, (v) arbitrary pure, and (vi) arbitrary randomised strategies.

The main results are:

1. Undecidability – For arbitrary randomised strategies as well as for pure strategies (even when limited to finite memory), the existence of a Nash equilibrium in which player 0 wins with probability 1 is undecidable. The proof reduces the halting problem of a Turing machine to this equilibrium existence problem by constructing a game that forces player 0 to achieve probability 1 exactly when the simulated machine halts. This shows that NE is undecidable for the most general strategy classes.

2. Positional strategies – When strategies are required to be positional, the NE problem becomes NP‑complete. A nondeterministic algorithm can guess a mapping from each vertex to an action and verify in polynomial time that the resulting profile is a Nash equilibrium with the desired payoff. NP‑hardness holds already for three‑player games with terminal payoffs.

3. Stationary strategies – Allowing randomisation but still forbidding memory yields a problem that is NP‑hard and lies in PSPACE. The authors give a polynomial‑time reduction from the classic SqrtSum problem to NE with stationary strategies, establishing that any improvement on the upper bound would also resolve the long‑standing open question about the exact complexity of SqrtSum.

4. Pure/Randomised finite‑state strategies – These intermediate classes inherit the undecidability of the unrestricted case; restricting to finite memory does not restore decidability.

5. Strictly qualitative fragment – The authors isolate a special case where the desired payoff vector is binary (each player either wins with probability 1 or loses with probability 0). For games whose objectives are parity conditions with a bounded number of priorities, this fragment can be decided in polynomial time. The algorithm exploits the limited priority structure to decompose the game into strongly connected components and solve each component by simple graph analysis.

6. Complexity landscape – The paper maps the complexity of NE across the six strategy classes, showing a dramatic shift from undecidable (general strategies) to NP (positional) to PSPACE (stationary), and finally to P for the bounded‑priority parity fragment.

The work also corrects an earlier claim by Chatterjee et al. regarding the existence of Nash equilibria in SMGs, providing a rigorous proof that every SMG does indeed have at least one equilibrium. Related literature on two‑player zero‑sum stochastic games, Markov decision processes, and the complexity of Nash equilibria in strategic‑form games is surveyed, highlighting how this paper extends those results to the multiplayer, stochastic, ω‑regular setting.

In practical terms, the findings suggest that system designers can obtain tractable verification or synthesis procedures by imposing appropriate strategy restrictions: using positional strategies yields NP‑level algorithms, stationary strategies remain feasible within PSPACE, and for many verification tasks (e.g., bounded‑priority parity specifications) a polynomial‑time solution is available. The undecidability results caution that without such restrictions, equilibrium analysis may be fundamentally infeasible.