An Automata-Based Approach to Games with $ω$-Automatic Preferences

This paper studies multiplayer turn-based games on graphs in which player preferences are modeled as $ω$-automatic relations given by deterministic parity automata. This contrasts with most existing work, which focuses on specific reward functions. We conduct a computational analysis of these games, starting with the threshold problem in the antagonistic zero-sum case. As in classical games, we introduce the concept of value, defined here as the set of plays a player can guarantee to improve upon, relative to their preference relation. We show that this set is recognized by an alternating parity automaton APW of polynomial size. We also establish the computational complexity of several problems related to the concepts of value and optimal strategy, taking advantage of the $ω$-automatic characterization of value. Next, we shift to multiplayer games and Nash equilibria, and revisit the threshold problem in this context. Based on an APW construction again, we close complexity gaps left open in the literature, and additionally show that cooperative rational synthesis is $\mathsf{PSPACE}$-complete, while it becomes undecidable in the non-cooperative case.

💡 Research Summary

The paper investigates turn‑based multiplayer games on finite graphs where each player’s preferences are expressed as ω‑automatic relations, i.e., binary relations on infinite words recognized by deterministic parity automata (DPW). This modeling departs from the usual quantitative reward functions or Boolean objectives and allows arbitrary, possibly non‑transitive or cyclic, preference structures.

In the two‑player zero‑sum setting, player 1 uses a preference relation ⋉ while player 2 uses its complement. The authors introduce a set‑valued notion of “value”: the collection of infinite plays that player 1 can guarantee to be at least as good as any opponent’s response according to ⋉. They prove (Theorem 5) that this value set is ω‑regular and can be recognized by an alternating parity automaton (APW) of polynomial size. Leveraging this automata‑theoretic representation, they analyze several decision problems: the threshold problem (does there exist a play that strictly improves a given lasso), existence of optimal strategies, and verification whether a given Mealy‑machine strategy is optimal. These problems are shown to be PSPACE‑complete or EXPTIME‑complete (Theorems 6‑7). Moreover, they establish that the value of player 1 is exactly the complement of the value of player 2 (Theorem 9), confirming the antagonistic nature of the game.

Extending to multiplayer games, the paper characterizes Nash equilibrium (NE) outcomes as the intersection of the values of all coalitions opposing each player. This intersection is again ω‑regular and recognized by a polynomial‑size APW (Theorem 12). Consequently, the NE existence problem, both unrestricted and with ω‑regular or LTL constraints, is PSPACE‑complete (Theorems 13‑14), improving over the previously known 2EXPTIME upper bound.

The authors then turn to rational synthesis, distinguishing cooperative synthesis (the system assumes the environment will select a favorable NE) from non‑cooperative synthesis (the system must satisfy the specification under all NE). Using the APW construction, they prove that cooperative rational synthesis is PSPACE‑complete, whereas non‑cooperative rational synthesis becomes undecidable even when the specification is a simple lasso path (Theorem 18). Verification of a given Mealy‑machine strategy for either synthesis variant is also PSPACE‑complete (Theorem 19).

Overall, the work provides a unified automata‑based framework for analyzing games with highly expressive preference relations. By translating value sets, NE outcomes, and synthesis requirements into alternating parity automata of polynomial size, the authors obtain tight complexity bounds for a range of fundamental problems, thereby extending the theoretical foundations of game theory, formal verification, and reactive synthesis beyond traditional quantitative or Boolean settings.