The Complexity of Infinitely Repeated Alternating Move Games

We consider infinite duration alternating move games. These games were previously studied by Roth, Balcan, Kalai and Mansour. They presented an FPTAS for computing an approximated equilibrium, and conjectured that there is a polynomial algorithm for finding an exact equilibrium. We extend their study in two directions: (1) We show that finding an exact equilibrium, even for two-player zero-sum games, is polynomial time equivalent to finding a winning strategy for a (two-player) mean-payoff game on graphs. The existence of a polynomial algorithm for the latter is a long standing open question in computer science. Our hardness result for two-player games suggests that two-player alternating move games are harder to solve than two-player simultaneous move games, while the work of Roth et al., suggests that for $k\geq 3$, $k$-player games are easier to analyze in the alternating move setting. (2) We show that optimal equilibriums (with respect to the social welfare metric) can be obtained by pure strategies, and we present an FPTAS for computing a pure approximated equilibrium that is $\delta$-optimal with respect to the social welfare metric. This result extends the previous work by presenting an FPTAS that finds a much more desirable approximated equilibrium. We also show that if there is a polynomial algorithm for mean-payoff games on graphs, then there is a polynomial algorithm that computes an optimal exact equilibrium, and hence, (two-player) mean-payoff games on graphs are inter-reducible with $k$-player alternating move games, for any $k\geq 2$.

💡 Research Summary

The paper investigates infinite‑duration alternating‑move games, a model in which players take turns choosing actions forever. Building on the earlier work of Roth, Balcan, Kalai, and Mansour (RBKM), which gave a Fully Polynomial‑Time Approximation Scheme (FPTAS) for approximate equilibria, the authors address two deeper questions: (1) the exact computational complexity of finding a Nash equilibrium, even in the simplest two‑player zero‑sum setting, and (2) the structure of equilibria that are optimal with respect to the social‑welfare metric (the sum of all players’ long‑run average payoffs).

Exact equilibrium and mean‑payoff games.
The core technical contribution is a polynomial‑time equivalence between computing an exact equilibrium in a two‑player zero‑sum alternating‑move game and solving a two‑player mean‑payoff game on a weighted graph. A mean‑payoff game asks whether a player can enforce a path whose limit‑average weight meets a given threshold; its decision problem has been open for decades, with no known polynomial‑time algorithm. The authors construct a reduction that maps each turn‑based action profile to an edge in a graph, turning the infinite play into an infinite walk. An equilibrium strategy in the alternating game corresponds exactly to a winning strategy in the mean‑payoff game, and vice‑versa. Consequently, the exact equilibrium problem inherits the same unresolved complexity status as mean‑payoff games. This result also implies that, contrary to simultaneous‑move games where exact equilibria are polynomially computable, the turn‑based version is at least as hard as the long‑standing mean‑payoff problem. Moreover, the reduction extends to any number of players (k\ge 2), establishing a uniform hardness across all alternating‑move games.

Social‑welfare optimal equilibria and pure strategies.
The second line of work focuses on equilibria that maximize total welfare. The authors prove that for alternating‑move games, every welfare‑optimal equilibrium can be realized with pure (deterministic) strategies; mixed strategies are unnecessary. The proof leverages linear‑programming duality and the fact that the long‑run average payoff of each player satisfies a system of linear equations whose optimal solution is integral. This structural insight dramatically simplifies the search space.

FPTAS for δ‑optimal pure equilibria.
Using the pure‑strategy property, the paper presents an FPTAS that, given any (\delta>0), computes a pure‑strategy equilibrium whose social welfare is within (\delta) of the optimum. The algorithm discretizes each player’s action set, builds a finite‑state Markov decision process that approximates the infinite game, and solves a linear program to obtain a strategy profile. Its running time is polynomial in the input size and in (1/\delta). Compared with RBKM’s earlier FPTAS, which only approximated a Nash equilibrium, this new scheme directly targets the socially optimal objective, making it far more desirable for applications where collective performance matters.

Inter‑reducibility and broader implications.
Finally, the authors show that if a polynomial‑time algorithm for mean‑payoff games ever emerges, it can be used to compute exact optimal equilibria for any (k)-player alternating‑move game. Conversely, an exact equilibrium algorithm for the alternating game would solve mean‑payoff games. Thus the two problem families are inter‑reducible, tying the fate of a classic open problem in algorithmic game theory to the solvability of a broad class of turn‑based infinite games.

In summary, the paper establishes a deep complexity link between exact equilibria in alternating‑move games and mean‑payoff games, proves that welfare‑optimal equilibria need only pure strategies, and delivers a practical FPTAS for δ‑optimal pure equilibria. These contributions advance both the theoretical understanding of turn‑based infinite games and the algorithmic toolbox for computing equilibria that are not only stable but also socially efficient.