We consider the problem of computing stationary points in min-max optimization, with a focus on the special case of Nash equilibria in (two-)team zero-sum games. We first show that computing ϵ-Nash equilibria in 3-player adversarial team games--wherein a team of 2players competes against a single adversary-- is CLS-complete, resolving the complexity of Nash equilibria in such settings. Our proof proceeds by reducing from symmetric ϵ-Nash equilibria in symmetric, identical-payoff, two-player games, by suitably leveraging the adversarial player so as to enforce symmetry--without disturbing the structure of the game. In particular, the class of instances we construct comprises solely polymatrix games, thereby also settling a question left open by Hollender, Maystre, and Nagarajan (2024). Moreover, we establish that computing symmetric (first-order) equilibria in symmetric min-max optimization is PPAD-complete, even for quadratic functions. Building on this reduction, we show that computing symmetric ϵ-Nash equilibria in symmetric, 6-player (3 vs. 3) team zero-sum games is also PPAD-complete, even for ϵ = poly(1/n). As a corollary, this precludes the existence of symmetric dynamics--which includes many of the algorithms considered in the literature-- converging to stationary points. Finally, we prove that computing a non-symmetric poly(1/n)-equilibrium in symmetric min-max optimization is FNP-hard.

Correlated equilibria--and their generalizations known as Φ-equilibria--are a fundamental object of study in game theory, offering a more tractable alternative to Nash equilibria in multi-player settings. While computational aspects of equilibrium computation are well-understood in some settings, fundamental questions are still open in generalized games, that is, games in which the set of strategies allowed to each player depends on the other players' strategies. These classes of games model fundamental settings in economics, and have been a cornerstone of economics research since the seminal paper of Arrow and Debreu [1954]. Recently, there has been growing interest, both in economics and in computer science, in studying correlated equilibria in generalized games. It is known that finding a social welfare maximizing correlated equilibrium in generalized games is NP-hard. However, the existence of efficient algorithms to find any equilibrium remains an important open question.

In this paper, we examine the robustness of Nash equilibria in continuous games, under both strategic and dynamic uncertainty. Starting with the former, we introduce the notion of a robust equilibrium as those equilibria that remain invariant to small--but otherwise arbitrary--perturbations to the game's payoff structure, and we provide a crisp geometric characterization thereof. Subsequently, we turn to the question of dynamic robustness, and we examine which equilibria may arise as stable limit points of the dynamics of "follow the regularized leader" (FTRL) in the presence of randomness and uncertainty. Despite their very distinct origins, we establish a structural correspondence between these two notions of robustness: strategic robustness implies dynamic robustness, and, conversely, the requirement of strategic robustness cannot be relaxed if dynamic robustness is to be maintained. Finally, we examine the rate of convergence to robust equilibria as a function of the underlying regularizer, and we show that entropically regularized learning converges at a geometric rate in games with affinely constrained action spaces.

Computing Nash equilibria of zero-sum games in classical and quantum settings is extensively studied. For general-sum games, computing Nash equilibria is PPAD-hard and the computing of a more general concept called correlated equilibria has been widely explored in game theory. In this paper, we initiate the study of quantum algorithms for computing $\varepsilon$-approximate correlated equilibria (CE) and coarse correlated equilibria (CCE) in multi-player normal-form games. Our approach utilizes quantum improvements to the multi-scale Multiplicative Weight Update (MWU) method for CE calculations, achieving a query complexity of $\tilde{O}(m\sqrt{n})$ for fixed $\varepsilon$. For CCE, we extend techniques from quantum algorithms for zero-sum games to multi-player settings, achieving query complexity $\tilde{O}(m\sqrt{n}/\varepsilon^{2.5})$. Both algorithms demonstrate a near-optimal scaling in the number of players $m$ and actions $n$, as confirmed by our quantum query lower bounds.

In this paper, we examine the long-run behavior of regularized, no-regret learning in1 finite games. A well-known result in the field states that the empirical frequencies2 of no-regret play converge to the game's set of coarse correlated equilibria; however,3 our understanding of how the players' actual strategies evolve over time is much4 more limited - and, in many cases, non-existent. This issue is exacerbated by5 a series of recent results showing that only strict Nash equilibria are stable and6 attracting under regularized learning, thus making the relation between learning7 and pointwise solution concepts particularly elusive. In lieu of this, we take a more8 general approach and instead seek to characterize the setwise rationality properties9 of the players' day-to-day play. To that end, we focus on one of the most stringent10 criteria of setwise strategic stability, namely that any unilateral deviation from the11 set in question incurs a cost to the deviator - a property known as closedness under12 better replies (club).

In two-player zero-sum extensive-form games, Nash equilibrium prescribes optimal strategies against perfectly rational opponents. However, it does not guarantee rational play in parts of the game tree that can only be reached by the players making mistakes. This can be problematic when operationalizing equilibria in the real world among imperfect players. Trembling-hand refinements are a sound remedy to this issue, and are subsets of Nash equilibria that are designed to handle the possibility that any of the players may make mistakes. In this paper, we initiate the study of equilibrium refinements for settings where one of the players is perfectly rational (the "machine") and the other may make mistakes.

In two-player zero-sum extensive-form games, Nash equilibrium prescribes optimal strategies against perfectly rational opponents. However, it does not guarantee rational play in parts of the game tree that can only be reached by the players making mistakes. This can be problematic when operationalizing equilibria in the real world among imperfect players. Trembling-hand refinements are a sound remedy to this issue, and are subsets of Nash equilibria that are designed to handle the possibility that any of the players may make mistakes. In this paper, we initiate the study of equilibrium refinements for settings where one of the players is perfectly rational (the "machine") and the other may make mistakes.

Neural Information Processing Systems http://nips.cc/

While Online Gradient Descent and other no-regret learning procedures are known to efficiently converge to a coarse correlated equilibrium in games where each agent's utility is concave in their own strategy, this is not the case when utilities are non-concave -- a common scenario in machine learning applications involving strategies parameterized by deep neural networks, or when agents' utilities are computed by neural networks, or both. Non-concave games introduce significant game-theoretic and optimization challenges: (i) Nash equilibria may not exist; (ii) local Nash equilibria, though they exist, are intractable; and (iii) mixed Nash, correlated, and coarse correlated equilibria generally have infinite support and are intractable. To sidestep these challenges, we revisit the classical solution concept of $\Phi$-equilibria introduced by Greenwald and Jafari [GJ03], which is guaranteed to exist for an arbitrary set of strategy modifications $\Phi$ even in non-concave games [SL07]. However, the tractability of $\Phi$-equilibria in such games remains elusive. In this paper, we initiate the study of tractable $\Phi$-equilibria in non-concave games and examine several natural families of strategy modifications. We show that when $\Phi$ is finite, there exists an efficient uncoupled learning algorithm that approximates the corresponding $\Phi$-equilibria. Additionally, we explore cases where $\Phi$ is infinite but consists of local modifications, showing that Online Gradient Descent can efficiently approximate $\Phi$-equilibria in non-trivial regimes.