General-Sum Games

Optimistic Hedge, a multiplicative weight update with one step of extrapolation, is one of the most natural regret-minimizing dynamics for games, and characterizing its regret has been a longstanding open question. Daskalakis, Fishelson and Golowich proved that if all $m$ players in a finite general-sum game follow Optimistic Hedge with a fixed learning rate, each player's regret is polylogarithmic in the horizon $T$. Subsequently, Ha showed that an appropriate choice of learning rate makes each player's regret bounded independently of $T$. This bound, however, is non-constructive and its dependence on the number of actions $n$ is not explicit, leaving open whether Optimistic Hedge can achieve a constant regret bound which is polynomial in the size of the game. We show that the answer hinges on the number of players. In two-player games, there is a learning rate for which Optimistic Hedge guarantees $O(n\log n)$ regret independent of the horizon. In three-player games, however, no such bound is possible: for any learning rate up to an absolute constant and for sufficiently large horizon $T$, there is a game with $n$ actions per player in which some player's regret is $2^{Ω(n)}$, and more generally, for every $m\ge3$ we construct $m$-player games in which some player's regret is $2^{Ω(mn)}$; for learning rates above $2^{-Ω(mn)}$, that player's regret even grows linearly in $T$. Our upper bound is based on a temporal approximation framework for the analysis of Optimistic Hedge. Starting from the variance-based RVU bound of Daskalakis et al., we approximate the losses over time-localized windows by well-behaved functions, whose total variation is a small fraction of their cumulative variance. In $m$-player games, the same framework gives a significantly simplified proof of the recent $O(\sqrt{m}\log n\log T)$ regret bound due to Ha.

We study the problem of computing the tightest $α$-potential approximation of a general-sum game over continuous action spaces, within a prescribed class of potential functions and when each player has access only to its own utility function. The difficulty is twofold: the approximation error involves a worst-case search over an infinite set of unilateral deviations, and the required utility information is distributed across players. For a linear-in-parameters potential class, we use an exact finite-tuple reformulation that separates the problem into a global outer search over deviation tuples and distributed convex inner problems. We develop a primal--dual inner oracle tailored to this structure and establish a uniform one-sided accuracy guarantee. This oracle can be combined with global outer search to obtain an end-to-end guarantee on the outer optimization error. We also develop a projected zeroth-order outer method as a computationally lighter alternative for higher-dimensional problems. Numerical experiments illustrate the accuracy--computation tradeoff between the two outer-search methods and show that the proposed optimization framework can improve upon analytical $α$-potential constructions.

Can simple learning rules keep their regret bounded in self-play? Recent work achieves constant regret bounds through modified regularization and higher-order prediction. Yet for Optimistic Hedge, arguably the most canonical method in games, the best known individual regret bound remains logarithmic. In this work, we prove that plain Optimistic Hedge with a constant step size can attain $O_{n,d}(1)$ individual regret in general-sum games with $n$ players and $d=(d_1,\ldots,d_n)$ actions, under expected loss-vector feedback. As a corollary, its time-averaged play enjoys an $O_{n,d}(1/T)$ coarse correlated equilibrium (CCE) gap. Our analysis represents Optimistic Hedge as a real-analytic recurrence on a compact space, which yields an exact finite-order difference relation that eliminates horizon dependence. Our proof hinges on nonconstructive Noetherianity argument of Frisch (1967), so the $(n,d)$-dependence remains implicit.

We introduce Multiplicatively Optimistic Regret Matching (MORM), an uncoupled learning rule for finite general-sum games. Under simultaneous full-information self-play, every player achieves external regret $O(\sqrt n\log d)$ uniformly over all horizons, using only one-step optimism. The analysis combines a potential-based regret-matching argument with multiplicative stability and Hellinger control of strategy movement. A learning-rate safeguard additionally gives $O(\sqrt{T\log d})$ regret in the face of adversarial utilities.

Game-theoretic analyses of cyber defence often compute equilibria of games whose payoffs exist only as the output of a simulator. Iterative equilibrium-finding methods grow a set of attacker and defender policies and need the payoff of every attacker--defender pair, so they are bottlenecked by payoff estimation: each payoff costs many simulator runs. We introduce Regret-Weighted Payoff Sampling (RWPS), which spends a fixed simulation budget on the payoffs the equilibrium actually depends on and predicts the rest with a model trained on every payoff measured so far. Standard error bounds for estimated games are driven by the worst-estimated payoff, so they cannot credit an estimator that is inaccurate only where accuracy does not matter. We prove a bound that weights payoff errors by the opponent's equilibrium strategy, a certificate that can be computed from simulated payoffs alone, and a condition under which errors in the predicted payoffs cannot change either player's regret. On three synthetic general-sum games, one of them a Colonel Blotto game of military resource allocation, the new bounds are four to six times tighter than the standard one, and RWPS finds less exploitable equilibria than minimum-regret-first search, information-gain search and progressive sampling at the same budget. On two cyber-defence simulators, CyGym and a new game whose hosts are LLM agents exposed to prompt injection, it gives the least exploitable equilibria at the smallest budgets.

We give deterministic and uncoupled learning dynamics for finite multiplayer general-sum games under full-information feedback that achieve constant individual swap regret in self-play, independent of the horizon $T$. With $n$ players and at most $m$ actions each, every player's individual swap regret is $O(\sqrt n\,m\log m\log^{5/2}(nm))$ at every finite horizon. The dynamics use the classical Blum-Mansour framework with optimism. Each player predicts the deviation gains, uses these predictions to update a row-stochastic transition matrix, and plays its stationary distribution. Our new ingredients include a tailored row normalization map and a two-scale higher-order predictor. An adversarially robust variant, obtained through a generic common-prefix switching wrapper, preserves the self-play bound up to a universal constant and guarantees individual swap regret at most $7\sqrt{mT\log m}$ in the adversarial setting.

Swap regret governs the rate at which uncoupled learning dynamics converge to correlated equilibria in multiplayer general-sum games. Under full-information feedback, the best previous guarantee when every player follows the same dynamics grows logarithmically in the horizon $T$. We construct uncoupled dynamics under which every player incurs only $O(nm^2\sqrt{\log m\log T})$ swap regret, where $n$ is the number of players and $m$ bounds the number of actions per player. To our knowledge, this is the first sublogarithmic individual guarantee in this setting, and it implies that the time-averaged product distribution of play is an $O(nm^2\sqrt{\log m\log T}/T)$-approximate correlated equilibrium. The key algorithmic choice is to combine the Blum--Mansour reduction with optimistic follow-the-regularized-leader using a hybrid regularizer that separately weights negative Shannon entropy and the log-barrier: the entropy controls the optimistic prediction error, whereas the log-barrier controls the transition-matrix movement through its Bregman divergence. A new sensitivity theorem for stationary distributions of Markov chains, which involves neither mixing parameters nor the smallest transition probability, transfers this control to the played strategies and yields a simpler analysis without local-norm or self-concordance arguments. The guarantee is preserved by an adversarially robust variant that additionally ensures $O(nm^2\sqrt{\log m\log T}+\sqrt{mT\log m})$ swap regret against arbitrary utility sequences, and by a horizon-free variant that requires no prior knowledge of $T$.

Offline multi-agent reinforcement learning in general-sum settings is challenged by the distribution shift between logged datasets and target equilibrium policies. While standard methods rely on manual pessimistic penalties, we demonstrate that KL regularization suffices to stabilize learning and achieve equilibrium recovery. We propose General-sum Anchored Nash Equilibrium (GANE), which recovers regularized Nash equilibria at an accelerated statistical rate of $\widetilde{O}(1/n)$. For computational tractability, we develop General-sum Anchored Mirror Descent (GAMD), an iterative algorithm converging to a Coarse Correlated Equilibrium at the standard rate of $\widetilde{O}(1/\sqrt{n}+1/T)$. These results establish KL regularization as a standalone mechanism for pessimism-free offline learning that achieves equivalent or accelerated rates in multi-player general-sum games.