Swap Regret

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 $T$.

We revisit the regret loss framework introduced in Park et al. (2025), which uses decision-theoretic regret as a direct loss function for training models to make better decisions, through the lens of probability-simplex policies. Our first result shows that a single-layer self-attention model trained with regret loss admits a stationary point whose forward-pass exactly matches smoothed fictitious play with the appropriate stepsize that ensures no-regret behavior-i.e., for any given policy input, the model outputs the same update that smoothed fictitious play would produce. In parallel, we also newly introduce a swap-regret loss function, which extends the regret-loss framework beyond external regret and enables models to directly optimize for swap-deviation robustness. We further show that this swap-regret loss admits a stationary point whose forward pass implements the corresponding swap-regret update induced by classical Blum-Mansour no-pass implementation algorithm, with each head implementing an external-regret update via smoothed fictitious play. Together, these results show that regret-trained attention can realize differentiable mechanisms whose deployment induces equilibrium behavior in games: external-regret dynamics lead to coarse correlated equilibrium, while swap-regret dynamics lead to correlated equilibrium. Thus, regret-based objectives steer minimal attention architectures toward online-learning dynamics with game-theoretic guarantees, without supervised traces of those algorithms.

We study the problem of forecasting for an arbitrary number of downstream agents with unknown objectives, each of whom best responds to the forecaster's predictions. We seek a single forecaster that guarantees sublinear swap regret for all downstream agents simultaneously. For two-dimensional outcome spaces, we give a polynomial time algorithm that guarantees $\tilde{O}(\sqrt{kT})$ swap regret for any downstream agent with $k$ actions. This improves over the previously known bound of $\tilde{O}(kT^{5/8})$ and avoids the exponential in $T$ runtime of prior algorithms in this setting. Our algorithm extends nicely to other low dimensional environments, retaining $\tilde{O}(\sqrt{T})$ downstream swap regret while the exponent of $k$ in the regret bound and the exponent of $T$ in the running time both grow with dimension. For arbitrary dimension $d$, we give a forecasting algorithm that guarantees $\tilde{O}(d\sqrt{kT})$ swap regret, assuming the forecaster knows an upper bound $k$ on the number of actions available to any downstream agent, albeit with a much longer runtime. This improves upon previous high dimensional guarantees that had $\tilde{O}(T^{2/3})$ dependence and required additional behavioral assumptions.

Real-world multi-agent systems, from traffic coordination to resource allocation, are often modeled as general-sum games where individual incentives conflict with collective welfare. In these settings, the central challenge is not merely finding an equilibrium, but selecting socially desirable outcomes among many suboptimal Nash equilibria. Standard deep multi-agent reinforcement learning (MARL) methods struggle with this problem, as value-decomposition approaches are constrained by monotonicity assumptions and policy-gradient methods often converge to stable but socially inefficient equilibria. To address this limitation, we propose $Φ$-Actor-Critic ($Φ$-AC), a framework that leverages swap regret minimization to steer learning toward high-welfare correlated equilibria (CE). To make counterfactual regret estimation tractable in deep MARL, $Φ$-AC employs a centralized attention critic that predicts vector-valued regrets in a single forward pass, avoiding computationally expensive counterfactual simulations. We further introduce a Lagrangian-based equilibrium selection mechanism that optimizes social welfare while enforcing stability through regret constraints. Experiments on matrix games, Multi-Agent Particle Environments (MPE), and the Melting Pot Harvest scenario demonstrate that $Φ$-AC learns efficient and stable coordination strategies across diverse mixed-motive settings while maintaining high collective return and competitive fairness.

AI generated predictions increasingly inform decision making in critical tasks, and therefore must be trustworthy. One widely used measure of trustworthiness is calibration, which requires that the predictions match the true frequencies and can be treated like real probabilities of a given outcome. However, defining calibration is subtle, and designing good measures of calibration error has been an active topic of recent research. The first goal is to find calibration measures that are actionable, meaning they can inform decision makers about their utility loss when predictions are treated as true probabilities, which is known as swap regret. The second goal is to find calibration measures that are testable, meaning that calibration error can be measured from a small sample of predictions and outcomes. Although these are very basic requirements, there is no existing calibration measure that fully satisfies both properties, and all existing measures relax actionability by bounding a weaker notion of swap regret, or relax testability by having suboptimal estimation error. We introduce a new calibration measure, Soft-Binned Calibration Decision Loss (SCDL), which we prove is fully actionable without weakening either requirement, and testable with nearly optimal error rate. In addition, SCDL satisfies other desired properties such as continuity and consistency. We also provide a set of experiments confirming that the theoretical advantages of SCDL compared to other measures lead to better performance in practice.

We study the discrete Bertrand pricing game with a non-increasing demand function. The game has $n \ge 2$ players who simultaneously choose prices from the set $\{1/k, 2/k, \ldots, 1\}$, where $k\in\mathbb{N}$. The player who sets the lowest price captures the entire demand; if multiple players tie for the lowest price, they split the demand equally. We study the Bertrand paradox, where classical theory predicts low prices, yet real markets often sustain high prices. To understand this gap, we analyze a repeated-game model in which firms set prices using no-regret learners. Our goal is to characterize the equilibrium outcomes that can arise under different no-regret learning guarantees. We are particularly interested in questions such as whether no-external-regret learners can converge to undesirable high-price outcomes, and how stronger guarantees such as no-swap regret shape the emergence of competitive low-price behavior. We address these and related questions through a theoretical analysis, complemented by experiments that support the theory and reveal surprising phenomena for no-swap regret learners.

In bandit settings, optimizing long-term regret metrics requires exploration, which corresponds to sometimes taking myopically sub-optimal actions. When a long-lived principal merely recommends actions to be executed by a sequence of different agents (as in an online recommendation platform) this provides an incentive misalignment: exploration is "worth it" for the principal but not for the agents. Prior work studies regret minimization under the constraint of Bayesian Incentive-Compatibility in a static stochastic setting with a fixed and common prior shared amongst the agents and the algorithm designer. We show that (weighted) swap regret bounds on their own suffice to cause agents to faithfully follow forecasts in an approximate Bayes Nash equilibrium, even in dynamic environments in which agents have conflicting prior beliefs and the mechanism designer has no knowledge of any agents beliefs. To obtain these bounds, it is necessary to assume that the agents have some degree of uncertainty not just about the rewards, but about their arrival time -- i.e. their relative position in the sequence of agents served by the algorithm. We instantiate our abstract bounds with concrete algorithms for guaranteeing adaptive and weighted regret in bandit settings.