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Theory · Game Theory

Haris Aziz, Yuhang Guo, Alexander Lam, Houyu Zhou

The augmentation of algorithms with predictions of the optimal solution, such as from a machine-learning algorithm, has garnered significant attention in recent years, particularly in facility location problems. Moving beyond the traditional focus on utilitarian and egalitarian objectives, we design learning-augmented facility location mechanisms for the envy ratio objective, a fairness metric defined as the maximum ratio between the utilities of any two agents. For the deterministic setting, we propose a mechanism which utilizes predictions to achieve $\alpha$-consistency and $\frac{\alpha}{\alpha - 1}$-robustness for a selected parameter $\alpha \in [1,2]$, and prove its optimality. We also resolve open questions raised by Ding et al. [2020], devising a randomized mechanism without predictions to improve upon the best-known approximation ratio from $2$ to $1.8944$. Building upon these advancements, we construct a novel randomized mechanism which incorporates predictions to achieve improved performance guarantees.

Theory · Game Theory

Benyamin Trachtenberg, Nir Rosenfeld

In strategic classification, the standard supervised learning setting is extended to support the notion of strategic user behavior in the form of costly feature manipulations made in response to a classifier. While standard learning supports a broad range of model classes, the study of strategic classification has, so far, been dedicated mostly to linear classifiers. This work aims to expand the horizon by exploring how strategic behavior manifests under non-linear classifiers and what this implies for learning. We take a bottom-up approach showing how non-linearity affects decision boundary points, classifier expressivity, and model class complexity. Our results show how, unlike the linear case, strategic behavior may either increase or decrease effective class complexity, and that the complexity decrease may be arbitrarily large. Another key finding is that universal approximators (e.g., neural nets) are no longer universal once the environment is strategic. We demonstrate empirically how this can create performance gaps even on an unrestricted model class.

Social Aspects

Jingru Jia, Zehua Yuan, Junhao Pan, Paul McNamara, Deming Chen

What does it truly mean for a language model to “reason” strategically, and can scaling up alone guarantee intelligent, context-aware decisions? Strategic decision-making requires adaptive reasoning, where agents anticipate and respond to others’ actions under uncertainty. Yet, most evaluations of large language models (LLMs) for strategic decision-making often rely heavily on Nash Equilibrium (NE) benchmarks, overlook reasoning depth, and fail to reveal the mechanisms behind model behavior. To address this gap, we introduce a behavioral game-theoretic evaluation framework that disentangles intrinsic reasoning from contextual influence. Using this framework, we evaluate 22 state-of-the-art LLMs across diverse strategic scenarios. We find models like GPT-o3-mini, GPT-o1, and DeepSeek-R1 lead in reasoning depth. Through thinking chain analysis, we identify distinct reasoning styles—such as maximin or belief-based strategies—and show that longer reasoning chains do not consistently yield better decisions. Furthermore, embedding demographic personas reveals context-sensitive shifts: some models (e.g., GPT-4o, Claude-3-Opus) improve when assigned female identities, while others (e.g., Gemini 2.0) show diminished reasoning under minority sexuality personas. These findings underscore that technical sophistication alone is insufficient; alignment with ethical standards, human expectations, and situational nuance is essential for the responsible deployment of LLMs in interactive settings.

Theory · Learning Theory

Shay Moran, Elizaveta Nesterova

Consider the task of locating an unknown target point using approximate distance queries: in each round, a reconstructor selects a reference point and receives a noisy version of its distance to the target. This problem arises naturally in various contexts—from localization in GPS and sensor networks to privacy-aware data access—making it relevant from the perspective of both the reconstructor (seeking accurate recovery) and the responder (aiming to limit information disclosure, e.g., for privacy or security reasons). We study this reconstruction game through a learning-theoretic lens, focusing on the rate and limits of the best possible reconstruction error. Our first result provides a tight geometric characterization of the optimal error in terms of the Chebyshev radius, a classical concept from geometry. This characterization applies to all compact metric spaces (in fact, to all totally bounded spaces) and yields explicit formulas for natural subsets of the Euclidean metric. Our second result addresses the asymptotic behavior of reconstruction, distinguishing between pseudo-finite spaces, where the optimal error is attained after finitely many queries, and spaces where the approximation curve exhibits a nontrivial decay. We characterize pseudo-finiteness for convex subsets of Euclidean spaces.

Theory · Learning Theory

Zhiyuan Fan, Arnab Maiti, Lillian Ratliff, Kevin Jamieson, Gabriele Farina

In this paper, we study the classical Hedge algorithm in combinatorial settings. In each round, the learner selects a vector $\mathbf{x}_t$ from a set $\mathcal{X} \subseteq$ {$0,1$}$^d$, observes a full loss vector $\mathbf{y}_t \in \mathbb{R}^d$, and incurs a loss $\langle \mathbf{x}_t, \mathbf{y}_t \rangle \in [-1,1]$. This setting captures several important problems, including extensive-form games, resource allocation, $m$-sets, online multitask learning, and shortest-path problems on directed acyclic graphs (DAGs). It is well known that Hedge achieves a regret of $\mathcal{O}\big(\sqrt{T \log |\mathcal{X}|}\big)$ after $T$ rounds of interaction. In this paper, we ask whether Hedge is optimal across all combinatorial settings. To that end, we show that for any $\mathcal{X} \subseteq$ {$0,1$}$^d$, Hedge is near-optimal—specifically, up to a $\sqrt{\log d}$ factor—by establishing a lower bound of $\Omega\big(\sqrt{T \log(|\mathcal{X}|)/\log d}\big)$ that holds for any algorithm. We then identify a natural class of combinatorial sets—namely, $m$-sets with $\log d \leq m \leq \sqrt{d}$—for which this lower bound is tight, and for which Hedge is provably suboptimal by a factor of exactly $\sqrt{\log d}$. At the same time, we show that Hedge is optimal for online multitask learning, a generalization of the classical $K$-experts problem. Finally, we leverage the near-optimality of Hedge to establish the existence of a near-optimal regularizer for online shortest-path problems in DAGs—a setting that subsumes a broad range of combinatorial domains. Specifically, we show that the classical Online Mirror Descent (OMD) algorithm, when instantiated with the dilated entropy regularizer, is iterate-equivalent to Hedge, and therefore inherits its near-optimal regret guarantees for DAGs.

Theory · Game Theory

Francesco Emanuele Stradi, Matteo Castiglioni, Alberto Marchesi, Nicola Gatti, Christian Kroer

We study online decision making problems under resource constraints, where both reward and cost functions are drawn from distributions that may change adversarially over time. We focus on two canonical settings: $(i)$ online resource allocation where rewards and costs are observed before action selection, and $(ii)$ online learning with resource constraints where they are observed after action selection, under full feedback or bandit feedback. It is well known that achieving sublinear regret in these settings is impossible when the rewards and cost distributions may change arbitrarily over time. To address this challenge, we analyze a framework in which the learner is guided by a spending plan—a sequence prescribing expected resource usage across rounds. We design general (primal-)dual methods that achieve sublinear regret with respect to baselines that follow the spending plan. Crucially, the performance of our algorithms improves when the spending plan ensures a well-balanced distribution of the budget across rounds. We additionally provide a robust variant of our methods to handle worst-case scenarios where the spending plan is highly imbalanced. To conclude, we study the regret of our algorithms when competing against benchmarks that deviate from the prescribed spending plan.

Theory · Game Theory

Kira Goldner, Taylor Lundy

We initiate the study of multidimensional Bayesian utility maximization, focusing on the unit-demand setting where values are i.i.d. across both items and buyers. The seminal result of Hartline and Roughgarden '08 studies simple, information-robust mechanisms that maximize utility for $n$ i.i.d. agents and $m$ identical items via an approximation to social welfare as an upper bound, and they prove this gap between optimal utility and social welfare is $\Theta(1+\log{n/m})$ in this setting. We extend these results to the multidimensional setting. To do so, we develop simple, prior-independent, approximately-optimal mechanisms, targeting the simplest benchmark of optimal welfare. We give a $(1-1/e)$-approximation when there are more items than buyers, and a $\Theta(\log{n/m})$-approximation when there are more buyers than items, and we prove that this bound is tight in both $n$ and $m$ by reducing the i.i.d. unit-demand setting to the identical items setting. Finally, we include an extensive discussion section on why Bayesian utility maximization is a promising research direction. In particular, we characterize complexities in this setting that defy our intuition from the welfare and revenue literature, and motivate why coming up with a better benchmark than welfare is a hard problem itself.

Theory · Game Theory

Noah Krever, Jakub Cerny, Moise Blanchard, Christian Kroer

Game-theoretic algorithms are commonly benchmarked on recreational games, classical constructs from economic theory such as congestion and dispersion games, or entirely random game instances. While the past two decades have seen the rise of security games -- grounded in real-world scenarios like patrolling and infrastructure protection -- their practical evaluation has been hindered by limited access to the datasets used to generate them. In particular, although the structural components of these games (e.g., patrol paths derived from maps) can be replicated, the critical data defining target values -- central to utility modeling -- remain inaccessible. In this paper, we introduce a flexible framework that leverages open-access datasets to generate realistic matrix and security game instances. These include animal movement data for modeling anti-poaching scenarios and demographic and infrastructure data for infrastructure protection. Our framework allows users to customize utility functions and game parameters, while also offering a suite of preconfigured instances. We provide theoretical results highlighting the degeneracy and limitations of benchmarking on random games, and empirically compare our generated games against random baselines across a variety of standard algorithms for computing Nash and Stackelberg equilibria, including linear programming, incremental strategy generation, and self-play with no-regret learners.

Deep Learning · Theory

Sofiane Ennadir, Levente Zólyomi, Oleg Smirnov, Tianze Wang, John Pertoft, Filip Cornell, Lele Cao

Transformer models have become the dominant backbone for sequence modeling, leveraging self-attention to produce contextualized token representations. These are typically aggregated into fixed-size vectors via pooling operations for downstream tasks. While much of the literature has focused on attention mechanisms, the role of pooling remains underexplored despite its critical impact on model behavior. In this paper, we introduce a theoretical framework that rigorously characterizes the expressivity of Transformer-based models equipped with widely used pooling methods by deriving closed-form bounds on their representational capacity and the ability to distinguish similar inputs. Our analysis extends to different variations of attention formulations, demonstrating that these bounds hold across diverse architectural variants. We empirically evaluate pooling strategies across tasks requiring both global and local contextual understanding, spanning three major modalities: computer vision, natural language processing, and time-series analysis. Results reveal consistent trends in how pooling choices affect accuracy, sensitivity, and optimization behavior. Our findings unify theoretical and empirical perspectives, providing practical guidance for selecting or designing pooling mechanisms suited to specific tasks. This work positions pooling as a key architectural component in Transformer models and lays the foundation for more principled model design beyond attention alone.

Theory · Game Theory

Francesco Bacchiocchi, Francesco Emanuele Stradi, Matteo Castiglioni, Alberto Marchesi, Nicola Gatti

In Bayesian persuasion, an informed sender strategically discloses information to a receiver so as to persuade them to undertake desirable actions. Recently, Markov persuasion processes (MPPs) have been introduced to capture sequential scenarios where a sender faces a stream of myopic receivers in a Markovian environment. The MPPs studied so far in the literature suffer from issues that prevent them from being fully operational in practice, e.g., they assume that the sender knows receivers' rewards. We fix such issues by addressing MPPs where the sender has no knowledge about the environment. We design a learning algorithm for the sender, working with partial feedback. We prove that its regret with respect to an optimal information-disclosure policy grows sublinearly in the number of episodes, as it is the case for the loss in persuasiveness cumulated while learning. Moreover, we provide lower bounds for our setting matching the guarantees of our algorithm.

Theory · Game Theory

Kshipra Bhawalkar, Marios Mertzanidis, Divyarthi Mohan, Alexandros Psomas

We study revenue maximization for agents with additive preferences, subject to downward-closed constraints on the set of feasible allocations. In seminal work,~\citet{alaei2014bayesian} introduced a powerful multi-to-single agent reduction based on an ex-ante relaxation of the multi-agent problem. This reduction employs a rounding procedure which is an online contention resolution scheme (OCRS) in disguise, a now widely-used method for rounding fractional solutions in online Bayesian and stochastic optimization problems. In this paper, we leverage our vantage point, 10 years after the work of Alaei, with a rich OCRS toolkit and modern approaches to analyzing multi-agent mechanisms; we introduce a general framework for designing non-sequential and sequential multi-agent, revenue-maximizing mechanisms, capturing a wide variety of problems Alaei's framework could not address. Our framework uses an \emph{interim} relaxation, that is rounded to a feasible mechanism using what we call a two-level OCRS, which allows for some structured dependence between the activation of its input elements. For a wide family of constraints, we can construct such schemes using existing OCRSs as a black box; for other constraints, such as knapsack, we construct such schemes from scratch. We demonstrate numerous applications of our framework, including a sequential mechanism that guarantees a $\frac{2e}{e-1} \approx 3.16$ approximation to the optimal revenue for the case of additive agents subject to matroid feasibility constraints. Finally, we show how our framework can be easily extended to multi-parameter procurement auctions, where we provide an OCRS for Stochastic Knapsack that might be of independent interest.

Theory · Game Theory

Linjian Meng, Tianpei Yang, Youzhi Zhang, Zhenxing Ge, Shangdong Yang, Tianyu Ding, Wenbin Li, Bo An, Yang Gao

To establish last-iterate convergence for Counterfactual Regret Minimization (CFR) algorithms in learning a Nash equilibrium (NE) of extensive-form games (EFGs), recent studies reformulate learning an NE of the original EFG as learning the NEs of a sequence of (perturbed) regularized EFGs. Hence, proving last-iterate convergence in solving the original EFG reduces to proving last-iterate convergence in solving (perturbed) regularized EFGs. However, these studies only establish last-iterate convergence for Online Mirror Descent (OMD)-based CFR algorithms instead of Regret Matching (RM)-based CFR algorithms in solving perturbed regularized EFGs, resulting in a poor empirical convergence rate, as RM-based CFR algorithms typically outperform OMD-based CFR algorithms. In addition, as solving multiple perturbed regularized EFGs is required, fine-tuning across multiple perturbed regularized EFGs is infeasible, making parameter-free algorithms highly desirable. This paper show that CFR$^+$, a classical parameter-free RM-based CFR algorithm, achieves last-iterate convergence in learning an NE of perturbed regularized EFGs. This is the first parameter-free last-iterate convergence for RM-based CFR algorithms in perturbed regularized EFGs. Leveraging CFR$^+$ to solve perturbed regularized EFGs, we get Reward Transformation CFR$^+$ (RTCFR$^+$). Importantly, we extend prior work on the parameter-free property of CFR$^+$, enhancing its stability, which is vital for the empirical convergence of RTCFR$^+$. Experiments show that RTCFR$^+$ exhibits a significantly faster empirical convergence rate than existing algorithms that achieve theoretical last-iterate convergence. Interestingly, RTCFR$^+$ show performance no worse than average-iterate convergence CFR algorithms. It is the first last-iterate convergence algorithm to achieve such performance. Our code is available at https://github.com/menglinjian/NeurIPS-2025-RTCFR.

Theory · Game Theory

Yuma Fujimoto, Kenshi Abe, Kaito Ariu

This study raises and addresses the problem of time-delayed feedback in learning in games. Because learning in games assumes that multiple agents independently learn their strategies, a discrepancy in optimization often emerges among the agents. To overcome this discrepancy, the prediction of the future reward is incorporated into algorithms, typically known as Optimistic Follow-the-Regularized-Leader (OFTRL). However, the time delay in observing the past rewards hinders the prediction. Indeed, this study firstly proves that even a single-step delay worsens the performance of OFTRL from the aspects of social regret and convergence. This study proposes the weighted OFTRL (WOFTRL), where the prediction vector of the next reward in OFTRL is weighted $n$ times. We further capture an intuition that the optimistic weight cancels out this time delay. We prove that when the optimistic weight exceeds the time delay, our WOFTRL recovers the good performances that social regret is constant in general-sum normal-form games, and the strategies last-iterate converge to the Nash equilibrium in poly-matrix zero-sum games. The theoretical results are supported and strengthened by our experiments.

Theory · Game Theory

Jiehua Chen, Jiong Guo, Yinghui Wen

We investigate congested assignment problems where agents have preferences over both resources and their associated congestion levels. These agents are \emph{averse} towards congestion, i.e., consistently preferring lower congestion for identical resources. Such scenarios are ubiquitous across domains including traffic management and school choice, where fair resource allocation is essential. We focus on the concept of \emph{competitiveness}, recently introduced by Bogomolnaia and Moulin [6], and contribute a polynomial-time algorithm that determines competitiveness, resolving their open question. Additionally, we explore two optimization variants of congested assignments by examining the problem of finding envy-free or maximally competitive assignments that guarantee a certain amount of social welfare for every agent, termed \emph{top-guarantees} [6]. While we prove that both problems are NP-hard, we develop parameterized algorithms with respect to the number of agents or resources.

Theory · Game Theory

Kyriakos Lotidis, Panayotis Mertikopoulos, Nicholas Bambos, Jose Blanchet

In this paper, we examine the convergence landscape of multi-agent learning under uncertainty. Specifically, we analyze two stochastic models of regularized learning in continuous games—one in continuous and one in discrete time—with the aim of characterizing the long run behavior of the induced sequence of play. In stark contrast to deterministic, full-information models of learning (or models with a vanishing learning rate), we show that the resulting dynamics do not converge in general. In lieu of this, we ask instead which actions are played more often in the long run, and by how much. We show that, in strongly monotone games, the dynamics of regularized learning may wander away from equilibrium infinitely often, but they always return to its vicinity in finite time (which we estimate), and their long-run distribution is sharply concentrated around a neighborhood thereof. We quantify the degree of this concentration, and we show that these favorable properties may all break down if the underlying game is not strongly monotone—underscoring in this way the limits of regularized learning in the presence of persistent randomness and uncertainty

Theory · Game Theory

Tongyang Li, Xinzhao Wang, Yexin Zhang

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.

Reinforcement Learning · Multi-agent

Rakshit Trivedi, Kartik Sharma, David Parkes

Effective human-AI coordination requires artificial agents capable of exhibiting and responding to human-like behaviors while adapting to changing contexts. Imitation learning has emerged as one of the prominent approaches to build such agents by training them to mimic human-demonstrated behaviors. However, current methods struggle to capture the inherent diversity and non-Markovian nature of human behavior and lack the ability to steer behavior at inference time. Drawing inspiration from the theory of human cognitive processes, where inner speech guides action selection before execution, we propose MIMIC (Modeling Inner Motivations for Imitation and Control), a framework that uses language as an internal representation of behavioral intent. MIMIC employs the novel use of vision-language models as linguistic scaffolding to train a conditional variational autoencoder capable of generating inner speech from observations. A diffusion-based behavior cloning policy then selects actions conditioned on current observations and the generated inner speech. MIMIC enables fine-grained steering of behavior at inference time by conditioning the agent on behavior-specific speech. Experiments across robotic manipulation tasks and human-AI collaboration games demonstrate that MIMIC significantly enhances both behavior diversity and fidelity to human demonstrations while enabling nuanced behavioral steering without training on additional demonstrations.

Theory · Game Theory

Vincent Leon, Iosif Sakos, Ryann Sim, Antonios Varvitsiotis

Concavity and its refinements underpin tractability in multiplayer games, where players independently choose actions to maximize their own payoffs which depend on other players’ actions. In *concave* games, where players' strategy sets are compact and convex, and their payoffs are concave in their own actions, strong guarantees follow: Nash equilibria always exist and decentralized algorithms converge to equilibria. If the game is furthermore *monotone*, an even stronger guarantee holds: Nash equilibria are unique under strictness assumptions. Unfortunately, we show that *certifying* concavity or monotonicity is NP-hard, already for games where utilities are multivariate polynomials and compact, convex basic semialgebraic strategy sets—an expressive class that captures extensive-form games with imperfect recall. On the positive side, we develop two hierarchies of sum-of-squares programs that certify concavity and monotonicity of a given game, and each level of the hierarchies can be solved in polynomial time. We show that almost all concave/monotone games are certified at some finite level of the hierarchies. Subsequently, we introduce the classes of SOS-concave/monotone games, which globally approximate concave/monotone games, and show that for any given game we can compute the closest SOS-concave/monotone game in polynomial time. Finally, we apply our techniques to canonical examples of extensive-form games with imperfect recall.

Deep Learning · Generative Models and Autoencoders

Weizhong Huang, Yuxin Zhang, Xiawu Zheng, Fei Chao, Rongrong Ji, Liujuan Cao

Mixture-of-Experts (MoE) architectures enable efficient scaling of large language models but face prohibitive memory demands due to massive parameterization. Existing pruning methods rely on heuristic metrics or impractical enumeration of expert subsets, leading to suboptimal performance or scalability. In this paper, we propose Shapley-MoE, an efficient pruning method for MoE models inspired by cooperative game theory. By quantifying each expert’s contribution via Shapley value, our method identifies important experts without exhaustive combination evaluations. To overcome the NP-hard complexity of exact Shapley computation, we introduce a Monte Carlo sampling strategy for efficient approximation that reduces complexity to quadratic time. However, vanilla Monte Carlo sampling still faces issues of insufficient estimation accuracy and low sampling efficiency. To address these issues, we further propose two novel methods to improve sampling accuracy and efficiency: (1) Early Truncation, which early terminates unstable sampling steps caused by overly small expert subsets, and (2) Router-Guided Importance Sampling, which prioritize sampling important expert subsets using gating activation probabilities. Both theoretical and experimental analyses show that both methods can accelerate Shapley value estimation and improve accuracy. Extensive empirical evaluations demonstrate that our pruned MoE models outperform existing expert pruning methods. Notably, when applied to the Qwen2-57B-A14B model, our method reduces the number of experts by 25% with only a 0.92 increase in perplexity and over 96.4% of the average zero-shot accuracy is maintained.

Theory · Game Theory

Ioannis Anagnostides, Ioannis Panageas, Tuomas Sandholm, Jingming Yan

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 $\epsilon$-Nash equilibria in $3$-player $\text{\emph{adversarial}}$ team games---wherein a team of $2$ players competes against a $\text{\emph{single}}$ adversary---is $\textsf{CLS}$-complete, resolving the complexity of Nash equilibria in such settings. Our proof proceeds by reducing from $\text{\emph{symmetric}}$ $\epsilon$-Nash equilibria in $\text{\emph{symmetric}}$, 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 $\text{\emph{symmetric}}$ (first-order) equilibria in $\text{\emph{symmetric}}$ min-max optimization is $\textsf{PPAD}$-complete, even for quadratic functions. Building on this reduction, we show that computing symmetric $\epsilon$-Nash equilibria in symmetric, $6$-player ($3$ vs. $3$) team zero-sum games is also $\textsf{PPAD}$-complete, even for $\epsilon = \text{poly}(1/n)$. 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 $\text{\emph{non-symmetric}}$ $\text{poly}(1/n)$-equilibrium in symmetric min-max optimization is $\textsf{FNP}$-hard.