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Tuomas Sandholm

Publications and source records attributed to Tuomas Sandholm.

At least 19 recordsLinked to original sources
arXiv2026

Test-time Reinforcement Learning in Imperfect Information Games

Test-time reasoning has significantly improved performance in domains ranging from games to language models. However, test-time policy changes with formal guarantees on the performance of the resulting strategy remain a challenge in two-player zero-sum imperfect-information games. Existing solutions are limited to tabular methods or single gradient step updates. In this work, we investigate policy-gradient algorithms as a method for scalable test-time reasoning. We extend the concept of gadget game, tabular technique for test-time search, to the reinforcement learning setting. Unlike prior approaches, we represent the gadget game implicitly by modified sampling and neural policy rather then explicitly by constructing it, thereby removing constraints on subgame size. Furthermore, we formally prove that, unlike prior tabular algorithms, regularized policy-gradient algorithms limit possible strategy degradation caused by test-time reasoning, even without the gadget games. Our evaluation across small- and large-scale games confirms that additional test-time training often substantially improves performance relative to the blueprint strategy.

cs.GT↗
arXiv2026

Watermarked Game Solving via Perturbed Regret Minimization

Many real-world interactions among self-interested parties can be modeled by game theory, and the rapid advancements in AI have raised concerns about the possible misuse---accidental or deliberate---of superhuman or human-level game-playing agents by bad actors. While AI watermarking has mainly been applied to LLM-generated texts, a recent line of work proposes developing watermarking techniques for agents in game-theoretic settings. However, existing watermarking techniques for game-theoretic agents are not readily applicable due to their limited scope or capabilities---they are tailored to perfect-information games and are thus inapplicable to richer game types. We propose a new approach to watermarking game-playing agents, which a) can be applied to imperfect-information settings; b) is directly integrated into the learning process itself; and c) incurs only a bounded cost in exploitability. For this purpose, we introduce perturbed regret minimization, which adds perturbations to the utilities prior to observation so as to encourage the learning algorithm to embed the watermark. Our experiments show that the watermark incurs only a small exploitability cost and can be detected within just a couple of hours of gameplay at human speed.

cs.GT↗
arXiv2026

Preference Elicitation for Policy Optimization and Application to Aligning Heart Transplantation with Human Values

Preference elicitation is essential for aligning AI systems with human values. Prior approaches (e.g., for organ allocation) often ask stakeholders to compare the decisions of an algorithm (e.g., patient A vs. patient B). Such a decision-level approach conflates the means with the ends. Instead, we elicit preferences directly over allocation outcomes to learn a utility function for policy optimization. We construct a novel preference elicitation algorithm for linear utilities that outperforms prior techniques in practice. Our algorithm has two phases. The first phase learns cutting planes through pairwise comparisons to rapidly shrink the space of possible attribute weights and warm-starts the second phase by eliminating dominated regions. The second phase then provably converges to the user's utility function. We apply our technique to heart transplant allocation where a policy must balance competing objectives such as post-transplant outcomes, waitlist mortality, geographic ease, and equity. Using our algorithm, we conduct a user study to learn and aggregate a community-aligned utility function, and use it to optimize heart transplant policies that are significantly better aligned with human values. Compared to the hindsight optimum, the status quo policy achieves a competitive ratio of just 0.54, while our method is near-optimal with a competitive ratio of 0.95.

cs.AI↗
arXiv2026

Distributed Constraint Optimization via Online Learning and Iterative Pricing with Application to Large-Scale Satellite Scheduling

Distributed constraint optimization problems (DCOPs) provide a popular framework for distributed decision making under limited communication, but many real-world instances are too large to solve monolithically. We address this challenge from two complementary directions. We revisit the connection between DCOPs and potential games, and adapt modern online learning algorithms for equilibrium finding to DCOPs. We show that these algorithms are competitive with representative incomplete DCOP algorithms. We then turn to decomposition frameworks for large-scale DCOPs, motivated by large-scale decentralized satellite scheduling. We propose a new framework that separates a DCOP into two interacting subproblems: a high-level meta-DCOP for task allocation, and independent local optimization problems for scheduling. To couple the two levels, we develop a novel iterative pricing method that updates the meta-level utilities using feedback from the local optimizers. Combining our online learning methods with our iterative pricing framework, we obtain near-optimal performance on real-world decentralized satellite scheduling problem instances, fulfilling over 99% of observation requests compared with 87% for state-of-the-art baselines.

cs.AI↗
arXiv2026

The Computational Complexity of Team Zero-Sum Games

A celebrated consequence of the minimax theorem is that two-player zero-sum games admit a tractable equilibrium characterization. In many central applications, however, each side comprises multiple independent agents who share a common objective but cannot perfectly coordinate their actions. Such settings can be modeled as \emph{team zero-sum games}, a natural generalization of both two-player zero-sum games and potential games -- the two most well-studied classes of games in algorithmic game theory. In this paper, we settle the complexity of team zero-sum games by establishing that computing Nash equilibria is \PPAD-complete. As a result, despite the global adversarial structure, team zero-sum games are as hard as general-sum games. Our hardness result holds even when i) the precision is inverse polynomial, thereby ruling out a fully polynomial-time approximation scheme (unless $\P = \PPAD$); ii) each team consists of only two players; and iii) the underlying class of games is polymatrix. As a byproduct, we resolve the complexity of group-wise zero-sum polymatrix games, a class introduced and examined in the seminal work of Cai and Daskalakis (SODA '11), and more recently highlighted by Hollender, Maystre, and Nagarajan (ICLR '25). Moreover, we show that computing a first-order stationary point in min-max optimization is \PPAD-complete even for quadratic (multilinear) objectives. From a technical standpoint, we develop a series of team zero-sum game gadgets that allow us to simulate the breakthrough reduction of Bernasconi and Castiglioni (STOC '26). Moreover, to obtain hardness results for quadratic objectives, we make use of a general technique based on linear local approximation, which is of independent interest.

cs.GT↗
arXiv2026

Aligning Data-Driven Predictors with Allocation: A Decision-Focused Approach to Survival Analysis

Machine learning predictors have become essential tools for guiding automated decision making. However, a major misalignment persists: predictive models are typically optimized in terms of standard statistical metrics in isolation from the algorithmic tasks they inform. We highlight this incongruity in the high-stakes domain of organ allocation by demonstrating that any algorithm relying on (even highly accurate) survival predictors optimized for standard metrics -- such as the Concordance index (C-index) -- can yield arbitrarily poor outcomes when used for allocation, failing to guarantee utility better than a uniform random selection. To bridge the gap between survival analysis and policy optimization, we introduce a decision-focused learning approach based on optimizing normalized discounted cumulative gain (NDCG), a mainstay metric in information retrieval. We establish the utility of NDCG in survival analysis by proving that it translates to guarantees on the performance of allocation. Empirically, we propose a bootstrapping approach to optimize the NDCG of existing survival models. Unlike prior work, we also address the challenge of right censorship when evaluating ranking. On historical heart transplant data from the US, our method dramatically boosts the NDCG of baseline models by 50-100%, which translates to tens of thousands of additional life years gained annually when deployed for transplant allocation. We anticipate that our framework will find broader applications in decision making with predictions.

cs.LG↗
arXiv2026

On the Complexity of Correlated Equilibria Beyond Normal-Form Games

Correlated equilibria are a fundamental solution concept in game theory. However, despite decades of research, the complexity beyond games of polynomial type -- such as extensive-form games, congestion or routing games, and more broadly concave games -- has remained a major open problem, first highlighted by Papadimitriou and Roughgarden (JACM '08). In this paper, we resolve several long-standing questions concerning the complexity of correlated equilibria and swap regret minimization. First, we show that computing a correlated equilibrium in concave quadratic games is as hard as computing the fixed point of a contraction mapping (Contr), providing the first strong evidence of intractability. Moreover, we establish an unconditional, information-theoretic lower bound ruling out the existence of a strongly sublinear swap regret minimizer: any online learning algorithm requires exponentially many iterations in the dimension $d$ to guarantee at most $1/\text{poly}(d)$ (average) swap regret. To circumvent these hardness results, we examine the complexity of $\Phi$-equilibria -- tractable relaxations of correlated equilibria. We obtain a fully polynomial-time approximation scheme (FPTAS) for computing poly-dimensional $\Phi$-equilibria in general concave games. We complement this by showing that Contr-hardness persists even under poly-dimensional swap deviations in the regime where the precision $\epsilon$ is exponentially small. Finally, we show that Contr-hardness can be bypassed in the canonical setting of concave \emph{quadratic games}, for which we provide a $\text{poly}(d, \log(1/\epsilon))$-time algorithm for computing poly-dimensional $\Phi$-equilibria. As a byproduct, we obtain an algorithm for computing fixed points of a mapping that is contracting with respect to an unknown Mahalanobis norm, which could be of independent interest.

cs.GT↗
arXiv2026

Domain-Independent Game Abstraction using Word Embedding Techniques

Many games of interest in the real world are often intractably large, thereby necessitating the use of game abstraction to shrink them in size, typically by many magnitudes. Over the last two decades, there have been significant advances in game abstraction; however, the domain-specific nature (usually poker) of much of the prior work prevents those techniques from being easily generalized to other settings without extensively analyzing the game at hand. In this paper, we propose a domain-independent approach to game abstraction, which applies word embedding techniques from the field of natural language processing. Treating each action as a word and gameplay data as a corpus, word vectors can be trained to represent each action as a real-valued vector, which can then be clustered to facilitate game abstraction. We also explore the use of foundational embedding models and show that action embeddings obtained this way can capture a surprising amount of information about the underlying game. Experimental results demonstrate that our proposed game abstraction technique is effective, although it does not outperform specialized algorithms tailored to specific games.

cs.GT↗
arXiv2026

Heuristic Pathologies and Further Variance Reduction via Uncertainty Propagation in the AIVAT Family of Techniques

How should an agent's performance in a multiagent environment be evaluated when there is a limited sample size or a high cost of running a trial? The AIVAT family of variance reduction techniques was proposed to address this challenge by introducing unbiased low-variance estimators of agents' expected payoffs. An important component of AIVAT is a heuristic value function that discriminates between potentially low- and high-value counterfactual histories. A notable gap in the literature is that there is little to no constraint or guideline on how the heuristic value function should be chosen or how uncertainty in its output should be handled. In our first contribution, we parameterize the heuristic value function to highlight AIVAT's potential vulnerabilities: a) the sample variance can be set pathologically low by directly applying gradient descent on the sample variance, and b) one can p-hack to draw a desired statistical conclusion via gradient descent/ascent on the test statistic. The main takeaway is that the heuristic value function should be fixed prior to observing the evaluation data! In our second contribution, we show how the heuristic uncertainty can be propagated to quantify the uncertainty of AIVAT estimates. It is then possible to further reduce the variance using inverse-variance weighted averaging, but AIVAT's unbiasedness guarantee may have to be sacrificed. In our experiments, we use a dataset of 10,000 poker hands to demonstrate our heuristic pathology and uncertainty results, with the latter yielding a 43.0% reduction in the number of samples (poker hands) needed to draw statistical conclusions.

cs.AI↗
arXiv2026

Parallelizing Counterfactual Regret Minimization

Parallelization has played an instrumental role in the field of artificial intelligence (AI), drastically reducing the time taken to train and evaluate large AI models. In contrast to its impact in the broader field of AI, applying parallelization to computational game solving is relatively unexplored, despite its great potential. In this paper, we parallelize the family of counterfactual regret minimization (CFR) algorithms, which were central to important breakthroughs for solving large imperfect-information games. We present a generalized parallelization framework, reframing CFR as a series of linear algebra operations. Then, existing techniques for parallelizing linear algebra operations can be applied to accelerate CFR. We also describe how our technique can be applied to other tabular members of the CFR family of algorithms, including the state-of-the-art, such as CFR+, discounted CFR, and predictive variants of CFR. Experimentally, we show that our CFR implementation on a GPU is up to four orders of magnitude faster than Google DeepMind OpenSpiel's CFR implementations on a CPU.

cs.AI↗
arXiv2026

Watermarking Game-Playing Agents in Perfect-Information Extensive-Form Games

Watermarking techniques for large language models (LLMs), which encode hidden information in the output so its source can be verified, have gained significant attention in recent days, thanks to their potential capability to detect accidental or deliberate misuse. Similar challenges involving model misuse also exist in the context of game-playing, such as when detecting the unauthorized use of AI tools in gaming platforms (e.g., cheating in online chess). In this paper, we initiate the study of how game-playing strategies can be watermarked. We show how the KGW watermark for LLMs can be adapted to watermark game-playing agents in perfect-information extensive-form games. The watermark can then be detected using a statistical test. We show that the degradation in the quality of the watermarked strategy profile, quantified by the expected utility, can be bounded, but there is a tradeoff between detectability and quality. In our experiments, we bootstrap the watermarking framework to various chess engines and demonstrate that a) the impact of the watermark on the quality of the strategy is negligible and b) the watermark can be detected with just a handful of games.

cs.GT↗
arXiv2026

Efficient representations for team and imperfect-recall equilibrium computation

Equilibrium finding in two-player zero-sum games with perfect recall is a well-studied topic that has led to many breakthroughs in computational game theory. This paper aims to generalize such techniques to (timeable) two-player zero-sum games with imperfect recall, or equivalently to two-team zero-sum games. In this setting, the problem of computing a mixed-strategy Nash equilibrium (or, equivalently, a team maxmin equilibrium with correlation) is known to be NP-hard. We connect the imperfect-recall setting with its perfect-recall counterpart through a novel construction we call the belief game. This is a perfect-recall game equivalent to a given (timeable) two-player zero-sum game with imperfect recall. The belief game may be exponentially larger than the original game but can be solved using any standard method. We then show that the strategy spaces of the two players in the belief game can be directly represented as a DAG, leading to a possibly exponential speedup. We call this the team belief DAG (TB-DAG). The TB-DAG simultaneously enjoys essentially optimal parameterized complexity bounds and the advantages of efficient regret minimization techniques. Along the way, we show $\Delta_2^P$-completeness and $\Sigma_2^P$-completeness of finding Nash equilibria in both mixed and behavioral strategies for the class of games we consider. Experimentally, we show that the TB-DAG, when paired with existing learning techniques, yields state-of-the-art performance on a wide variety of benchmark team games.

cs.GT↗
arXiv2026

A Polynomial-Time Algorithm for Variational Inequalities under the Minty Condition

Solving (Stampacchia) variational inequalities (SVIs) is a foundational problem at the heart of optimization. However, this expressivity comes at the cost of computational hardness. As a result, most research has focused on carving out specific subclasses that elude those intractability barriers. A classical property that goes back to the 1960s is the Minty condition, which postulates that the Minty VI (MVI) problem admits a solution. In this paper, we establish the first polynomial-time algorithm -- with complexity growing polynomially in the dimension $d$ and $\log(1/ε)$ -- for solving $ε$-SVIs for Lipschitz continuous mappings under the Minty condition. Prior approaches either incurred an exponentially worse dependence on $1/ε$ (and other natural parameters of the problem) or made more restrictive assumptions, such as monotonicity. To do so, we introduce a new variant of the ellipsoid algorithm whereby separating hyperplanes are obtained after taking a descent step from the center of the ellipsoid. It succeeds even though the set of SVIs can be nonconvex and not fully dimensional. Moreover, when our algorithm is applied to an instance with no MVI solution and fails to identify an SVI solution, it produces a succinct certificate of MVI infeasibility. We also show that deciding whether the Minty condition holds is $\mathsf{coNP}$-complete, thereby establishing that the disjunction of those two problems is polynomial-time solvable even though each problem is individually intractable. We provide several extensions and new applications of our main results. Most notably, we obtain the first polynomial-time algorithms for computing Nash equilibria in multi-player harmonic games. Finally, in two-player general-sum concave games, we give the first polynomial-time algorithm that outputs either a Nash equilibrium or a strict coarse correlated equilibrium.

math.OC↗
arXiv2026

Steering No-Regret Learners to a Desired Equilibrium

A mediator observes no-regret learners playing an extensive-form game repeatedly across $T$ rounds. The mediator attempts to steer players toward some desirable predetermined equilibrium by giving (nonnegative) payments to players. We call this the steering problem. The steering problem captures problems several problems of interest, among them equilibrium selection and information design (persuasion). If the mediator's budget is unbounded, steering is trivial because the mediator can simply pay the players to play desirable actions. We study two bounds on the mediator's payments: a total budget and a per-round budget. If the mediator's total budget does not grow with $T$, we show that steering is impossible. However, we show that it is enough for the total budget to grow sublinearly with $T$, that is, for the average payment to vanish. When players' full strategies are observed at each round, we show that constant per-round budgets permit steering. In the more challenging setting where only trajectories through the game tree are observable, we show that steering is impossible with constant per-round budgets in general extensive-form games, but possible in normal-form games or if the per-round budget may itself depend on $T$. We also show how our results can be generalized to the case when the equilibrium is being computed online while steering is happening. We supplement our theoretical positive results with experiments highlighting the efficacy of steering in large games.

cs.GT↗
arXiv2026

Hidden-Role Games: Equilibrium Concepts and Computation

In this paper, we study the class of games known as hidden-role games in which players are assigned privately to teams and are faced with the challenge of recognizing and cooperating with teammates. This model includes both popular recreational games such as the Mafia/Werewolf family and The Resistance (Avalon) and many real-world settings, such as distributed systems where nodes need to work together to accomplish a goal in the face of possible corruptions. There has been little to no formal mathematical grounding of such settings in the literature, and it was previously not even clear what the right solution concepts (notions of equilibria) should be. A suitable notion of equilibrium should take into account the communication channels available to the players (e.g., can they communicate? Can they communicate in private?). Defining such suitable notions turns out to be a nontrivial task with several surprising consequences. In this paper, we provide the first rigorous definition of equilibrium for hidden-role games, which overcomes serious limitations of other solution concepts not designed for hidden-role games. We then show that in certain cases, including the above recreational games, optimal equilibria can be computed efficiently. In most other cases, we show that computing an optimal equilibrium is at least NP-hard or coNP-hard. Lastly, we experimentally validate our approach by computing exact equilibria for complete 5- and 6-player Avalon instances whose size in terms of number of information sets is larger than $10^{56}$.

cs.GT↗
arXiv2026

General search techniques without common knowledge for imperfect-information games, and application to superhuman Fog of War chess

Since the advent of AI, games have served as progress benchmarks. Meanwhile, imperfect-information variants of chess have existed for over a century, present extreme challenges, and have been the focus of decades of AI research. Beyond calculation needed in regular chess, they require reasoning about information gathering, the opponent's knowledge, signaling, etc. The most popular variant, Fog of War (FoW) chess (a.k.a. dark chess), has been a major challenge problem in imperfect-information game solving since superhuman performance was reached in no-limit Texas hold'em poker. We present Obscuro, the first superhuman AI for FoW chess. It introduces advances to search in imperfect-information games, enabling strong, scalable reasoning. Experiments against the prior state-of-the-art AI and human players -- including the world's best -- show that Obscuro is significantly stronger. FoW chess is the largest (by amount of imperfect information) turn-based zero-sum game in which superhuman performance has been achieved and the largest zero-sum game in which imperfect-information search has been successfully applied.

cs.GT↗
arXiv2026

Scale-Invariant Regret Matching and Online Learning with Optimal Convergence: Bridging Theory and Practice in Zero-Sum Games

A considerable chasm has been looming for decades between theory and practice in zero-sum game solving through first-order methods. Although a convergence rate of $T^{-1}$ has long been established, the most effective paradigm in practice is counterfactual regret minimization (CFR), which is based on regret matching and its modern variants. In particular, the state of the art across most benchmarks is predictive regret matching$^+$ (PRM$^+$). Yet, such algorithms can exhibit slower $T^{-1/2}$ convergence even in self-play. In this paper, we close the gap between theory and practice. We propose a new scale-invariant and parameter-free variant of PRM$^+$, which we call IREG-PRM$^+$. We show that it achieves $T^{-1/2}$ best-iterate and $T^{-1}$ (i.e., optimal) average-iterate convergence guarantees, while also being on par or even better relative to PRM$^+$ on benchmark games. From a technical standpoint, we draw an analogy between (IREG-)PRM$^+$ and optimistic gradient descent with adaptive learning rate. Reflecting this theoretical bridge, we find that the adaptive version of optimistic gradient descent we consider performs on par with IREG-PRM$^+$. This demystifies the effectiveness of the regret matching family vis-a-vis more standard optimization techniques. Moreover, we extend our analysis beyond zero-sum games to a family of variational inequality problems that includes harmonic games, as well as extensive-form games with fully-mixed equilibria, via a new and intriguing connection between CFR and harmonic games. Unlike prior work in harmonic games, our algorithms do not require knowing the underlying weights by virtue of scale invariance. Under the weighted Minty condition, we show that any algorithm satisfying a scale-invariant RVU property (such as IREG-PRM$^+$) has constant regret (in self-play) and $T^{-1/2}$ iterate convergence.

cs.GT↗
arXiv2026

Decision Making under Imperfect Recall: Algorithms and Benchmarks

In game theory, imperfect-recall decision problems model situations in which an agent forgets information it held before. They encompass games such as the ``absentminded driver'' and team games with limited communication. In this paper, we introduce the first benchmark suite for imperfect-recall decision problems. Our benchmarks capture a variety of problem types, including ones concerning privacy in AI systems that elicit sensitive information, and AI safety via testing of agents in simulation. Across 61 problem instances generated using this suite, we evaluate the performance of different algorithms for finding first-order optimal strategies in such problems. In particular, we introduce the family of regret matching (RM) algorithms for nonlinear constrained optimization. This class of parameter-free algorithms has enjoyed tremendous success in solving large two-player zero-sum games, but, surprisingly, they were hitherto relatively unexplored beyond that setting. Our key finding is that RM algorithms consistently outperform commonly employed first-order optimizers such as projected gradient descent, often by orders of magnitude. This establishes, for the first time, the RM family as a formidable approach to large-scale constrained optimization problems.

cs.GT↗