arXiv subjects

Tuomas Sandholm

Publications and source records attributed to Tuomas Sandholm.

At least 55 records · Page 3Linked to original sources

arXiv2025

The Complexity of Symmetric Equilibria in Min-Max Optimization and Team Zero-Sum Games

We consider the problem of computing stationary points in min-max optimization, with a particular focus on the special case of computing Nash equilibria in (two-)team zero-sum games. We first show that computing $ε$-Nash equilibria in $3$-player \emph{adversarial} team games -- wherein a team of $2$ players competes against a \emph{single} adversary -- is \textsf{CLS}-complete, resolving the complexity of Nash equilibria in such settings. Our proof proceeds by reducing from \emph{symmetric} $ε$-Nash equilibria in \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). We also provide some further results concerning equilibrium computation in adversarial team games. Moreover, we establish that computing \emph{symmetric} (first-order) equilibria in \emph{symmetric} min-max optimization is \textsf{PPAD}-complete, even for quadratic functions. Building on this reduction, we further show that computing symmetric $ε$-Nash equilibria in symmetric, $6$-player ($3$ vs. $3$) team zero-sum games is also \textsf{PPAD}-complete, even for $ε= \text{poly}(1/n)$. As an immediate 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 \emph{non-symmetric} $\text{poly}(1/n)$-equilibrium in symmetric min-max optimization is \textsf{FNP}-hard.

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arXiv2025

Game-Theoretic Multiagent Reinforcement Learning

Tremendous advances have been made in multiagent reinforcement learning (MARL). MARL corresponds to the learning problem in a multiagent system in which multiple agents learn simultaneously. It is an interdisciplinary field of study with a long history that includes game theory, machine learning, stochastic control, psychology, and optimization. Despite great successes in MARL, there is a lack of a self-contained overview of the literature that covers game-theoretic foundations of modern MARL methods and summarizes the recent advances. The majority of existing surveys are outdated and do not fully cover the recent developments since 2010. In this work, we provide a monograph on MARL that covers both the fundamentals and the latest developments on the research frontier. The goal of this monograph is to provide a self-contained assessment of the current state-of-the-art MARL techniques from a game-theoretic perspective. We expect this work to serve as a stepping stone for both new researchers who are about to enter this fast-growing field and experts in the field who want to obtain a panoramic view and identify new directions based on recent advances.

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arXiv2025

On the Power of Perturbation under Sampling in Solving Extensive-Form Games

We investigate how perturbation does and does not improve the Follow-the-Regularized-Leader (FTRL) algorithm in solving imperfect-information extensive-form games under sampling, where payoffs are estimated from sampled trajectories. While optimistic algorithms are effective under full feedback, they often become unstable in the presence of sampling noise. Payoff perturbation offers a promising alternative for stabilizing learning and achieving \textit{last-iterate convergence}. We present a unified framework for \textit{Perturbed FTRL} algorithms and study two variants: PFTRL-KL (standard KL divergence) and PFTRL-RKL (Reverse KL divergence), the latter featuring an estimator with both unbiasedness and conditional zero variance. While PFTRL-KL generally achieves equivalent or better performance across benchmark games, PFTRL-RKL consistently outperforms it in Leduc poker, whose structure is more asymmetric than the other games in a sense. Given the modest advantage of PFTRL-RKL, we design the second experiment to isolate the effect of conditional zero variance, showing that the variance-reduction property of RKL improve last-iterate performance.

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arXiv2025

Weakest Bidder Types and New Core-Selecting Combinatorial Auctions

Core-selecting combinatorial auctions are popular auction designs that constrain prices to eliminate the incentive for any group of bidders -- with the seller -- to renegotiate for a better deal. They help overcome the low-revenue issues of classical combinatorial auctions. We introduce a new class of core-selecting combinatorial auctions that leverage bidder information available to the auction designer. We model such information through constraints on the joint type space of the bidders -- these are constraints on bidders' private valuations that are known to hold by the auction designer before bids are elicited. First, we show that type space information can overcome the well-known impossibility of incentive-compatible core-selecting combinatorial auctions. We present a revised and generalized version of that impossibility result that depends on how much information is conveyed by the type spaces. We then devise a new family of core-selecting combinatorial auctions and show that they minimize the sum of bidders' incentives to deviate from truthful bidding. We develop new constraint generation techniques -- and build upon existing quadratic programming techniques -- to compute core prices, and conduct experiments to evaluate the incentive, revenue, fairness, and computational merits of our new auctions. Our new core-selecting auctions directly improve upon existing designs that have been used in many high-stakes auctions around the world. We envision that they will be a useful addition to any auction designer's toolkit.

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arXiv2025

Revenue-Optimal Efficient Mechanism Design with General Type Spaces

We derive the revenue-optimal efficient (welfare-maximizing) mechanism in a general multidimensional mechanism design setting when type spaces -- that is, the underlying domains from which agents' values come from -- can capture arbitrarily complex informational constraints about the agents. Type spaces can encode information about agents representing, for example, machine learning predictions of agent behavior, institutional knowledge about feasible market outcomes (such as item substitutability or complementarity in auctions), and correlations between multiple agents. Prior work has only dealt with connected type spaces, which are not expressive enough to capture many natural kinds of constraints such as disjunctive constraints. We provide two characterizations of the optimal mechanism based on allocations and connected components; both make use of an underlying network flow structure to the mechanism design. Our results significantly generalize and improve the prior state of the art in revenue-optimal efficient mechanism design. They also considerably expand the scope of what forms of agent information can be expressed and used to improve revenue.

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arXiv2025

Expected Variational Inequalities

Variational inequalities (VIs) encompass many fundamental problems in diverse areas ranging from engineering to economics and machine learning. However, their considerable expressivity comes at the cost of computational intractability. In this paper, we introduce and analyze a natural relaxation -- which we refer to as expected variational inequalities (EVIs) -- where the goal is to find a distribution that satisfies the VI constraint in expectation. By adapting recent techniques from game theory, we show that, unlike VIs, EVIs can be solved in polynomial time under general (nonmonotone) operators. EVIs capture the seminal notion of correlated equilibria, but enjoy a greater reach beyond games. We also employ our framework to capture and generalize several existing disparate results, including from settings such as smooth games, and games with coupled constraints or nonconcave utilities.

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arXiv2025

Computing Game Symmetries and Equilibria That Respect Them

Strategic interactions can be represented more concisely, and analyzed and solved more efficiently, if we are aware of the symmetries within the multiagent system. Symmetries also have conceptual implications, for example for equilibrium selection. We study the computational complexity of identifying and using symmetries. Using the classical framework of normal-form games, we consider game symmetries that can be across some or all players and/or actions. We find a strong connection between game symmetries and graph automorphisms, yielding graph automorphism and graph isomorphism completeness results for characterizing the symmetries present in a game. On the other hand, we also show that the problem becomes polynomial-time solvable when we restrict the consideration of actions in one of two ways. Next, we investigate when exactly game symmetries can be successfully leveraged for Nash equilibrium computation. We show that finding a Nash equilibrium that respects a given set of symmetries is PPAD- and CLS-complete in general-sum and team games respectively -- that is, exactly as hard as Brouwer fixed point and gradient descent problems. Finally, we present polynomial-time methods for the special cases where we are aware of a vast number of symmetries, or where the game is two-player zero-sum and we do not even know the symmetries.

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arXiv2025

Automated Design of Affine Maximizer Mechanisms in Dynamic Settings

Dynamic mechanism design is a challenging extension to ordinary mechanism design in which the mechanism designer must make a sequence of decisions over time in the face of possibly untruthful reports of participating agents. Optimizing dynamic mechanisms for welfare is relatively well understood. However, there has been less work on optimizing for other goals (e.g. revenue), and without restrictive assumptions on valuations, it is remarkably challenging to characterize good mechanisms. Instead, we turn to automated mechanism design to find mechanisms with good performance in specific problem instances. In fact, the situation is similar even in static mechanism design. However, in the static case, optimization/machine learning-based automated mechanism design techniques have been successful in finding high-revenue mechanisms in cases beyond the reach of analytical results. We extend the class of affine maximizer mechanisms to MDPs where agents may untruthfully report their rewards. This extension results in a challenging bilevel optimization problem in which the upper problem involves choosing optimal mechanism parameters, and the lower problem involves solving the resulting MDP. Our approach can find truthful dynamic mechanisms that achieve strong performance on goals other than welfare, and can be applied to essentially any problem setting-without restrictions on valuations-for which RL can learn optimal policies.

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arXiv2025

A Multiagent Path Search Algorithm for Large-Scale Coalition Structure Generation

Coalition structure generation (CSG), i.e. the problem of optimally partitioning a set of agents into coalitions to maximize social welfare, is a fundamental computational problem in multiagent systems. This problem is important for many applications where small run times are necessary, including transportation and disaster response. In this paper, we develop SALDAE, a multiagent path finding algorithm for CSG that operates on a graph of coalition structures. Our algorithm utilizes a variety of heuristics and strategies to perform the search and guide it. It is an anytime algorithm that can handle large problems with hundreds and thousands of agents. We show empirically on nine standard value distributions, including disaster response and electric vehicle allocation benchmarks, that our algorithm enables a rapid finding of high-quality solutions and compares favorably with other state-of-the-art methods.

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arXiv2025

Efficient $Φ$-Regret Minimization with Low-Degree Swap Deviations in Extensive-Form Games

Recent breakthrough results by Dagan, Daskalakis, Fishelson and Golowich [2023] and Peng and Rubinstein [2023] established an efficient algorithm attaining at most $ε$ swap regret over extensive-form strategy spaces of dimension $N$ in $N^{\tilde O(1/ε)}$ rounds. On the other extreme, Farina and Pipis [2023] developed an efficient algorithm for minimizing the weaker notion of linear-swap regret in $\mathsf{poly}(N)/ε^2$ rounds. In this paper, we develop efficient parameterized algorithms for regimes between these two extremes. We introduce the set of $k$-mediator deviations, which generalize the untimed communication deviations recently introduced by Zhang, Farina and Sandholm [2024] to the case of having multiple mediators, and we develop algorithms for minimizing the regret with respect to this set of deviations in $N^{O(k)}/ε^2$ rounds. Moreover, by relating $k$-mediator deviations to low-degree polynomials, we show that regret minimization against degree-$k$ polynomial swap deviations is achievable in $N^{O(kd)^3}/ε^2$ rounds, where $d$ is the depth of the game, assuming a constant branching factor. For a fixed degree $k$, this is polynomial for Bayesian games and quasipolynomial more broadly when $d = \mathsf{polylog} N$ -- the usual balancedness assumption on the game tree.

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arXiv2025

Solving Infinite-Player Games with Player-to-Strategy Networks

We present a new approach to solving games with a countably or uncountably infinite number of players. Such games are often used to model multiagent systems with a large number of agents. The latter are frequently encountered in economics, financial markets, crowd dynamics, congestion analysis, epidemiology, and population ecology, among other fields. Our two primary contributions are as follows. First, we present a way to represent strategy profiles for an infinite number of players, which we name a Player-to-Strategy Network (P2SN). Such a network maps players to strategies, and exploits the generalization capabilities of neural networks to learn across an infinite number of inputs (players) simultaneously. Second, we present an algorithm, which we name Shared-Parameter Simultaneous Gradient (SPSG), for training such a network, with the goal of finding an approximate Nash equilibrium. This algorithm generalizes simultaneous gradient ascent and its variants, which are classical equilibrium-seeking dynamics used for multiagent reinforcement learning. We test our approach on infinite-player games and observe its convergence to approximate Nash equilibria. Our method can handle games with infinitely many states, infinitely many players, infinitely many actions (and mixed strategies on them), and discontinuous utility functions.

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arXiv2025

On the Interplay between Social Welfare and Tractability of Equilibria

Computational tractability and social welfare (aka. efficiency) of equilibria are two fundamental but in general orthogonal considerations in algorithmic game theory. Nevertheless, we show that when (approximate) full efficiency can be guaranteed via a smoothness argument à la Roughgarden, Nash equilibria are approachable under a family of no-regret learning algorithms, thereby enabling fast and decentralized computation. We leverage this connection to obtain new convergence results in large games -- wherein the number of players $n \gg 1$ -- under the well-documented property of full efficiency via smoothness in the limit. Surprisingly, our framework unifies equilibrium computation in disparate classes of problems including games with vanishing strategic sensitivity and two-player zero-sum games, illuminating en route an immediate but overlooked equivalence between smoothness and a well-studied condition in the optimization literature known as the Minty property. Finally, we establish that a family of no-regret dynamics attains a welfare bound that improves over the smoothness framework while at the same time guaranteeing convergence to the set of coarse correlated equilibria. We show this by employing the clairvoyant mirror descent algortihm recently introduced by Piliouras et al.

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arXiv2024

The Value of Recall in Extensive-Form Games

Imperfect-recall games, in which players may forget previously acquired information, have found many practical applications, ranging from game abstractions to team games and testing AI agents. In this paper, we quantify the utility gain by endowing a player with perfect recall, which we call the value of recall (VoR). While VoR can be unbounded in general, we parameterize it in terms of various game properties, namely the structure of chance nodes and the degree of absentmindedness (the number of successive times a player enters the same information set). Further, we identify several pathologies that arise with VoR, and show how to circumvent them. We also study the complexity of computing VoR, and how to optimally apportion partial recall. Finally, we connect VoR to other previously studied concepts in game theory, including the price of anarchy. We use that connection in conjunction with the celebrated smoothness framework to characterize VoR in a broad class of games.

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arXiv2024

Barriers to Welfare Maximization with No-Regret Learning

A celebrated result in the interface of online learning and game theory guarantees that the repeated interaction of no-regret players leads to a coarse correlated equilibrium (CCE) -- a natural game-theoretic solution concept. Despite the rich history of this foundational problem and the tremendous interest it has received in recent years, a basic question still remains open: how many iterations are needed for no-regret players to approximate an equilibrium? In this paper, we establish the first computational lower bounds for that problem in two-player (general-sum) games under the constraint that the CCE reached approximates the optimal social welfare (or some other natural objective). From a technical standpoint, our approach revolves around proving lower bounds for computing a near-optimal $T$-sparse CCE -- a mixture of $T$ product distributions, thereby circumscribing the iteration complexity of no-regret learning even in the centralized model of computation. Our proof proceeds by extending a classical reduction of Gilboa and Zemel [1989] for optimal Nash to sparse (approximate) CCE. In particular, we show that the inapproximability of maximum clique precludes attaining any non-trivial sparsity in polynomial time. Moreover, we strengthen our hardness results to apply in the low-precision regime as well via the planted clique conjecture.

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arXiv2024

Computational Lower Bounds for Regret Minimization in Normal-Form Games

A celebrated connection in the interface of online learning and game theory establishes that players minimizing swap regret converge to correlated equilibria (CE) -- a seminal game-theoretic solution concept. Despite the long history of this problem and the renewed interest it has received in recent years, a basic question remains open: how many iterations are needed to approximate an equilibrium under the usual normal-form representation? In this paper, we provide evidence that existing learning algorithms, such as multiplicative weights update, are close to optimal. In particular, we prove lower bounds for the problem of computing a CE that can be expressed as a uniform mixture of $T$ product distributions -- namely, a uniform $T$-sparse CE; such lower bounds immediately circumscribe (computationally bounded) regret minimization algorithms in games. Our results are obtained in the algorithmic framework put forward by Kothari and Mehta (STOC 2018) in the context of computing Nash equilibria, which consists of the sum-of-squares (SoS) relaxation in conjunction with oracle access to a verification oracle; the goal in that framework is to lower bound either the degree of the SoS relaxation or the number of queries to the verification oracle. Here, we obtain two such hardness results, precluding computing i) uniform $\text{log }n$-sparse CE when $ε=\text{poly}(1/\text{log }n)$ and ii) uniform $n^{1 - o(1)}$-sparse CE when $ε= \text{poly}(1/n)$.

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arXiv2024

Convergence of $\text{log}(1/ε)$ for Gradient-Based Algorithms in Zero-Sum Games without the Condition Number: A Smoothed Analysis

Gradient-based algorithms have shown great promise in solving large (two-player) zero-sum games. However, their success has been mostly confined to the low-precision regime since the number of iterations grows polynomially in $1/ε$, where $ε> 0$ is the duality gap. While it has been well-documented that linear convergence -- an iteration complexity scaling as $\textsf{log}(1/ε)$ -- can be attained even with gradient-based algorithms, that comes at the cost of introducing a dependency on certain condition number-like quantities which can be exponentially large in the description of the game. To address this shortcoming, we examine the iteration complexity of several gradient-based algorithms in the celebrated framework of smoothed analysis, and we show that they have polynomial smoothed complexity, in that their number of iterations grows as a polynomial in the dimensions of the game, $\textsf{log}(1/ε)$, and $1/σ$, where $σ$ measures the magnitude of the smoothing perturbation. Our result applies to optimistic gradient and extra-gradient descent/ascent, as well as a certain iterative variant of Nesterov's smoothing technique. From a technical standpoint, the proof proceeds by characterizing and performing a smoothed analysis of a certain error bound, the key ingredient driving linear convergence in zero-sum games. En route, our characterization also makes a natural connection between the convergence rate of such algorithms and perturbation-stability properties of the equilibrium, which is of interest beyond the model of smoothed complexity.

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arXiv2024

Optimal Correlated Equilibria in General-Sum Extensive-Form Games: Fixed-Parameter Algorithms, Hardness, and Two-Sided Column-Generation

We study the problem of finding optimal correlated equilibria of various sorts in extensive-form games: normal-form coarse correlated equilibrium (NFCCE), extensive-form coarse correlated equilibrium (EFCCE), and extensive-form correlated equilibrium (EFCE). We make two primary contributions. First, we introduce a new algorithm for computing optimal equilibria in all three notions. Its runtime depends exponentially only on a parameter related to the information structure of the game. We also prove a fundamental complexity gap: while our size bounds for NFCCE are similar to those achieved in the case of team games by Zhang et al., this is impossible to achieve for the other two concepts under standard complexity assumptions. Second, we propose a two-sided column generation approach for use when the runtime or memory usage of the previous algorithm is prohibitive. Our algorithm improves upon the one-sided approach of Farina et al. by means of a new decomposition of correlated strategies which allows players to re-optimize their sequence-form strategies with respect to correlation plans which were previously added to the support. Experiments show that our techniques outperform the prior state of the art for computing optimal general-sum correlated equilibria.

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arXiv2024

Verifying Approximate Equilibrium in Auctions

In practice, most auction mechanisms are not strategy-proof, so equilibrium analysis is required to predict bidding behavior. In many auctions, though, an exact equilibrium is not known and one would like to understand whether -- manually or computationally generated -- bidding strategies constitute an approximate equilibrium. We develop a framework and methods for estimating the distance of a strategy profile from equilibrium, based on samples from the prior and either bidding strategies or sample bids. We estimate an agent's utility gain from deviating to strategies from a constructed finite subset of the strategy space. We use PAC-learning to give error bounds, both for independent and interdependent prior distributions. The primary challenge is that one may miss large utility gains by considering only a finite subset of the strategy space. Our work differs from prior research in two critical ways. First, we explore the impact of bidding strategies on altering opponents' perceived prior distributions -- instead of assuming the other agents to bid truthfully. Second, we delve into reasoning with interdependent priors, where the type of one agent may imply a distinct distribution for other agents. Our main contribution lies in establishing sufficient conditions for strategy profiles and a closeness criterion for conditional distributions to ensure that utility gains estimated through our finite subset closely approximate the maximum gains. To our knowledge, ours is the first method to verify approximate equilibrium in any auctions beyond single-item ones. Also, ours is the first sample-based method for approximate equilibrium verification.

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