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Theory · Game Theory

MohammadTaghi Hajiaghayi, Sébastien Lahaie, Keivan Rezaei, Suho Shin

In the field of computational advertising, the integration of ads into the outputs of large language models (LLMs) presents an opportunity to support these services without compromising content integrity. This paper introduces novel auction mechanisms for ad allocation and pricing within the textual outputs of LLMs, leveraging retrieval-augmented generation (RAG). We propose a \emph{segment auction} where an ad is probabilistically retrieved for each discourse segment (paragraph, section, or entire output) according to its bid and relevance, following the RAG framework, and priced according to competing bids. We show that our auction maximizes logarithmic social welfare, a new notion of welfare that balances allocation efficiency and fairness, and we characterize the associated incentive-compatible pricing rule. These results are extended to multi-ad allocation per segment. An empirical evaluation validates the feasibility and effectiveness of our approach over several ad auction scenarios, and exhibits inherent tradeoffs in metrics as we allow the LLM more flexibility to allocate ads.

Theory · Game Theory

Gabriele Farina, Charilaos Pipis

It is a well-known fact that correlated equilibria can be computed in polynomial time in a large class of concisely represented games using the celebrated Ellipsoid Against Hope algorithm \citep{Papadimitriou2008:Computing, Jiang2015:Polynomial}. However, the landscape of efficiently computable equilibria in sequential (extensive-form) games remains unknown. The Ellipsoid Against Hope does not apply directly to these games, because they do not have the required ``polynomial type'' property. Despite this barrier, \citet{Huang2008:Computing} altered the algorithm to compute exact extensive-form correlated equilibria.In this paper, we generalize the Ellipsoid Against Hope and develop a simple algorithmic framework for efficiently computing saddle-points in bilinear zero-sum games, even when one of the dimensions is exponentially large. Moreover, the framework only requires a ``good-enough-response'' oracle, which is a weakened notion of a best-response oracle.Using this machinery, we develop a general algorithmic framework for computing exact linear $\Phi$-equilibria in any polyhedral game (under mild assumptions), including correlated equilibria in normal-form games, and extensive-form correlated equilibria in extensive-form games. This enables us to give the first polynomial-time algorithm for computing exact linear-deviation correlated equilibria in extensive-form games, thus resolving an open question by \citet{Farina2023:Polynomial}. Furthermore, even for the cases for which a polynomial time algorithm for exact equilibria was already known, our framework provides a conceptually simpler solution.

Theory · Game Theory

Angelos Assos, Yuval Dagan, Constantinos Daskalakis

Learning algorithms are often used to make decisions in sequential decision-making environments. In multi-agent settings, the decisions of each agent can affect the utilities/losses of the other agents. Therefore, if an agent is good at anticipating the behavior of the other agents, in particular how they will make decisions in each round as a function of their experience that far, it could try to judiciously make its own decisions over the rounds of the interaction so as to influence the other agents to behave in a way that ultimately benefits its own utility. In this paper, we study repeated two-player games involving two types of agents: a learner, which employs an online learning algorithm to choose its strategy in each round; and an optimizer, which knows the learner's utility function and the learner's online learning algorithm. The optimizer wants to plan ahead to maximize its own utility, while taking into account the learner's behavior. We provide two results: a positive result for repeated zero-sum games and a negative result for repeated general-sum games. Our positive result is an algorithm for the optimizer, which exactly maximizes its utility against a learner that plays the Replicator Dynamics --- the continuous-time analogue of Multiplicative Weights Update (MWU). Additionally, we use this result to provide an algorithm for the optimizer against MWU, i.e.~for the discrete-time setting, which guarantees an average utility for the optimizer that is higher than the value of the one-shot game. Our negative result shows that, unless P=NP, there is no Fully Polynomial Time Approximation Scheme (FPTAS) for maximizing the utility of an optimizer against a learner that best-responds to the history in each round. Yet, this still leaves open the question of whether there exists a polynomial-time algorithm that optimizes the utility up to $o(T)$.

Theory · Game Theory

Po-Wei Huang, Patrick Rebentrost

Classical algorithms for market equilibrium computation such as proportional response dynamics face scalability issues with Internet-based applications such as auctions, recommender systems, and fair division, despite having an almost linear runtime in terms of the product of buyers and goods. In this work, we provide the first quantum algorithm for market equilibrium computation with sub-linear performance. Our algorithm provides a polynomial runtime speedup in terms of the product of the number of buyers and goods while reaching the same optimization objective value as the classical algorithm. Numerical simulations of a system with 16384 buyers and goods support our theoretical results that our quantum algorithm provides a significant speedup.

Theory · Game Theory

Ilan Cohen, Alon Eden, Talya Eden, Arsen Vasilyan

We study truthful mechanisms for approximating the Maximin-Share (MMS) allocation of agents with additive valuations for indivisible goods. Algorithmically, constant factor approximations exist for the problem for any number of agents. When adding incentives to the mix, a jarring result by Amanatidis, Birmpas, Christodoulou, and Markakis [EC 2017] shows that the best possible approximation for two agents and $m$ items is $\lfloor \frac{m}{2} \rfloor$. We adopt a learning-augmented framework to investigate what is possible when some prediction on the input is given. For two agents, we give a truthful mechanism that takes agents' ordering over items as prediction. When the prediction is accurate, we give a $2$-approximation to the MMS (consistency), and when the prediction is off, we still get an $\lceil \frac{m}{2} \rceil$-approximation to the MMS (robustness). We further show that the mechanism's performance degrades gracefully in the number of ``mistakes" in the prediction; i.e., we interpolate (up to constant factors) between the two extremes: when there are no mistakes, and when there is a maximum number of mistakes. We also show an impossibility result on the obtainable consistency for mechanisms with finite robustness. For the general case of $n\ge 2$ agents, we give a 2-approximation mechanism for accurate predictions, with relaxed fallback guarantees. Finally, we give experimental results which illustrate when different components of our framework, made to insure consistency and robustness, come into play.

Theory · Game Theory

Parand A. Alamdari, Soroush Ebadian, Ariel Procaccia

We consider the challenge of AI value alignment with multiple individuals that have different reward functions and optimal policies in an underlying Markov decision process. We formalize this problem as one of *policy aggregation*, where the goal is to identify a desirable collective policy. We argue that an approach informed by social choice theory is especially suitable. Our key insight is that social choice methods can be reinterpreted by identifying ordinal preferences with volumes of subsets of the *state-action occupancy polytope*. Building on this insight, we demonstrate that a variety of methods — including approval voting, Borda count, the proportional veto core, and quantile fairness — can be practically applied to policy aggregation.

Theory · Game Theory

Phuong Nam Tran, The Anh Ta, shuqing shi, Debmalya Mandal, Yali Du, Long Tran-Thanh

Reward allocation, also known as the credit assignment problem, has been an important topic in economics, engineering, and machine learning. An important concept in reward allocation is the core, which is the set of stable allocations where no agent has the motivation to deviate from the grand coalition. In previous works, computing the core requires either knowledge of the reward function in deterministic games or the reward distribution in stochastic games. However, this is unrealistic, as the reward function or distribution is often only partially known and may be subject to uncertainty. In this paper, we consider the core learning problem in stochastic cooperative games, where the reward distribution is unknown. Our goal is to learn the expected core, that is, the set of allocations that are stable in expectation, given an oracle that returns a stochastic reward for an enquired coalition each round. Within the class of strictly convex games, we present an algorithm named \texttt{Common-Points-Picking} that returns a point in the expected core given a polynomial number of samples, with high probability. To analyse the algorithm, we develop a new extension of the separation hyperplane theorem for multiple convex sets.t.

Theory · Game Theory

Antoine Scheid, Aymeric Capitaine, Etienne Boursier, Eric Moulines, Michael Jordan, Alain Durmus

In Economics, the concept of externality refers to any indirect effect resulting from an interaction between players and affecting a third party without compensation. Most of the models within which externality has been studied assume that agents have perfect knowledge of their environment and preferences. This is a major hindrance to the practical implementation of many proposed solutions. To adress this issue, we consider a two-players bandit game setting where the actions of one of the player affect the other one. Building upon this setup, we extend the Coase theorem [Coase, 2013], which suggests that the optimal approach for maximizing the social welfare in the presence of externality is to establish property rights, i.e., enabling transfers and bargaining between the players. Nonetheless, this fundamental result relies on the assumption that bargainers possess perfect knowledge of the underlying game. We first demonstrate that in the absence of property rights in the considered online scenario, the social welfare breaks down. We then provide a policy for the players, which allows them to learn a bargaining strategy which maximizes the total welfare, recovering the Coase theorem under uncertainty.

Theory · Game Theory

Yiling Chen, Shi Feng, Fang-Yi Yu

Comparison data elicited from people are fundamental to many machine learning tasks, including reinforcement learning from human feedback for large language models and estimating ranking models. They are typically subjective and not directly verifiable. How to truthfully elicit such comparison data from rational individuals? We design peer prediction mechanisms for eliciting comparison data using a bonus-penalty payment. Our design leverages on the strong stochastic transitivity for comparison data to create symmetrically strongly truthful mechanisms such that truth-telling 1) forms a strict Bayesian Nash equilibrium, and 2) yields the highest payment among all symmetric equilibria. Each individual only needs to evaluate one pair of items and report her comparison in our mechanism.We further extend the bonus-penalty payment concept to eliciting networked data, designing a symmetrically strongly truthful mechanism when agents’ private signals are sampled according to the Ising models. We provide the necessary and sufficient conditions for our bonus-penalty payment to have truth-telling as a strict Bayesian Nash equilibrium. Experiments on two real-world datasets further support our theoretical discoveries.

Theory · Game Theory

Kyriakos Lotidis, Angeliki Giannou, Panayotis Mertikopoulos, Nicholas Bambos

Motivated by the success of Nesterov's accelerated gradient algorithm for convex minimization problems, we examine whether it is possible to achieve similar performance gains in the context of online learning in games.To that end, we introduce a family of accelerated learning methods, which we call “follow the accelerated leader” (FTXL), and which incorporates the use of momentum within the general framework of regularized learning - and, in particular, the exponential / multiplicative weights algorithm and its variants.Drawing inspiration and techniques from the continuous-time analysis of Nesterov's algorithm, we show that FTXL converges locally to strict Nash equilibria at a superlinear rate, achieving in this way an exponential speed-up over vanilla regularized learning methods (which, by comparison, converge to strict equilibria at a geometric, linear rate).Importantly, the FTXL maintains its superlinear convergence rate in a broad range of feedback structures, from deterministic, full information models to stochastic, realization-based ones, and even bandit, payoff-based information, where players are only able to observe their individual realized payoffs.

Theory · Game Theory

Jiehua Chen, Christian Hatschka, Sofia Simola

We introduce a multi-winner reconfiguration model to examine how to transition between subsets of alternatives (aka. committees) through a sequence of minor yet impactful modifications, called reconfiguration path. We analyze this model under four approval-based voting rules: Chamberlin-Courant (CC), Proportional Approval Voting (PAV), Approval Voting (AV), and Satisfaction Approval Voting (SAV). The problem exhibits computational intractability for CC and PAV, and polynomial solvability for AV and SAV. We provide a detailed multivariate complexity analysis for CC and PAV, demonstrating that although the problem remains challenging in many scenarios, there are specific cases that allow for efficient parameterized algorithms.

Theory · Game Theory

Yang Cai, Gabriele Farina, Julien Grand-Clément, Christian Kroer, Chung-Wei Lee, Haipeng Luo, Weiqiang Zheng

Self play via online learning is one of the premier ways to solve large-scale zero-sum games, both in theory and practice. Particularly popular algorithms include optimistic multiplicative weights update (OMWU) and optimistic gradient-descent-ascent (OGDA). While both algorithms enjoy $O(1/T)$ ergodic convergence to Nash equilibrium in two-player zero-sum games, OMWU offers several advantages, including logarithmic dependence on the size of the payoff matrix and $\tilde{O}(1/T)$ convergence to coarse correlated equilibria even in general-sum games. However, in terms of last-iterate convergence in two-player zero-sum games, an increasingly popular topic in this area, OGDA guarantees that the duality gap shrinks at a rate of $(1/\sqrt{T})$, while the best existing last-iterate convergence for OMWU depends on some game-dependent constant that could be arbitrarily large. This begs the question: is this potentially slow last-iterate convergence an inherent disadvantage of OMWU, or is the current analysis too loose? Somewhat surprisingly, we show that the former is true. More generally, we prove that a broad class of algorithms that do not forget the past quickly all suffer the same issue: for any arbitrarily small $\delta>0$, there exists a $2\times 2$ matrix game such that the algorithm admits a constant duality gap even after $1/\delta$ rounds. This class of algorithms includes OMWU and other standard optimistic follow-the-regularized-leader algorithms.

Theory · Game Theory

Qi Ju, Falin Hei, Ting Feng, Dengbing Yi, Zhemei Fang, YunFeng Luo

Counterfactual Regret Minimization (CFR) and its variants are widely recognized as effective algorithms for solving extensive-form imperfect information games. Recently, many improvements have been focused on enhancing the convergence speed of the CFR algorithm. However, most of these variants are not applicable under Monte Carlo (MC) conditions, making them unsuitable for training in large-scale games. We introduce a new MC-based algorithm for solving extensive-form imperfect information games, called MCCFVFP (Monte Carlo Counterfactual Value-Based Fictitious Play). MCCFVFP combines CFR’s counterfactual value calculations with fictitious play’s best response strategy, leveraging the strengths of fictitious play to gain significant advantages in games with a high proportion of dominated strategies. Experimental results show that MCCFVFP achieved convergence speeds approximately 20\%$\sim$50\% faster than the most advanced MCCFR variants in games like poker and other test games.

Interpretability

Maximilian Muschalik, Hubert Baniecki, Fabian Fumagalli, Patrick Kolpaczki, Barbara Hammer, Eyke Hüllermeier

Originally rooted in game theory, the Shapley Value (SV) has recently become an important tool in machine learning research. Perhaps most notably, it is used for feature attribution and data valuation in explainable artificial intelligence. Shapley Interactions (SIs) naturally extend the SV and address its limitations by assigning joint contributions to groups of entities, which enhance understanding of black box machine learning models. Due to the exponential complexity of computing SVs and SIs, various methods have been proposed that exploit structural assumptions or yield probabilistic estimates given limited resources. In this work, we introduce shapiq, an open-source Python package that unifies state-of-the-art algorithms to efficiently compute SVs and any-order SIs in an application-agnostic framework. Moreover, it includes a benchmarking suite containing 11 machine learning applications of SIs with pre-computed games and ground-truth values to systematically assess computational performance across domains. For practitioners, shapiq is able to explain and visualize any-order feature interactions in predictions of models, including vision transformers, language models, as well as XGBoost and LightGBM with TreeSHAP-IQ. With shapiq, we extend shap beyond feature attributions and consolidate the application of SVs and SIs in machine learning that facilitates future research. The source code and documentation are available at https://github.com/mmschlk/shapiq.

Datasets and Benchmarks

Junlin Xie, Ruifei Zhang, Zhihong Chen, Xiang Wan, Guanbin Li

Recently, large language models (LLMs) have achieved superior performance, empowering the development of large multimodal agents (LMAs). An LMA is anticipated to execute practical tasks requires various capabilities including multimodal perception, interaction, reasoning, and decision making. However, existing benchmarks are limited in assessing compositional skills and actions demanded by practical scenarios, where they primarily focused on single tasks and static scenarios. To bridge this gap, we introduce WhodunitBench, a benchmark rooted from murder mystery games, where players are required to utilize the aforementioned skills to achieve their objective (i.e., identifying the `murderer' or hiding themselves), providing a simulated dynamic environment for evaluating LMAs. Specifically, WhodunitBench includes two evaluation modes. The first mode, the arena-style evaluation, is constructed from 50 meticulously curated scripts featuring clear reasoning clues and distinct murderers; The second mode, the chain of evaluation, consists of over 3000 curated multiple-choice questions and open-ended questions, aiming to assess every facet of the murder mystery games for LMAs. Experiments show that although current LMAs show acceptable performance in basic perceptual tasks, they are insufficiently equipped for complex multi-agent collaboration and multi-step reasoning tasks. Furthermore, the full application of the theory of mind to complete games in a manner akin to human behavior remains a significant challenge. We hope this work can illuminate the path forward, providing a solid foundation for the future development of LMAs. Our WhodunitBench is open-source and accessible at: https://github.com/jun0wanan/WhodunitBench-Murder_Mystery_Games

Theory · Game Theory

George Christodoulou, Alkmini Sgouritsa, Ioannis Vlachos

Our work revisits the design of mechanisms via the learning-augmented framework. In this model, the algorithm is enhanced with imperfect (machine-learned) information concerning the input, usually referred to as prediction. The goal is to design algorithms whose performance degrades gently as a function of the prediction error and, in particular, perform well if the prediction is accurate, but also provide a worst-case guarantee under any possible error. This framework has been successfully applied recently to various mechanism design settings, where in most cases the mechanism is provided with a prediction about the types of the players.We adopt a perspective in which the mechanism is provided with an output recommendation. We make no assumptions about the quality of the suggested outcome, and the goal is to use the recommendation to design mechanisms with low approximation guarantees whenever the recommended outcome is reasonable, but at the same time to provide worst-case guarantees whenever the recommendation significantly deviates from the optimal one. We propose a generic, universal measure, which we call quality of recommendation, to evaluate mechanisms across various information settings. We demonstrate how this new metric can provide refined analysis in existing results.This model introduces new challenges, as the mechanism receives limited information comparing to settings that use predictions about the types of the agents. We study, through this lens, several well-studied mechanism design paradigms, devising new mechanisms, but also providing refined analysis for existing ones, using as a metric the quality of recommendation. We complement our positive results, by exploring the limitations of known classes of strategyproof mechanisms that can be devised using output recommendation.

Theory · Game Theory

Alexander Braun, Sherry Sarkar

The secretary problem is one of the fundamental problems in online decision making; a tight competitive ratio for this problem of $1/e \approx 0.368$ has been known since the 1960s. Much more recently, the study of algorithms with predictions was introduced: The algorithm is equipped with a (possibly erroneous) additional piece of information upfront which can be used to improve the algorithm's performance. Complementing previous work on secretary problems with prior knowledge, we tackle the following question: _What is the weakest piece of information that allows us to break the $1/e$ barrier?_To this end, we introduce the secretary problem with predicted additive gap. As in the classical problem, weights are fixed by an adversary and elements appear in random order. In contrast to previous variants of predictions, our algorithm only has access to a much weaker piece of information: an _additive gap_ $c$. This gap is the difference between the highest and $k$-th highest weight in the sequence.Unlike previous pieces of advice, knowing an exact additive gap does not make the problem trivial. Our contribution is twofold. First, we show that for any index $k$ and any gap $c$, we can obtain a competitive ratio of $0.4$ when knowing the exact gap (even if we do not know $k$), hence beating the prevalent bound for the classical problem by a constant. Second, a slightly modified version of our algorithm allows to prove standard robustness-consistency properties as well as improved guarantees when knowing a range for the error of the prediction.

Theory · Game Theory

Ahmed Dönmez, Yüksel Arslantaş, Muhammed Sayin

Multi-team games, prevalent in robotics and resource management, involve team members striving for a joint best response against other teams. Team-Nash equilibrium (TNE) predicts the outcomes of such coordinated interactions. However, can teams of self-interested agents reach TNE? We introduce Team-Fictitious Play (Team-FP), a new variant of fictitious play where agents respond to the last actions of team members and the beliefs formed about other teams with some inertia in action updates. This design is essential in team coordination beyond the classical fictitious play dynamics. We focus on zero-sum potential team games (ZSPTGs) where teams can interact pairwise while the team members do not necessarily have identical payoffs. We show that Team-FP reaches near TNE in ZSPTGs with a quantifiable error bound. We extend Team-FP dynamics to multi-team Markov games for model-based and model-free cases. The convergence analysis tackles the challenge of non-stationarity induced by evolving opponent strategies based on the optimal coupling lemma and stochastic differential inclusion approximation methods. Our work strengthens the foundation for using TNE to predict the behavior of decentralized teams and offers a practical rule for team learning in multi-team environments. We provide extensive simulations of Team-FP dynamics and compare its performance with other widely studied dynamics such as smooth fictitious play and multiplicative weights update. We further explore how different parameters impact the speed of convergence.

Theory · Game Theory

Felipe Garrido Lucero, Benjamin Heymann, Maxime Vono, Patrick Loiseau, Vianney Perchet

We consider the dataset valuation problem, that is the problem of quantifying the incremental gain, to some relevant pre-defined utility of a machine learning task, of aggregating an individual dataset to others.The Shapley value is a natural tool to perform dataset valuation due to its formal axiomatic justification, which can be combined with Monte Carlo integration to overcome the computational tractability challenges. Such generic approximation methods, however, remain expensive in some cases. In this paper, we exploit the knowledge about the structure of the dataset valuation problem to devise more efficient Shapley value estimators. We propose a novel approximation, referred to as discrete uniform Shapley, which is expressed as an expectation under a discrete uniform distribution with support of reasonable size. We justify the relevancy of the proposed framework via asymptotic and non-asymptotic theoretical guarantees and illustrate its benefits via an extensive set of numerical experiments.

Theory · Game Theory

Guru Guruganesh, Yoav Kolumbus, Jon Schneider, Inbal Talgam-Cohen, Emmanouil-Vasileios Vlatakis-Gkaragkounis, Joshua Wang, S. Weinberg

Real-life contractual relations typically involve repeated interactions between the principal and agent, where, despite theoretical appeal, players rarely use complex dynamic strategies and instead manage uncertainty through learning algorithms.In this paper, we initiate the study of repeated contracts with learning agents, focusing on those achieving no-regret outcomes. For the canonical setting where the agent’s actions result in success or failure, we present a simple, optimal solution for the principal: Initially provide a linear contract with scalar $\alpha > 0$, then switch to a zero-scalar contract. This shift causes the agent to “free-fall” through their action space, yielding non-zero rewards for the principal at zero cost. Interestingly, despite the apparent exploitation, there are instances where our dynamic contract can make \emph{both} players better off compared to the best static contract. We then broaden the scope of our results to general linearly-scaled contracts, and, finally, to the best of our knowledge, we provide the first analysis of optimization against learning agents with uncertainty about the time horizon.