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Theory/Game Theory

Gabriele Farina, Julien Grand-Clément, Christian Kroer, Chung-Wei Lee, Haipeng Luo

Regret Matching$^+$ (RM$^+$) and its variants are important algorithms for solving large-scale games.However, a theoretical understanding of their success in practice is still a mystery.Moreover, recent advances on fast convergence in games are limited to no-regret algorithms such as online mirror descent, which satisfy stability.In this paper, we first give counterexamples showing that RM+ and its predictive version can be unstable, which might cause other players to suffer large regret. We then provide two fixes: restarting and chopping off the positive orthant that RM$^+$ works in.We show that these fixes are sufficient to get $O(T^{1/4})$ individual regret and $O(1)$ social regret in normal-form games via RM$^+$ with predictions.We also apply our stabilizing techniques to clairvoyant updates in the uncoupled learning setting for RM$^+$ and prove desirable results akin to recent works for Clairvoyant online mirror descent. Our experiments show the advantages of our algorithms over vanilla RM$^+$-based algorithms in matrix and extensive-form games.

Theory/Game Theory

Yang Cai, Haipeng Luo, Chen-Yu Wei, Weiqiang Zheng

We revisit the problem of learning in two-player zero-sum Markov games, focusing on developing an algorithm that is *uncoupled*, *convergent*, and *rational*, with non-asymptotic convergence rates to Nash equilibrium. We start from the case of stateless matrix game with bandit feedback as a warm-up, showing an $\tilde{\mathcal{O}}(t^{-\frac{1}{8}})$ last-iterate convergence rate. To the best of our knowledge, this is the first result that obtains finite last-iterate convergence rate given access to only bandit feedback. We extend our result to the case of irreducible Markov games, providing a last-iterate convergence rate of $\tilde{\mathcal{O}}(t^{-\frac{1}{9+\varepsilon}})$ for any $\varepsilon>0$. Finally, we study Markov games without any assumptions on the dynamics, and show a *path convergence* rate, a new notion of convergence we defined, of $\tilde{\mathcal{O}}(t^{-\frac{1}{10}})$. Our algorithm removes the synchronization and prior knowledge requirement of Wei et al. (2021), which pursued the same goals as us for irreducible Markov games. Our algorithm is related to Chen et al. (2021) and Cen et al. (2021)'s and also builds on the entropy regularization technique. However, we remove their requirement of communications on the entropy values, making our algorithm entirely uncoupled.

Theory/Game Theory

Nikita Tsoy, Nikola Konstantinov

Collaborative learning techniques have significantly advanced in recent years, enabling private model training across multiple organizations. Despite this opportunity, firms face a dilemma when considering data sharing with competitors—while collaboration can improve a company’s machine learning model, it may also benefit competitors and hence reduce profits. In this work, we introduce a general framework for analyzing this data-sharing trade-off. The framework consists of three components, representing the firms’ production decisions, the effect of additional data on model quality, and the data-sharing negotiation process, respectively. We then study an instantiation of the framework, based on a conventional market model from economic theory, to identify key factors that affect collaboration incentives. Our findings indicate a profound impact of market conditions on the data-sharing incentives. In particular, we find that reduced competition, in terms of the similarities between the firms’ products, and harder learning tasks foster collaboration.

Theory/Game Theory

Mengzi Amy Guo, Donghao Ying, Javad Lavaei, Zuo-Jun Shen

This work is dedicated to the algorithm design in a competitive framework, with the primary goal of learning a stable equilibrium. We consider the dynamic price competition between two firms operating within an opaque marketplace, where each firm lacks information about its competitor. The demand follows the multinomial logit (MNL) choice model, which depends on the consumers' observed price and their reference price, and consecutive periods in the repeated games are connected by reference price updates. We use the notion of stationary Nash equilibrium (SNE), defined as the fixed point of the equilibrium pricing policy for the single-period game, to simultaneously capture the long-run market equilibrium and stability. We propose the online projected gradient ascent algorithm (OPGA), where the firms adjust prices using the first-order derivatives of their log-revenues that can be obtained from the market feedback mechanism. Despite the absence of typical properties required for the convergence of online games, such as strong monotonicity and variational stability, we demonstrate that under diminishing step-sizes, the price and reference price paths generated by OPGA converge to the unique SNE, thereby achieving the no-regret learning and a stable market. Moreover, with appropriate step-sizes, we prove that this convergence exhibits a rate of $\mathcal{O}(1/t)$.

Theory/Game Theory

Hedi Hadiji, Sarah Sachs, Tim van Erven, Wouter Koolen

In the first-order query model for zero-sum $K\times K$ matrix games, players observe the expected pay-offs for all their possible actions under the randomized action played by their opponent. This classical model has received renewed interest after the discovery by Rakhlin and Sridharan that $\epsilon$-approximate Nash equilibria can be computed efficiently from $O(\frac{\ln K}{\epsilon})$ instead of $O(\frac{\ln K}{\epsilon^2})$ queries. Surprisingly, the optimal number of such queries, as a function of both $\epsilon$ and $K$, is not known. We make progress on this question on two fronts. First, we fully characterise the query complexity of learning exact equilibria ($\epsilon=0$), by showing that they require a number of queries that is linear in $K$, which means that it is essentially as hard as querying the whole matrix, which can also be done with $K$ queries. Second, for $\epsilon > 0$, the current query complexity upper bound stands at $O(\min(\frac{\ln(K)}{\epsilon} , K))$. We argue that, unfortunately, obtaining a matching lower bound is not possible with existing techniques: we prove that no lower bound can be derived by constructing hard matrices whose entries take values in a known countable set, because such matrices can be fully identified by a single query. This rules out, for instance, reducing to an optimization problem over the hypercube by encoding it as a binary payoff matrix. We then introduce a new technique for lower bounds, which allows us to obtain lower bounds of order $\tilde\Omega(\log(\frac{1}{K\epsilon})$ for any $\epsilon \leq 1 / (cK^4)$, where $c$ is a constant independent of $K$. We further discuss possible future directions to improve on our techniques in order to close the gap with the upper bounds.

Theory/Game Theory

Yixuan Xu, Steven Jecmen, Zimeng Song, Fei Fang

The assignment of papers to reviewers is a crucial part of the peer review processes of large publication venues, where organizers (e.g., conference program chairs) rely on algorithms to perform automated paper assignment. As such, a major challenge for the organizers of these processes is to specify paper assignment algorithms that find appropriate assignments with respect to various desiderata. Although the main objective when choosing a good paper assignment is to maximize the expertise of each reviewer for their assigned papers, several other considerations make introducing randomization into the paper assignment desirable: robustness to malicious behavior, the ability to evaluate alternative paper assignments, reviewer diversity, and reviewer anonymity. However, it is unclear in what way one should randomize the paper assignment in order to best satisfy all of these considerations simultaneously. In this work, we present a practical, one-size-fits-all method for randomized paper assignment intended to perform well across different motivations for randomness. We show theoretically and experimentally that our method outperforms currently-deployed methods for randomized paper assignment on several intuitive randomness metrics, demonstrating that the randomized assignments produced by our method are general-purpose.

Theory/Game Theory

Darshan Chakrabarti, Jelena Diakonikolas, Christian Kroer

Coordinate descent methods are popular in machine learning and optimization for their simple sparse updates and excellent practical performance. In the context of large-scale sequential game solving, these same properties would be attractive, but until now no such methods were known, because the strategy spaces do not satisfy the typical separable block structure exploited by such methods.We present the first cyclic coordinate-descent-like method for the polytope of sequence-form strategies, which form the strategy spaces for the players in an extensive-form game (EFG). Our method exploits the recursive structure of the proximal update induced by what are known as dilated regularizers, in order to allow for a pseudo block-wise update.We show that our method enjoys a O(1/T) convergence rate to a two-player zero-sum Nash equilibrium, while avoiding the worst-case polynomial scaling with the number of blocks common to cyclic methods. We empirically show that our algorithm usually performs better than other state-of-the-art first-order methods (i.e., mirror prox), and occasionally can even beat CFR$^+$, a state-of-the-art algorithm for numerical equilibrium computation in zero-sum EFGs. We then introduce a restarting heuristic for EFG solving. We show empirically that restarting can lead to speedups, sometimes huge, both for our cyclic method, as well as for existing methods such as mirror prox and predictive CFR$^+$.

Theory/Game Theory

Minbo Gao, Zhengfeng Ji, Tongyang Li, Qisheng Wang

We propose the first online quantum algorithm for zero-sum games with $\widetilde O(1)$ regret under the game setting. Moreover, our quantum algorithm computes an $\varepsilon$-approximate Nash equilibrium of an $m \times n$ matrix zero-sum game in quantum time $\widetilde O(\sqrt{m+n}/\varepsilon^{2.5})$. Our algorithm uses standard quantum inputs and generates classical outputs with succinct descriptions, facilitating end-to-end applications. Technically, our online quantum algorithm "quantizes" classical algorithms based on the optimistic multiplicative weight update method. At the heart of our algorithm is a fast quantum multi-sampling procedure for the Gibbs sampling problem, which may be of independent interest.

Theory/Game Theory

Zhijian Duan, Haoran Sun, Yurong Chen, Xiaotie Deng

Automated auction design aims to find empirically high-revenue mechanisms through machine learning. Existing works on multi item auction scenarios can be roughly divided into RegretNet-like and affine maximizer auctions (AMAs) approaches. However, the former cannot strictly ensure dominant strategy incentive compatibility (DSIC), while the latter faces scalability issue due to the large number of allocation candidates. To address these limitations, we propose AMenuNet, a scalable neural network that constructs the AMA parameters (even including the allocation menu) from bidder and item representations. AMenuNet is always DSIC and individually rational (IR) due to the properties of AMAs, and it enhances scalability by generating candidate allocations through a neural network. Additionally, AMenuNet is permutation equivariant, and its number of parameters is independent of auction scale. We conduct extensive experiments to demonstrate that AMenuNet outperforms strong baselines in both contextual and non-contextual multi-item auctions, scales well to larger auctions, generalizes well to different settings, and identifies useful deterministic allocations. Overall, our proposed approach offers an effective solution to automated DSIC auction design, with improved scalability and strong revenue performance in various settings.

Theory/Game Theory

Han Shao, Avrim Blum, Omar Montasser

We study the fundamental mistake bound and sample complexity in the strategic classification, where agents can strategically manipulate their feature vector up to an extent in order to be predicted as positive. For example, given a classifier determining college admission, student candidates may try to take easier classes to improve their GPA, retake SAT and change schools in an effort to fool the classifier. *Ball manipulations* are a widely studied class of manipulations in the literature, where agents can modify their feature vector within a bounded radius ball. Unlike most prior work, our work consider manipulations to be *personalized*, meaning that agents can have different levels of manipulation abilities (e.g., varying radii for ball manipulations), and *unknown* to the learner.We formalize the learning problem in an interaction model where the learner first deploys a classifier and the agent manipulates the feature vector within their manipulation set to game the deployed classifier. We investigate various scenarios in terms of the information available to the learner during the interaction, such as observing the original feature vector before or after deployment, observing the manipulated feature vector, or not seeing either the original or the manipulated feature vector. We begin by providing online mistake bounds and PAC sample complexity in these scenarios for ball manipulations. We also explore non-ball manipulations and show that, even in the simplest scenario where both the original and the manipulated feature vectors are revealed, the mistake bounds and sample complexity are lower bounded by $\Omega(|\mathcal H|)$ when the target function belongs to a known class $\mathcal H$.

Theory/Game Theory

Yiding Chen, Jerry Zhu, Kirthevasan Kandasamy

We study collaborative normal mean estimation, where $m$ strategic agents collect i.i.d samples from a normal distribution $\mathcal{N}(\mu, \sigma^2)$ at a cost. They all wish to estimate the mean $\mu$. By sharing data with each other, agents can obtain better estimates while keeping the cost of data collection small. To facilitate this collaboration, we wish to design mechanisms that encourage agents to collect a sufficient amount of data and share it truthfully, so that they are all better off than working alone. In naive mechanisms, such as simply pooling and sharing all the data, an individual agent might find it beneficial to under-collect and/or fabricate data, which can lead to poor social outcomes. We design a novel mechanism that overcomes these challenges via two key techniques: first, when sharing the others' data with an agent, the mechanism corrupts this dataset proportional to how much the data reported by the agent differs from the others; second, we design minimax optimal estimators for the corrupted dataset. Our mechanism, which is Nash incentive compatible and individually rational, achieves a social penalty (sum of all agents' estimation errors and data collection costs) that is at most a factor 2 of the global minimum. When applied to high dimensional (non-Gaussian) distributions with bounded variance, this mechanism retains these three properties, but with slightly weaker results. Finally, in two special cases where we restrict the strategy space of the agents, we design mechanisms that essentially achieve the global minimum.

Optimization/Bilevel optimization

Zihan Zhu, Ethan Fang, Zhuoran Yang

We study the multi-agent game within the innovative framework of decision-dependent games, which establishes a feedback mechanism that population data reacts to agents’ actions and further characterizes the strategic interactions between agents. We focus on finding the Nash equilibrium of decision-dependent games in the bandit feedback setting. However, since agents are strategically coupled, traditional gradient-based methods are infeasible without the gradient oracle. To overcome this challenge, we model the strategic interactions by a general parametric model and propose a novel online algorithm, Online Performative Gradient Descent (OPGD), which leverages the ideas of online stochastic approximation and projected gradient descent to learn the Nash equilibrium in the context of function approximation for the unknown gradient. In particular, under mild assumptions on the function classes defined in the parametric model, we prove that OPGD can find the Nash equilibrium efficiently for strongly monotone decision-dependent games. Synthetic numerical experiments validate our theory.

Theory/Game Theory

Tao Lin, Yiling Chen

We consider a Bayesian forecast aggregation model where $n$ experts, after observing private signals about an unknown binary event, report their posterior beliefs about the event to a principal, who then aggregates the reports into a single prediction for the event. The signals of the experts and the outcome of the event follow a joint distribution that is unknown to the principal, but the principal has access to i.i.d. "samples" from the distribution, where each sample is a tuple of the experts' reports (not signals) and the realization of the event. Using these samples, the principal aims to find an $\varepsilon$-approximately optimal aggregator, where optimality is measured in terms of the expected squared distance between the aggregated prediction and the realization of the event. We show that the sample complexity of this problem is at least $\tilde \Omega(m^{n-2} / \varepsilon)$ for arbitrary discrete distributions, where $m$ is the size of each expert's signal space. This sample complexity grows exponentially in the number of experts $n$. But, if the experts' signals are independent conditioned on the realization of the event, then the sample complexity is significantly reduced, to $\tilde O(1 / \varepsilon^2)$, which does not depend on $n$. Our results can be generalized to non-binary events. The proof of our results uses a reduction from the distribution learning problem and reveals the fact that forecast aggregation is almost as difficult as distribution learning.

Theory/Game Theory

Bo Li, Fangxiao Wang, Yu Zhou

We study the maximin share (MMS) fair allocation of $m$ indivisible tasks to $n$ agents who have costs for completing the assigned tasks.It is known that exact MMS fairness cannot be guaranteed, and so far the best-known approximation for additive cost functions is $\frac{13}{11}$ by Huang and Segal-Halevi [EC, 2023]; however, beyond additivity, very little is known. In this work, we first prove that no algorithm can ensure better than $\min\{n,\frac{\log m}{\log \log m}\}$-approximation if the cost functions are submodular. This result also shows a sharp contrast with the allocation of goods where constant approximations exist as shown by Barman and Krishnamurthy [TEAC, 2020] and Ghodsi et al. [AIJ, 2022]. We then prove that for subadditive costs, there always exists an allocation that is $\min\{n,\lceil\log m\rceil\}$-approximation, and thus the approximation ratio is asymptotically tight.Besides multiplicative approximation, we also consider the ordinal relaxation, 1-out-of-$d$ MMS, which was recently proposed by Hosseini et al. [JAIR and AAMAS, 2022]. Our impossibility result implies that for any $d\ge 2$, a 1-out-of-$d$ MMS allocation may not exist.Due to these hardness results for general subadditive costs, we turn to studying two specific subadditive costs, namely, bin packing and job scheduling. For both settings, we show that constant approximate allocations exist for both multiplicative and ordinal relaxations of MMS.

Theory/Game Theory

William Brown, Jon Schneider, Kiran Vodrahalli

We consider a number of questions related to tradeoffs between reward and regret in repeated gameplay between two agents. To facilitate this, we introduce a notion of generalized equilibrium which allows for asymmetric regret constraints, and yields polytopes of feasible values for each agent and pair of regret constraints, where we show that any such equilibrium is reachable by a pair of algorithms which maintain their regret guarantees against arbitrary opponents. As a central example, we highlight the case one agent is no-swap and the other's regret is unconstrained. We show that this captures an extension of Stackelberg equilibria with a matching optimal value, and that there exists a wide class of games where a player can significantly increase their utility by deviating from a no-swap-regret algorithm against a no-swap learner (in fact, almost any game without pure Nash equilibria is of this form). Additionally, we make use of generalized equilibria to consider tradeoffs in terms of the opponent's algorithm choice. We give a tight characterization for the maximal reward obtainable against some no-regret learner, yet we also show a class of games in which this is bounded away from the value obtainable against the class of common "mean-based" no-regret algorithms. Finally, we consider the question of learning reward-optimal strategies via repeated play with a no-regret agent when the game is initially unknown. Again we show tradeoffs depending on the opponent's learning algorithm: the Stackelberg strategy is learnable in exponential time with any no-regret agent (and in polynomial time with any no-adaptive-regret agent) for any game where it is learnable via queries, and there are games where it is learnable in polynomial time against any no-swap-regret agent but requires exponential time against a mean-based no-regret agent.

Theory/Game Theory

Ioannis Panageas, Nikolas Patris, Stratis Skoulakis, Volkan Cevher

Fictitious Play (FP) is a simple and natural dynamic for repeated play with many applications in game theory and multi-agent reinforcement learning. It was introduced by Brown and its convergence properties for two-player zero-sum games was established later by Robinson. Potential games [Monderer and Shapley 1996] is another class of games which exhibit the FP property [Monderer and Shapley 1996], i.e., FP dynamics converges to a Nash equilibrium if all agents follows it. Nevertheless, except for two-player zero-sum games and for specific instances of payoff matrices [Abernethy et. al. 2021] or for adversarial tie-breaking rules [Daskalakis and Pan, 2014], the \textit{convergence rate} of FP is unknown. In this work, we focus on the rate of convergence of FP when applied to potential games and more specifically identical payoff games. We prove that FP can take exponential time (in the number of strategies) to reach a Nash equilibrium, even if the game is restricted to \textit{two agents}. To prove this, we recursively construct a two-player coordination game with a unique Nash equilibrium. Moreover, every approximate Nash equilibrium in the constructed game must be close to the pure Nash equilibrium in $\ell_1$-distance.

Theory/Game Theory

Christos Boutsikas, Petros Drineas, Marios Mertzanidis, Alexandros Psomas, Paritosh Verma

We consider the problem of a revenue-maximizing seller with a large number of items $m$ for sale to $n$ strategic bidders, whose valuations are drawn independently from high-dimensional, unknown prior distributions. It is well-known that optimal and even approximately-optimal mechanisms for this setting are notoriously difficult to characterize or compute, and, even when they can be found, are often rife with various counter-intuitive properties. In this paper, following a model introduced recently by Cai and Daskalakis [CD22], we consider the case that bidders' prior distributions can be well-approximated by a topic model. We design an active learning component, responsible for interacting with the bidders and outputting low-dimensional approximations of their types, and a mechanism design component, responsible for robustifying mechanisms for the low-dimensional model to work for the approximate types of the former component. On the active learning front, we cast our problem in the framework of Randomized Linear Algebra (RLA) for regression problems, allowing us to import several breakthrough results from that line of research, and adapt them to our setting. On the mechanism design front, we remove many restrictive assumptions of prior work on the type of access needed to the underlying distributions and the associated mechanisms. To the best of our knowledge, our work is the first to formulate connections between mechanism design, and RLA for active learning of regression problems, opening the door for further applications of randomized linear algebra primitives to mechanism design.

Theory/Game Theory

Nika Haghtalab, Chara Podimata, Kunhe Yang

In this paper, we introduce a generalization of the standard Stackelberg Games (SGs) framework: _Calibrated Stackelberg Games_. In CSGs, a principal repeatedly interacts with an agent who (contrary to standard SGs) does not have direct access to the principal's action but instead best responds to _calibrated forecasts_ about it. CSG is a powerful modeling tool that goes beyond assuming that agents use ad hoc and highly specified algorithms for interacting in strategic settings to infer the principal's actions and thus more robustly addresses real-life applications that SGs were originally intended to capture. Along with CSGs, we also introduce a stronger notion of calibration, termed _adaptive calibration_, that provides fine-grained any-time calibration guarantees against adversarial sequences. We give a general approach for obtaining adaptive calibration algorithms and specialize them for finite CSGs. In our main technical result, we show that in CSGs, the principal can achieve utility that converges to the optimum Stackelberg value of the game both in _finite_ and _continuous_ settings and that no higher utility is achievable. Two prominent and immediate applications of our results are the settings of learning in Stackelberg Security Games and strategic classification, both against _calibrated_ agents.

Theory/Game Theory

Meena Jagadeesan, Michael Jordan, Jacob Steinhardt, Nika Haghtalab

As the scale of machine learning models increases, trends such as scaling laws anticipate consistent downstream improvements in predictive accuracy. However, these trends take the perspective of a single model-provider in isolation, while in reality providers often compete with each other for users. In this work, we demonstrate that competition can fundamentally alter the behavior of these scaling trends, even causing overall predictive accuracy across users to be non-monotonic or decreasing with scale. We define a model of competition for classification tasks, and use data representations as a lens for studying the impact of increases in scale. We find many settings where improving data representation quality (as measured by Bayes risk) decreases the overall predictive accuracy across users (i.e., social welfare) for a marketplace of competing model-providers. Our examples range from closed-form formulas in simple settings to simulations with pretrained representations on CIFAR-10. At a conceptual level, our work suggests that favorable scaling trends for individual model-providers need not translate to downstream improvements in social welfare in marketplaces with multiple model providers.

Theory/Game Theory

Meena Jagadeesan, Nikhil Garg, Jacob Steinhardt

Algorithmic recommender systems such as Spotify and Netflix affect not only consumer behavior but also *producer incentives*. Producers seek to create content that will be shown by the recommendation algorithm, which can impact both the diversity and quality of their content. In this work, we investigate the resulting supply-side equilibria in personalized content recommender systems. We model the decisions of producers as choosing *multi-dimensional* content vectors and users as having *heterogenous* preferences, which contrasts with classical low-dimensional models. Multi-dimensionality and heterogeneity creates the potential for *specialization*, where different producers create different types of content at equilibrium. Using a duality argument, we derive necessary and sufficient conditions for whether specialization occurs. Then, we characterize the distribution of content at equilibrium in concrete settings with two populations of users. Lastly, we show that specialization can enable producers to achieve *positive profit at equilibrium*, which means that specialization can reduce the competitiveness of the marketplace. At a conceptual level, our analysis of supply-side competition takes a step towards elucidating how personalized recommendations shape the marketplace of digital goods.