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1. Learning Zero-Sum Simultaneous-Move Markov Games Using Function Approximation and Correlated Equilibrium

   https://doi.org/10.1287/moor.2022.1268  

   Xie, Qiaomin ; Chen, Yudong ; Wang, Zhaoran ; Yang, Zhuoran ( June 2022 , Mathematics of Operations Research)

   We develop provably efficient reinforcement learning algorithms for two-player zero-sum finite-horizon Markov games with simultaneous moves. To incorporate function approximation, we consider a family of Markov games where the reward function and transition kernel possess a linear structure. Both the offline and online settings of the problems are considered. In the offline setting, we control both players and aim to find the Nash equilibrium by minimizing the duality gap. In the online setting, we control a single player playing against an arbitrary opponent and aim to minimize the regret. For both settings, we propose an optimistic variant of the least-squares minimax value iteration algorithm. We show that our algorithm is computationally efficient and provably achieves an [Formula: see text] upper bound on the duality gap and regret, where d is the linear dimension, H the horizon and T the total number of timesteps. Our results do not require additional assumptions on the sampling model. Our setting requires overcoming several new challenges that are absent in Markov decision processes or turn-based Markov games. In particular, to achieve optimism with simultaneous moves, we construct both upper and lower confidence bounds of the value function, and then compute the optimistic policy by solving a general-sum matrix game with these bounds as the payoff matrices. As finding the Nash equilibrium of a general-sum game is computationally hard, our algorithm instead solves for a coarse correlated equilibrium (CCE), which can be obtained efficiently. To our best knowledge, such a CCE-based scheme for optimism has not appeared in the literature and might be of interest in its own right. 

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2. Hindsight and Sequential Rationality of Correlated Play

   Morrill, D ; D'Orazio, R ; Sarfati, R ; Lanctot, M ; Wright, J.R. ; Greenwald, A.R ; Bowling, M. ( May 2021 , Proceedings of the AAAI Conference on Artificial Intelligence)

   null (Ed.)

   Driven by recent successes in two-player, zero-sum game solving and playing, artificial intelligence work on games has increasingly focused on algorithms that produce equilibrium-based strategies. However, this approach has been less effective at producing competent players in general-sum games or those with more than two players than in two-player, zero-sum games. An appealing alternative is to consider adaptive algorithms that ensure strong performance in hindsight relative to what could have been achieved with modified behavior. This approach also leads to a game-theoretic analysis, but in the correlated play that arises from joint learning dynamics rather than factored agent behavior at equilibrium. We develop and advocate for this hindsight rationality framing of learning in general sequential decision-making settings. To this end, we re-examine mediated equilibrium and deviation types in extensive-form games, thereby gaining a more complete understanding and resolving past misconceptions. We present a set of examples illustrating the distinct strengths and weaknesses of each type of equilibrium in the literature, and prove that no tractable concept subsumes all others. This line of inquiry culminates in the definition of the deviation and equilibrium classes that correspond to algorithms in the counterfactual regret minimization (CFR) family, relating them to all others in the literature. Examining CFR in greater detail further leads to a new recursive definition of rationality in correlated play that extends sequential rationality in a way that naturally applies to hindsight evaluation. 

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3. Function Approximation for Solving Stackelberg Equilibrium in Large Perfect Information Games

   https://doi.org/10.1609/aaai.v37i5.25715  

   Ling, Chun Kai ; Kolter, J. Zico ; Fang, Fei ( June 2023 , Proceedings of the AAAI Conference on Artificial Intelligence)

   Function approximation (FA) has been a critical component in solving large zero-sum games. Yet, little attention has been given towards FA in solving general-sum extensive-form games, despite them being widely regarded as being computationally more challenging than their fully competitive or cooperative counterparts. A key challenge is that for many equilibria in general-sum games, no simple analogue to the state value function used in Markov Decision Processes and zero-sum games exists. In this paper, we propose learning the Enforceable Payoff Frontier (EPF)---a generalization of the state value function for general-sum games. We approximate the optimal Stackelberg extensive-form correlated equilibrium by representing EPFs with neural networks and training them by using appropriate backup operations and loss functions. This is the first method that applies FA to the Stackelberg setting, allowing us to scale to much larger games while still enjoying performance guarantees based on FA error. Additionally, our proposed method guarantees incentive compatibility and is easy to evaluate without having to depend on self-play or approximate best-response oracles.

    

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4. Price Discrimination with Fairness Constraints

   https://doi.org/10.1287/mnsc.2022.4317  

   Cohen, Maxime C. ; Elmachtoub, Adam N. ; Lei, Xiao ( December 2022 , Management Science)

   Price discrimination strategies, which offer different prices to customers based on differences in their valuations, have become common practice. Although it allows sellers to increase their profits, it also raises several concerns in terms of fairness (e.g., by charging higher prices (or denying access) to protected minorities in case they have higher (or lower) valuations than the general population). This topic has received extensive attention from media, industry, and regulatory agencies. In this paper, we consider the problem of setting prices for different groups under fairness constraints. We first propose four definitions: fairness in price, demand, consumer surplus, and no-purchase valuation. We prove that satisfying more than one of these fairness constraints is impossible even under simple settings. We then analyze the pricing strategy of a profit-maximizing seller and the impact of imposing fairness on the seller’s profit, consumer surplus, and social welfare. Under a linear demand model, we find that imposing a small amount of price fairness increases social welfare, whereas too much price fairness may result in a lower welfare relative to imposing no fairness. On the other hand, imposing fairness in demand or consumer surplus always decreases social welfare. Finally, no-purchase valuation fairness always increases social welfare. We observe similar patterns under several extensions and for other common demand models numerically. Our results and insights provide a first step in understanding the impact of imposing fairness in the context of discriminatory pricing. This paper was accepted by Jayashankar Swaminathan, operations management. Funding: A. N. Elmachtoub was supported by the Division of Civil, Mechanical and Manufacturing Innovation [Grants 1763000 and 1944428]. Supplemental Material: The data files and online appendix are available at https://doi.org/10.1287/mnsc.2022.4317 . 

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5. Diversified Strategies for Mitigating Adversarial Attacks in Multiagent Systems

   Blum, Avrim ; Balcan, Maria-Florina ; Chen, Shang-Tse ( July 2018 , AAMAS)

   In this work we consider online decision-making in settings where players want to guard against possible adversarial attacks or other catastrophic failures. To address this, we propose a solution concept in which players have an additional constraint that at each time step they must play a diversified mixed strategy: one that does not put too much weight on any one action. This constraint is motivated by applications such as finance, routing, and resource allocation, where one would like to limit one’s exposure to adversarial or catastrophic events while still performing well in typical cases. We explore properties of diversified strategies in both zero-sum and general-sum games, and provide algorithms for minimizing regret within the family of diversified strategies as well as methods for using taxes or fees to guide standard regret-minimizing players towards diversified strategies. We also analyze equilibria produced by diversified strategies in general-sum games. We show that surprisingly, requiring diversification can actually lead to higher-welfare equilibria, and give strong guarantees on both price of anarchy and the social welfare produced by regret-minimizing diversified agents. We additionally give algorithms for finding optimal diversified strategies in distributed settings where one must limit communication overhead. 

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