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1. No-Regret Learning Dynamics for Extensive-Form Correlated Equilibrium

   Celli, A. ; Marchesi, A. ; Farina, G. ; Gatti, N. ( January 2020 , NeurIPS)

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   The existence of simple, uncoupled no-regret dynamics that converge to correlated equilibria in normal-form games is a celebrated result in the theory of multi-agent systems. Specifically, it has been known for more than 20 years that when all players seek to minimize their internal regret in a repeated normal-form game, the empirical frequency of play converges to a normal-form correlated equilibrium. Extensive-form (that is, tree-form) games generalize normal-form games by modeling both sequential and simultaneous moves, as well as private information. Because of the sequential nature and presence of partial information in the game, extensive-form correlation has significantly different properties than the normal-form counterpart, many of which are still open research directions. Extensive-form correlated equilibrium (EFCE) has been proposed as the natural extensive-form counterpart to normal-form correlated equilibrium. However, it was currently unknown whether EFCE emerges as the result of uncoupled agent dynamics. In this paper, we give the first uncoupled no-regret dynamics that converge to the set of EFCEs in n-player general-sum extensive-form games with perfect recall. First, we introduce a notion of trigger regret in extensive-form games, which extends that of internal regret in normal-form games. When each player has low trigger regret, the empirical frequency of play is close to an EFCE. Then, we give an efficient no-trigger-regret algorithm. Our algorithm decomposes trigger regret into local subproblems at each decision point for the player, and constructs a global strategy of the player from the local solutions at each decision point. 

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2. Simple Uncoupled No-Regret Learning Dynamics for Extensive-Form Correlated Equilibrium

   Farina, G. ; Celli, A. ; Marchesi, A. ; Gatti, N. ( January 2021 , ArXivorg)

   null (Ed.)

   The existence of simple uncoupled no-regret learning dynamics that converge to correlated equilibria in normal-form games is a celebrated result in the theory of multi-agent systems. Specifically, it has been known for more than 20 years that when all players seek to minimize their internal regret in a repeated normal-form game, the empirical frequency of play converges to a normal-form correlated equilibrium. Extensive-form games generalize normal-form games by modeling both sequential and simultaneous moves, as well as imperfect information. Because of the sequential nature and presence of private information in the game, correlation in extensive-form games possesses significantly different properties than its counterpart in normal-form games, many of which are still open research directions. Extensive-form correlated equilibrium (EFCE) has been proposed as the natural extensive-form counterpart to the classical notion of correlated equilibrium in normal-form games. Compared to the latter, the constraints that define the set of EFCEs are significantly more complex, as the correlation device must keep into account the evolution of beliefs of each player as they make observations throughout the game. Due to that significant added complexity, the existence of uncoupled learning dynamics leading to an EFCE has remained a challenging open research question for a long time. In this article, we settle that question by giving the first uncoupled no-regret dynamics that converge to the set of EFCEs in n-player general-sum extensive-form games with perfect recall. We show that each iterate can be computed in time polynomial in the size of the game tree, and that, when all players play repeatedly according to our learning dynamics, the empirical frequency of play is proven to be a O(T^-0.5)-approximate EFCE with high probability after T game repetitions, and an EFCE almost surely in the limit. 

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3. 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)

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   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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4. Model-Free Online Learning in Unknown Sequential Decision Making Problems and Games

   Farina, G. ; Sandholm, T. ( January 2021 , Proceedings of the AAAI Conference on Artificial Intelligence)

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   Regret minimization has proved to be a versatile tool for tree- form sequential decision making and extensive-form games. In large two-player zero-sum imperfect-information games, mod- ern extensions of counterfactual regret minimization (CFR) are currently the practical state of the art for computing a Nash equilibrium. Most regret-minimization algorithms for tree-form sequential decision making, including CFR, require (i) an exact model of the player’s decision nodes, observation nodes, and how they are linked, and (ii) full knowledge, at all times t, about the payoffs—even in parts of the decision space that are not encountered at time t. Recently, there has been growing interest towards relaxing some of those restric- tions and making regret minimization applicable to settings for which reinforcement learning methods have traditionally been used—for example, those in which only black-box access to the environment is available. We give the first, to our knowl- edge, regret-minimization algorithm that guarantees sublinear regret with high probability even when requirement (i)—and thus also (ii)—is dropped. We formalize an online learning setting in which the strategy space is not known to the agent and gets revealed incrementally whenever the agent encoun- ters new decision points. We give an efficient algorithm that achieves O(T 3/4) regret with high probability for that setting, even when the agent faces an adversarial environment. Our experiments show it significantly outperforms the prior algo- rithms for the problem, which do not have such guarantees. It can be used in any application for which regret minimization is useful: approximating Nash equilibrium or quantal response equilibrium, approximating coarse correlated equilibrium in multi-player games, learning a best response, learning safe opponent exploitation, and online play against an unknown opponent/environment. 

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5. Polynomial-Time Computation of Optimal Correlated Equilibria in Two-Player Extensive-Form Games with Public Chance Moves and Beyond

   Farina, G. ; Sandholm, T. ( January 2021 , AAAI Workshop on Reinforcement Learning in Games)

   null (Ed.)

   Unlike normal-form games, where correlated equilibria have been studied for more than 45 years, extensive-form correlation is still generally not well understood. Part of the reason for this gap is that the sequential nature of extensive-form games allows for a richness of behaviors and incentives that are not possible in normal-form settings. This richness translates to a significantly different complexity landscape surrounding extensive-form correlated equilibria. As of today, it is known that finding an optimal extensive-form correlated equilibrium (EFCE), extensive-form coarse correlated equilibrium (EFCCE), or normal-form coarse correlated equilibrium (NFCCE) in a two-player extensive-form game is computationally tractable when the game does not include chance moves, and intractable when the game involves chance moves. In this paper we significantly refine this complexity threshold by showing that, in two-player games, an optimal correlated equilibrium can be computed in polynomial time, provided that a certain condition is satisfied. We show that the condition holds, for example, when all chance moves are public, that is, both players observe all chance moves. This implies that an optimal EFCE, EFCCE and NFCCE can be computed in polynomial time in the game size in two-player games with public chance moves, providing the biggest positive complexity result surrounding extensive-form correlation in more than a decade. 

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