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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)

   null (Ed.)

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

   Celli, A. ; Marchesi, A. ; Farina, G. ; Gatti, N. ( January 2020 , NeurIPS Workshop on Cooperative AI)

   null (Ed.)

   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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3. 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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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)

   null (Ed.)

   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. Dynamic Set Values for Nonzero-Sum Games with Multiple Equilibriums

   https://doi.org/10.1287/moor.2021.1143  

   Feinstein, Zachary ; Rudloff, Birgit ; Zhang, Jianfeng ( February 2022 , Mathematics of Operations Research)

   Nonzero sum games typically have multiple Nash equilibriums (or no equilibrium), and unlike the zero-sum case, they may have different values at different equilibriums. Instead of focusing on the existence of individual equilibriums, we study the set of values over all equilibriums, which we call the set value of the game. The set value is unique by nature and always exists (with possible value [Formula: see text]). Similar to the standard value function in control literature, it enjoys many nice properties, such as regularity, stability, and more importantly, the dynamic programming principle. There are two main features in order to obtain the dynamic programming principle: (i) we must use closed-loop controls (instead of open-loop controls); and (ii) we must allow for path dependent controls, even if the problem is in a state-dependent (Markovian) setting. We shall consider both discrete and continuous time models with finite time horizon. For the latter, we will also provide a duality approach through certain standard PDE (or path-dependent PDE), which is quite efficient for numerically computing the set value of the game. 

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