DOCSLIB.ORG

Stable-Predictive Optimistic Counterfactual Regret Minimization

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Stable-Predictive Optimistic Counterfactual Regret Minimization Stable-Predictive Optimistic Counterfactual Regret Minimization Gabriele Farina 1 Christian Kroer 2 Noam Brown 1 Tuomas Sandholm 1 3 4 5 Abstract were used as an essential ingredient for all recent milestones in the benchmark domain of poker (Bowling et al., 2015; The CFR framework has been a powerful tool Moravcˇ´ık et al., 2017; Brown & Sandholm, 2017b). Despite for solving large-scale extensive-form games in this practical success all known CFR variants have a signif- practice. However, the theoretical rate at which icant theoretical drawback: their worst-case convergence 1=2 past CFR-based algorithms converge to the Nash rate is on the order of O(T − ), where T is the number of 1=2 equilibrium is on the order of O(T − ), where T iterations. In contrast to this, there exist first-order methods 1 is the number of iterations. In contrast, first-order that converge at a rate of O(T − ) (Hoda et al., 2010; Kroer 1 methods can be used to achieve a O(T − ) depen- et al., 2015; 2018b). However, these methods have been dence on iterations, yet these methods have been found to perform worse than newer CFR algorithms such as less successful in practice. In this work we present CFR+, in spite of their theoretical advantage (Kroer et al., the first CFR variant that breaks the square-root 2018b;a). dependence on iterations. By combining and ex- tending recent advances on predictive and stable In this paper we present the first CFR variant which breaks regret minimizers for the matrix-game setting we the square-root dependence on the number of iterations. By show that it is possible to leverage “optimistic” leveraging recent theoretical breakthroughs on “optimistic” 3=4 regret minimizers to achieve a O(T − ) conver- regret minimizers for the matrix-game setting, we show how gence rate within CFR. This is achieved by intro- to set up optimistic counterfactual regret minimizers at each ducing a new notion of stable-predictivity, and by information set such that the overall algorithm retains the setting the stability of each counterfactual regret properties needed in order to accelerate convergence. In minimizer relative to its location in the decision particular, this leads to a predictive and stable variant of 3=4 tree. Experiments show that this method is faster CFR that converges at a rate of O(T − ). than the original CFR algorithm, although not as Typical analysis of regret-minimization leads to a conver- 1=2 fast as newer variants, in spite of their worst-case gence rate of O(T − ) for solving zero-sum matrix games. 1=2 O(T − ) dependence on iterations. However, by leveraging the idea of optimistic learning (Chi- ang et al., 2012; Rakhlin & Sridharan, 2013a;b; Syrgka- nis et al., 2015; Wang & Abernethy, 2018), Rakhlin and 1. Introduction Sridharan show in a series of papers that it is possible to 1 converge at a rate of O(T − ) when leveraging cancellations Counterfactual regret minimization (CFR) (Zinkevich et al., that occur due to the optimistic mirror descent (OMD) al- 2007) and later variants such as Monte-Carlo CFR (Lanctot gorithm (Rakhlin & Sridharan, 2013a;b). Syrgkanis et al. + et al., 2009), CFR (Tammelin et al., 2015), and Discounted (2015) build on this idea, and introduce the optimistic follow- CFR (Brown & Sandholm, 2019), have been the practical the-regularized-leader (OFTRL) algorithm; they show that state-of-the-art in solving large-scale zero-sum extensive- even when the players do not employ the same algorithm, a form games (EFGs) for the last decade. These algorithms 3=4 rate of O(T − ) can be achieved as long as each algorithm 1Computer Science Department, Carnegie Mellon Uni- belongs to a class of algorithms that satisfy a stability crite- versity, Pittsburgh PA 15213 2IEOR Department, Columbia rion and leverage predictability of loss inputs. We build on University, New York NY 10027 3Strategic Machine, Inc. this latter generalization. Because we can only perform the 4 5 Strategy Robot, Inc. Optimized Markets, Inc.. Corre- optimistic updates locally with respect to counterfactual re- < > spondence to: Gabriele Farina [email protected] , grets we cannot achieve the cancellations that leads to a rate Christian Kroer <[email protected]>, Noam 1 Brown <[email protected]>, Tuomas Sandholm <sand- of O(T − ); instead we show that by carefully instantiating [email protected]>. each counterfactual regret minimizer it is possible to main- tain predictability and stability with respect to the overall th Proceedings of the 36 International Conference on Machine decision-tree structure, thus leading to a convergence rate of Learning , Long Beach, California, PMLR 97, 2019. Copyright O(T 3=4). In order to achieve these results we introduce a 2019 by the author(s). − Stable-Predictive Optimistic Counterfactual Regret Minimization new variant of stable-predictivity, and show that each local decision node j 2 J , the agent chooses a strategy from the nj counterfactual regret minimizer must have its stability set simplex ∆ of all probability distributions over the set Aj relative to its location in the overall strategy space, with of nj = jAjj actions available at that decision node. An regret minimizers deeper in the decision tree requiring more action is sampled according to the chosen distribution, and stability. the agent then waits to play again. While waiting, the agent might receive a signal (observation) from the process; this In addition to our theoretical results we investigate the prac- possibility is represented with an observation node. At a tical performance of our algorithm on several poker sub- generic observation point k 2 K, the agent might receive games from the Libratus AI which beat top poker profes- n signals; the set of signals that the agent can observe is sionals (Brown & Sandholm, 2017b). We find that our CFR k denoted as S . The observation node that is reached by the variant coupled with the OFTRL algorithm and the entropy k agent after picking action a 2 A at decision point j 2 J is regularizer leads to better convergence rate than the vanilla j denoted by ρ(j; a). Likewise, the decision node reached by CFR algorithm with regret matching, while it does not out- the agent after observing signal s 2 S at observation point perform the newer state-of-the-art algorithm Discounted k k 2 K is denoted by ρ(k; s). The set of all observation points CFR (DCFR) (Brown & Sandholm, 2019). This latter fact reachable from j 2 J is denoted as C := fρ(j; a) : a 2 A g. is not too surprising, as it has repeatedly been observed that j j Similarly, the set of all decision points reachable from k 2 K CFR+, and the newer and faster DCFR, converges at a rate is denoted as C := fρ(k; s) : s 2 S g. To ease the notation, better than O(T 1) for many practical games of interest, in k k − sometimes we will use the notation C to mean C .A spite of the worst-case rate of O(T 1=2). ja ρ(j;a) − concrete example of a decision process is given in the next The reader may wonder why we care about breaking the subsection. square-root barrier within the CFR framework. It is well- At each decision point j 2 J in a sequential decision pro- known that a convergence rate of O(T 1) can be achieved − cess, the decision x^ 2 ∆nj of the agent incurs an (expected) outside the CFR framework. As mentioned previously, this j linear loss h` ; x^ i. The expected loss throughout the whole can be done with first-order methods such as the excessive j j process is therefore P π h` ; x^ i; where π is the prob- gap technique (Nesterov, 2005) or mirror prox (Nemirovski, j j j j j ability of the agent reaching2J decision point j, defined as the 2004) combined with a dilated distance-generating func- product of the probability with which the agent plays each tion (Hoda et al., 2010; Kroer et al., 2015; 2018b). Despite action on the path from the root of the process to j. this, there has been repeated interest in optimistic regret min- imization within the CFR framework, due to the strong prac- In extensive-form games where all players have perfect re- tical performance of CFR algorithms. Burch (2017) tries call (that is, they never forget about their past moves or their 1 to implement CFR-like features in the context of O(T − ) observations), all players face a sequential decision process. FOMs and regret minimizers, while Brown & Sandholm The loss vectors f`jg are defined based on the strategies of (2019) experimentally tries optimistic variants of regret min- the opponent(s) as well as the chance player. However, as imizers in CFR. We stress that these prior results are only already observed by Farina et al. (2019), sequential decision experimental; our results are the first to rigorously incorpo- processes are more general and can model other settings as rate optimistic regret minimization in CFR, and the first to well, such as POMDPs and MDPs when the decision maker achieve a theoretical speedup. conditions on the entire history of observations and actions. Notation. Throughout the paper, we use the following no- n 2.1. Example: Sequential Decision Process for the First tation when dealing with R . We use hx; yi to denote the Player in Kuhn Poker dot product x>y of two vectors x and y. We assume that a pair of dual norms k · k; k · k has been chosen. These norms ∗ As an illustration, consider the game of Kuhn poker (Kuhn, need not be induced by inner products.
Load more

Recommended publications
  • Game Theory 10
    Game Theory 10. Poker Albert-Ludwigs-Universität Freiburg Bernhard Nebel and Robert Mattmüller 1 Motivation Motivation Kuhn Poker Real Poker: Problems and techniques Counterfac- tual regret minimization B. Nebel, R. Mattmüller – Game Theory 3 / 25 Motivation The system Libratus played a Poker tournament (heads up no-limit Texas hold ’em) from January 11 to 31, 2017 Motivation against four world-class Poker players. Kuhn Poker Heads up: One-on-One, i.e., a zero-sum game. Real Poker: No-limit: There is no limit in betting, only the stack the Problems and user has. techniques Texas hold’em: Each player gets two private cards, then Counterfac- tual regret open cards are dealt: first three, then one, and finally minimization another one. One combines the best 5 cards. Betting before the open cards are dealt and in the end: check, call, raise, or fold. Two teams (reversing the dealt cards). Libratus won the tournament with more than 1.7 Million US-$ (which neither the system nor the programming team got). B. Nebel, R. Mattmüller – Game Theory 4 / 25 The humans behind the scene Motivation Kuhn Poker Real Poker: Problems and techniques Counterfac- tual regret minimization Professional player Jason Les and Prof. Tuomas Sandholm (CMU) B. Nebel, R. Mattmüller – Game Theory 5 / 25 2 Kuhn Poker Motivation Kuhn Poker Real Poker: Problems and techniques Counterfac- tual regret minimization B. Nebel, R. Mattmüller – Game Theory 7 / 25 Kuhn Poker Motivation Minimal form of heads-up Poker, with only three cards: Kuhn Poker Real Poker: Jack, Queen, King. Problems and Each player is dealt one card and antes 1 chip (forced bet techniques in the beginning).
    [Show full text]
  • An Essay on Assignment Games
    GRAU DE MATEMÀTIQUES Facultat de Matemàtiques i Informàtica Universitat de Barcelona DEGREE PROJECT An essay on assignment games Rubén Ureña Martínez Advisor: Dr. Javier Martínez de Albéniz Dept. de Matemàtica Econòmica, Financera i Actuarial Barcelona, January 2017 Abstract This degree project studies the main results on the bilateral assignment game. This is a part of cooperative game theory and models a market with indivisibilities and money. There are two sides of the market, let us say buyers and sellers, or workers and firms, such that when we match two agents from different sides, a profit is made. We show some good properties of the core of these games, such as its non-emptiness and its lattice structure. There are two outstanding points: the buyers-optimal core allocation and the sellers-optimal core allocation, in which all agents of one sector get their best possible outcome. We also study a related non-cooperative mechanism, an auction, to implement the buyers- optimal core allocation. Resumen Este trabajo de fin de grado estudia los resultados principales acerca de los juegos de asignación bilaterales. Corresponde a una parte de la teoría de juegos cooperativos y proporciona un modelo de mercado con indivisibilidades y dinero. Hay dos lados del mercado, digamos compradores y vendedores, o trabajadores y empresas, de manera que cuando se emparejan dos agentes de distinto lado, se produce un cierto beneficio. Se muestran además algunas buenas propiedades del núcleo de estos juegos, tales como su condición de ser siempre no vacío y su estructura de retículo. Encontramos dos puntos destacados: la distribución óptima para los compradores en el núcleo y la distribución óptima para los vendedores en el núcleo, en las cuales todos los agentes de cada sector obtienen simultáneamente el mejor resultado posible en el núcleo.
    [Show full text]
  • Improving Fictitious Play Reinforcement Learning with Expanding Models
    Improving Fictitious Play Reinforcement Learning with Expanding Models Rong-Jun Qin1;2, Jing-Cheng Pang1, Yang Yu1;y 1National Key Laboratory for Novel Software Technology, Nanjing University, China 2Polixir emails: [email protected], [email protected], [email protected]. yTo whom correspondence should be addressed Abstract Fictitious play with reinforcement learning is a general and effective framework for zero- sum games. However, using the current deep neural network models, the implementation of fictitious play faces crucial challenges. Neural network model training employs gradi- ent descent approaches to update all connection weights, and thus is easy to forget the old opponents after training to beat the new opponents. Existing approaches often maintain a pool of historical policy models to avoid the forgetting. However, learning to beat a pool in stochastic games, i.e., a wide distribution over policy models, is either sample-consuming or insufficient to exploit all models with limited amount of samples. In this paper, we pro- pose a learning process with neural fictitious play to alleviate the above issues. We train a single model as our policy model, which consists of sub-models and a selector. Everytime facing a new opponent, the model is expanded by adding a new sub-model, where only the new sub-model is updated instead of the whole model. At the same time, the selector is also updated to mix up the new sub-model with the previous ones at the state-level, so that the model is maintained as a behavior strategy instead of a wide distribution over policy models.
    [Show full text]
  • Using Counterfactual Regret Minimization to Create Competitive Multiplayer Poker Agents
    Using Counterfactual Regret Minimization to Create Competitive Multiplayer Poker Agents Nick Abou Risk Duane Szafron University of Alberta University of Alberta Department of Computing Science Department of Computing Science Edmonton, AB Edmonton, AB 780-492-5468 780-492-5468 [email protected] [email protected] ABSTRACT terminal choice node exists for each player’s possible decisions. Games are used to evaluate and advance Multiagent and Artificial The directed edges leaving a choice node represent the legal Intelligence techniques. Most of these games are deterministic actions for that player. Each terminal node contains the utilities of with perfect information (e.g. Chess and Checkers). A all players for one potential game result. Stochastic events deterministic game has no chance element and in a perfect (Backgammon dice rolls or Poker cards) are represented as special information game, all information is visible to all players. nodes, for a player called the chance player. The directed edges However, many real-world scenarios with competing agents are leaving a chance node represent the possible chance outcomes. stochastic (non-deterministic) with imperfect information. For Since extensive games can model a wide range of strategic two-player zero-sum perfect recall games, a recent technique decision-making scenarios, they have been used to study poker. called Counterfactual Regret Minimization (CFR) computes Recent advances in computer poker have led to substantial strategies that are provably convergent to an ε-Nash equilibrium. advancements in solving general two player zero-sum perfect A Nash equilibrium strategy is useful in two-player games since it recall extensive games. Koller and Pfeffer [10] used linear maximizes its utility against a worst-case opponent.
    [Show full text]
  • Games Lectures.Key
    Imperfect Information • So far, all games we’ve developed solutions for have perfect information Lecture 10: Imperfect Information • No hidden information such as individual cards • Hidden information often represented as chance AI For Traditional Games nodes Prof. Nathan Sturtevant Winter 2011 • Could be a play by one player that is hidden until the end of the game Example Tree What is the size of a game with ii? • Simple betting game (Kuhn Poker) • Ante 1 chip • 2-player game, 3-card deck, 1 card each • First player can check/bet • Second player can bet/check or call/fold • If 2nd player bets, 1st player can call/fold 1 1111-1-1 -1 -1 -1 111 -1 -1 -1 • 3 hands each / 6 total combinations • [Exercise: Draw top portion of tree in class] Simple Approach: Perfect-Info Monte-Carlo Drawbacks of Monte-Carlo • We have good perfect information-solvers • May be too many worlds to sample • How can we use them for imperfect information • May get probabilities on worlds incorrect games? • World prob. based on previous actions in the game • Sample all unknown information (eg a world) • May reveal information in actions • For each world: • Good probabilities needed for information hiding • Solve perfectly with alpha-beta • Program has no sense of information seeking/hiding • Take the average best move moves • If too many worlds, sample a reasonable subset • Analysis may be incorrect (see work by Frank and Basin) Strategy Fusion Non-locality World 2 World 1 c 1 c c' World 1 & 2 -1 1 b a a b a' b' -1 1 World 1 1 -1 World 2 1 -1 -1 1 Strengths of Monte-Carlo Analysis of PIMC • Simple to implement • Understanding the Success of Perfect Information Monte Carlo Sampling in Game Tree Search • Relatively fast • Jeffrey Long and Nathan R.
    [Show full text]
  • Successful Nash Equilibrium Agent for a 3-Player Imperfect-Information Game
    Successful Nash Equilibrium Agent for a 3-Player Imperfect-Information Game Sam Ganzfried www.ganzfriedresearch.com 1 2 3 4 Scope and applicability of game theory • Strategic multiagent interactions occur in all fields – Economics and business: bidding in auctions, offers in negotiations – Political science/law: fair division of resources, e.g., divorce settlements – Biology/medicine: robust diabetes management (robustness against “adversarial” selection of parameters in MDP) – Computer science: theory, AI, PL, systems; national security (e.g., deploying officers to protect ports), cybersecurity (e.g., determining optimal thresholds against phishing attacks), internet phenomena (e.g., ad auctions) 5 Game theory background rock paper scissors Rock 0,0 -1, 1 1, -1 Paper 1,-1 0, 0 -1,1 Scissors -1,1 1,-1 0,0 • Players • Actions (aka pure strategies) • Strategy profile: e.g., (R,p) • Utility function: e.g., u1(R,p) = -1, u2(R,p) = 1 6 Zero-sum game rock paper scissors Rock 0,0 -1, 1 1, -1 Paper 1,-1 0, 0 -1,1 Scissors -1,1 1,-1 0,0 • Sum of payoffs is zero at each strategy profile: e.g., u1(R,p) + u2(R,p) = 0 • Models purely adversarial settings 7 Mixed strategies • Probability distributions over pure strategies • E.g., R with prob. 0.6, P with prob. 0.3, S with prob. 0.1 8 Best response (aka nemesis) • Any strategy that maximizes payoff against opponent’s strategy • If P2 plays (0.6, 0.3, 0.1) for r,p,s, then a best response for P1 is to play P with probability 1 9 Nash equilibrium • Strategy profile where all players simultaneously play a best response • Standard solution concept in game theory – Guaranteed to always exist in finite games [Nash 1950] • In Rock-Paper-Scissors, the unique equilibrium is for both players to select each pure strategy with probability 1/3 10 • Theorem [Nash 1950]: Every game in strategic form G, with a finite number of players and in which every player has a finite number of pure strategies, has an equilibrium in mixed strategies.
    [Show full text]
  • Game Theory 10
    Game Theory 10. Poker Albert-Ludwigs-Universität Freiburg Bernhard Nebel and Robert Mattmüller Motivation Kuhn Poker Real Poker: Problems and techniques Counterfac- Motivation tual regret minimization B. Nebel, R. Mattmüller – Game Theory 2 / 25 Motivation The system Libratus played a Poker tournament (heads up no-limit Texas hold ’em) from January 11 to 31, 2017 Motivation against four world-class Poker players. Kuhn Poker Heads up: One-on-One, i.e., a zero-sum game. Real Poker: No-limit: There is no limit in betting, only the stack the Problems and user has. techniques Texas hold’em: Each player gets two private cards, then Counterfac- tual regret open cards are dealt: first three, then one, and finally minimization another one. One combines the best 5 cards. Betting before the open cards are dealt and in the end: check, call, raise, or fold. Two teams (reversing the dealt cards). Libratus won the tournament with more than 1.7 Million US-$ (which neither the system nor the programming team got). B. Nebel, R. Mattmüller – Game Theory 4 / 25 The humans behind the scene Motivation Kuhn Poker Real Poker: Problems and techniques Counterfac- tual regret minimization Professional player Jason Les and Prof. Tuomas Sandholm (CMU) B. Nebel, R. Mattmüller – Game Theory 5 / 25 Motivation Kuhn Poker Real Poker: Problems and techniques Counterfac- Kuhn Poker tual regret minimization B. Nebel, R. Mattmüller – Game Theory 6 / 25 Kuhn Poker Motivation Minimal form of heads-up Poker, with only three cards: Kuhn Poker Real Poker: Jack, Queen, King. Problems and Each player is dealt one card and antes 1 chip (forced bet techniques in the beginning).
    [Show full text]
  • Online Enhancement of Existing Nash Equilibrium Poker Agents
    Online Enhancement of existing Nash Equilibrium Poker Agents Online Verbesserung bestehender Nash-Equilibrium Pokeragenten Master-Thesis von Suiteng Lu Tag der Einreichung: 1. Gutachten: Prof. Dr. Johannes Fürnkranz 2. Gutachten: Dr. Eneldo Loza Mencía 3. Gutachten: Christian Wirth Fachbereich Informatik Fachgebiet Knowledge Engineering Prof. Dr. Johannes Fürnkranz Online Enhancement of existing Nash Equilibrium Poker Agents Online Verbesserung bestehender Nash-Equilibrium Pokeragenten Vorgelegte Master-Thesis von Suiteng Lu 1. Gutachten: Prof. Dr. Johannes Fürnkranz 2. Gutachten: Dr. Eneldo Loza Mencía 3. Gutachten: Christian Wirth Tag der Einreichung: Bitte zitieren Sie dieses Dokument als: URN: urn:nbn:de:tuda-tuprints-12345 URL: http://tuprints.ulb.tu-darmstadt.de/1234 Dieses Dokument wird bereitgestellt von tuprints, E-Publishing-Service der TU Darmstadt http://tuprints.ulb.tu-darmstadt.de [email protected] Die Veröffentlichung steht unter folgender Creative Commons Lizenz: Namensnennung – Keine kommerzielle Nutzung – Keine Bearbeitung 2.0 Deutschland http://creativecommons.org/licenses/by-nc-nd/2.0/de/ Erklärung zur Master-Thesis Hiermit versichere ich, die vorliegende Master-Thesis ohne Hilfe Dritter nur mit den an- gegebenen Quellen und Hilfsmitteln angefertigt zu haben. Alle Stellen, die aus Quellen entnommen wurden, sind als solche kenntlich gemacht. Diese Arbeit hat in gleicher oder ähnlicher Form noch keiner Prüfungsbehörde vorgelegen. Darmstadt, den December 18, 2016 (Suiteng Lu) Contents 1 Introduction 1 2 Poker 3 2.1 No-Limit Texas Hold’em . .3 2.2 Kuhn Poker . .5 3 Background and Related Work 6 3.1 Extensive-form Games . .6 3.2 Finding Nash Equilibrium . .9 3.2.1 LP Approach . .9 3.2.2 Counterfactual Regret Minimization .
    [Show full text]
  • ECE 586BH: Problem Set 4: Problems and Solutions Extensive Form Games
    University of Illinois Spring 2013 ECE 586BH: Problem Set 4: Problems and Solutions Extensive form games: Normal form representation, behavioral strategies, sequential rationality Due: Thursday, March 28 at beginning of class Reading: Fudendberg and Triole, Sections 3.1-3.4, & 8.3 Section III of Osborne and Rubenstein, available online, covers this material too. Lectures also relied on Myerson's book. 1. [Ultimatum game] Consider the following two stage game, about how two players split a pile of 100 gold coins. The action (strategy) set of player 1 is given by S1 = f0;:::; 100g; with choice i meaning that player 1 proposes to keep i of the gold coins. Player 2 learns the choice of player one, and then takes one of two actions in response: A (accept) or R (reject). If player two plays accept, the payoff vector is (i; 100 − i): If player two plays reject, the payoff vector is (0; 0): (a) Describe the extensive form version of the game using a tree labeled as in class. Solution: 2.100 accept 100,0 reject 0,0 100 2.99 accept 99,1 99 reject 0,0 1.0 . 0 2.0 accept 0,100 reject 0,0 (b) Describe the normal form of the game. It suffices to specify the strategy spaces and payoff functions. (Hint: Player 2 has 2101 pure strategies.) 100 Solution: The strategy spaces are S1 = f0; 1;:::; 100g and S2 = f0; 1g : The inter- pretation of d = (d(i) : 0 ≤ i ≤ 100) 2 S2 is d(i) = 1 if player 2 accepts when player 1 plays i, and d(i) = 0 if player 2 rejects when player 1 plays i: The payoff functions are given by u1(i; d) = i · d(i) and u2(i; d) = (100 − i)d(i): (c) Identify a Nash equilibria of the normal form game with payoff vector (50; 50): Solution: The pair (50; d50) is such an NE, where d50(i) = 1 for i ≤ 50 and d50(i) = 0 for i > 50: That is, when player 2 uses the strategy d50; she accepts if player 1 plays i ≤ 50 and she rejects otherwise.
    [Show full text]
  • Heads-Up Limit Hold'em Poker Is Solved
    Heads-up Limit Hold’em Poker is Solved Michael Bowling,1∗ Neil Burch,1 Michael Johanson,1 Oskari Tammelin2 1Department of Computing Science, University of Alberta, Edmonton, Alberta, T6G2E8, Canada 2Unaffiliated, http://jeskola.net ∗To whom correspondence should be addressed; E-mail: [email protected] Poker is a family of games that exhibit imperfect information, where players do not have full knowledge of past events. Whereas many perfect informa- tion games have been solved (e.g., Connect Four and checkers), no nontrivial imperfect information game played competitively by humans has previously been solved. Here, we announce that heads-up limit Texas hold’em is now es- sentially weakly solved. Furthermore, this computation formally proves the common wisdom that the dealer in the game holds a substantial advantage. This result was enabled by a new algorithm, CFR+, which is capable of solv- ing extensive-form games orders of magnitude larger than previously possible. Games have been intertwined with the earliest developments in computation, game theory, and artificial intelligence (AI). At the very conception of computing, Babbage had detailed plans for an “automaton” capable of playing tic-tac-toe and dreamt of his Analytical Engine playing chess (1). Both Turing (2) and Shannon (3) — on paper and in hardware, respectively — de- veloped programs to play chess as a means of validating early ideas in computation and AI. For more than a half century, games have continued to act as testbeds for new ideas and the result- ing successes have marked important milestones in the progress of AI. Examples include the checkers-playing computer program Chinook becoming the first to win a world championship title against humans (4), Deep Blue defeating Kasparov in chess (5), and Watson defeating Jen- nings and Rutter on Jeopardy! (6).
    [Show full text]
  • Α-Rank: Multi-Agent Evaluation by Evolution
    www.nature.com/scientificreports OPEN α-Rank: Multi-Agent Evaluation by Evolution Shayegan Omidshafei1, Christos Papadimitriou5, Georgios Piliouras 4, Karl Tuyls1, Mark Rowland2, Jean-Baptiste Lespiau1, Wojciech M. Czarnecki2, Marc Lanctot3, 2 1 Received: 18 February 2019 Julien Perolat & Remi Munos Accepted: 11 June 2019 We introduce α-Rank, a principled evolutionary dynamics methodology, for the evaluation and ranking Published: xx xx xxxx of agents in large-scale multi-agent interactions, grounded in a novel dynamical game-theoretic solution concept called Markov-Conley chains (MCCs). The approach leverages continuous-time and discrete-time evolutionary dynamical systems applied to empirical games, and scales tractably in the number of agents, in the type of interactions (beyond dyadic), and the type of empirical games (symmetric and asymmetric). Current models are fundamentally limited in one or more of these dimensions, and are not guaranteed to converge to the desired game-theoretic solution concept (typically the Nash equilibrium). α-Rank automatically provides a ranking over the set of agents under evaluation and provides insights into their strengths, weaknesses, and long-term dynamics in terms of basins of attraction and sink components. This is a direct consequence of the correspondence we establish to the dynamical MCC solution concept when the underlying evolutionary model’s ranking- intensity parameter, α, is chosen to be large, which exactly forms the basis of α-Rank. In contrast to the Nash equilibrium, which is a static solution concept based solely on fxed points, MCCs are a dynamical solution concept based on the Markov chain formalism, Conley’s Fundamental Theorem of Dynamical Systems, and the core ingredients of dynamical systems: fxed points, recurrent sets, periodic orbits, and limit cycles.
    [Show full text]
  • Optimistic Regret Minimization for Extensive-Form Games Via Dilated Distance-Generating Functions∗
    Optimistic Regret Minimization for Extensive-Form Games via Dilated Distance-Generating Functions∗ Gabriele Farina Christian Kroer Computer Science Department IEOR Department Carnegie Mellon University Columbia University [email protected] [email protected] Tuomas Sandholm Computer Science Department, CMU Strategic Machine, Inc. Strategy Robot, Inc. Optimized Markets, Inc. [email protected] Abstract We study the performance of optimistic regret-minimization algorithms for both minimizing regret in, and computing Nash equilibria of, zero-sum extensive-form games. In order to apply these algorithms to extensive-form games, a distance- generating function is needed. We study the use of the dilated entropy and dilated Euclidean distance functions. For the dilated Euclidean distance function we prove the first explicit bounds on the strong-convexity parameter for general treeplexes. Furthermore, we show that the use of dilated distance-generating functions enable us to decompose the mirror descent algorithm, and its optimistic variant, into local mirror descent algorithms at each information set. This decomposition mirrors the structure of the counterfactual regret minimization framework, and enables important techniques in practice, such as distributed updates and pruning of cold 1 parts of the game tree. Our algorithms provably converge at a rate of T − , which is superior to prior counterfactual regret minimization algorithms. We experimentally compare to the popular algorithm CFR+, which has a theoretical convergence rate 0:5 1 of T − in theory, but is known to often converge at a rate of T − , or better, in practice. We give an example matrix game where CFR+ experimentally converges 0:74 at a relatively slow rate of T − , whereas our optimistic methods converge faster 1 than T − .
    [Show full text]