DOCSLIB.ORG

Effective Short-Term Opponent Exploitation in Simplified Poker

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Effective Short-Term Opponent Exploitation in Simplified Poker Effective Short-Term Opponent Exploitation in Simplified Poker Bret Hoehn Valeriy Bulitko Finnegan Southey Centre for Science, Athabasca University Robert C. Holte University of Alberta, Dept. of Computing Science Abstract game trees and the strategies involve many parameters (e.g. two-player, limit Texas Hold'em requires O(1018) Uncertainty in poker stems from two key sources, the shuffled deck and an adversary whose strategy is un- parameters (Billings et al. 2003)). The game also has known. One approach is to find a pessimistic game theo- high variance, stemming from the deck and stochas- retic solution (i.e. a Nash equilibrium), but human play- tic opponents, and folding gives rise to partial obser- ers have idiosyncratic weaknesses that can be exploited vations. Strategically complex, the aim is not simply if a model of their strategy can be learned by observing to win but to maximize winnings by enticing a weakly- their play. However, games against humans last for at positioned opponent to bet. Finally, we cannot expect most a few hundred hands so learning must be fast to a large amount of data when playing human opponents. be effective. We explore two approaches to opponent You may play only 50 or 100 hands against a given op- modelling in the context of Kuhn poker, a small game ponent and want to quickly learn how to exploit them. for which game theoretic solutions are known. Param- eter estimation and expert algorithms are both studied. This research explores how rapidly we can gain an Experiments demonstrate that, even in this small game, advantage by observing opponent play given that only convergence to maximally exploitive solutions in a small a small number of hands will be played in total. Two number of hands is impractical, but that good (i.e. better learning approaches are studied: maximum a posteri- than Nash or breakeven) performance can be achieved in ori parameter estimation (parameter learning), and an a short period of time. Finally, we show that amongst a “experts” method derived from Exp3 (Auer et al. 1995) set of strategies with equal game theoretic value, in par- (strategy learning). Both will be described in detail. ticular the set of Nash equilibrium strategies, some are preferable because they speed learning of the opponent's While existing poker opponent modelling research strategy by exploring it more effectively. focuses on real-world games (Korb & Nicholson 1999; Billings et al. ), we systematically study a simpler ver- Introduction sion, reducing the game's intrinsic difficulty to show that, even in what might be considered a best case, the Poker is a game of imperfect information against an ad- problem is still hard. We start by assuming that the op- versary with an unknown, stochastic strategy. It rep- ponent's strategy is fixed. Tracking a non-stationary resents a tough challenge to artificial intelligence re- strategy is a hard problem and learning to exploit a search. Game theoretic approaches seek to approximate fixed strategy is clearly the first step. Next, we con- the Nash equilibrium (i.e. minimax) strategies of the sider the game of Kuhn poker (Kuhn 1950), a tiny game game (Koller & Pfeffer 1997; Billings et al. 2003), for which complete game theoretic analysis is available. but this represents a pessimistic worldview where we Finally, we evaluate learning in a two-phase manner; assume optimality in our opponent. Human players the first phase exploring and learning, while the sec- have weaknesses that can be exploited to obtain win- ond phase switches to pure exploitation based on what nings higher than the game-theoretic value of the game. was learned. We use this simplified framework to show Learning by observing their play allows us to exploit that learning to maximally exploit an opponent in a their idiosyncratic weaknesses. This can be done ei- small number of hands is not feasible. However, we ther directly, by learning a model of their strategy, or also demonstrate that some advantage can be rapidly at- indirectly, by identifying an effective counter-strategy. tained, making short-term learning a winning proposi- Several factors render this difficult in practice. First, tion. Finally, we observe that, amongst the set of Nash real-world poker games like Texas Hold'em have huge strategies for the learner (which are “safe” strategies), Copyright c 2005, American Association for Artificial Intel- the exploration inherent in some strategies facilitates ligence (www.aaai.org). All rights reserved. faster learning compared with other members of the set. AAAI-05 / 783 J|Q J|K Q|J Q|K K|J K|Q pass bet pass bet pass pass pass bet pass bet α 1−γ γ 1−γ γ 1−α α 1−α 1 1 pass pass bet bet bet pass bet bet pass bet pass pass pass bet 1 1−η η 1 1 1−ξ ξ 1 1−ξ ξ 1 1 1−η η -1 +1 -2 -2 +1 +1 +1 +1 +1 +2 pass pass bet pass bet bet 1 1−β β 1−β β 1 -1 -1 +2 -1 -2 +2 Player 1 Choice/Leaf Node Player 2 Choice Node Figure 1: Kuhn Poker game tree with dominated strategies removed Kuhn Poker ξ). The decisions governed by these parameters are Kuhn poker (Kuhn 1950) is a very simple, two-player shown in Figure 1. Kuhn determined that the set of game (P1 - Player 1, P2 - Player 2). The deck consists equilibrium strategies for P1 has the form (α, β; γ) = of three cards (J - Jack, Q - Queen, and K - King). There (γ=3; (1 + γ)=3; γ) for 0 ≤ γ ≤ 1. Thus, there is a are two actions available: bet and pass. The value of continuum of Nash strategies for P1 governed by a sin- each bet is 1. In the event of a showdown (players have gle parameter. There is only one Nash strategy for P2, matched bets), the player with the higher card wins the η = 1=3 and ξ = 1=3; all other P2 strategies can be pot (the King is highest and the Jack is lowest). A game exploited by P1. If either player plays an equilibrium proceeds as follows: strategy (and neither play dominated strategies), then P1 expects to lose at a rate of −1=18 per hand. Thus • Both players initially put an ante of 1 into the pot. P1 can only hope to win in the long run if P2 is playing • Each player is dealt a single card and the remaining suboptimally and P1 deviates from playing equilibrium card is unseen by either player. strategies to exploit errors in P2's play. Our discussion focuses on playing as P1 and exploiting P2, so all ob- • After the deal, P1 has the opportunity to bet or pass. servations and results are from this perspective. – If P1 bets in round one, then in round two P2 can: The strategy-space for P2 can be partitioned into the ∗ bet (calling P1's bet) and the game then ends in a 6 regions shown in Figure 2. Within each region, a sin- showdown, or gle P1 pure strategy gives maximal value to P1. For ∗ pass (folding) and forfeit the pot to P1. points on the lines dividing the regions, the bordering – If P1 passes in round one, then in round two P2 maximal strategies achieve the same value. The in- can: tersection of the three dividing lines is the Nash strat- egy for P2. Therefore, to maximally exploit P2, it is ∗ bet (in which case there is a third action where P1 sufficient to identify the region in which their strat- can bet and go to showdown, or pass and forfeit egy lies and then to play the corresponding P1 pure to P2), or strategy. Note that there are 8 pure strategies for P1: ∗ pass (game proceeds to a showdown). S1 = (0; 0; 0); S2 = (0; 0; 1); S3 = (0; 1; 0); : : : ; S7 = Figure 1 shows the game tree with P1's value for each (1; 1; 0); S8 = (1; 1; 1). Two of these (S1 and S8) are outcome. Note that the dominated strategies have been never the best response to any P2 strategy, so we need removed from this tree already. Informally, a dominated only consider the remaining six. strategy is one for which there exists an alternative strat- This natural division of P2's strategy space was used egy that offers equal or better value in any given situa- to obtain the suboptimal opponents for our study. Six tion. We eliminate these obvious sources of suboptimal opponent strategies were created by selecting a point play but note that non-dominated suboptimal strategies at random from each of the six regions. They are remain, so it is still possible to play suboptimally with O1 = (:25; :67); O2 = (:75; :8); O3 = (:67; :4); O4 = respect to a specific opponent. (:5; :29); O5 = (:25; :17); O6 = (:17; 2). All experi- The game has a well-known parametrization, in ments were run against these six opponents, although which P1's strategy can be summarized by three pa- we only have space to show results against representa- rameters (α, β, γ), and P2's by two parameters (η, tive opponents here. AAAI-05 / 784 ξ of how they play. A Beta distribution is characterized by two parameters, θ ≥ 0 and ! ≥ 0. The distribu- 1 tion can be understood as pretending that we have ob- served the opponent's choices several times in the past, θ ! S and that we observed choices one way and choices S 3 the other way. Thus, low values for this pair of param- 7 eters (e.g.
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]