DOCSLIB.ORG

Modeling Replicator Dynamics in Stochastic Games Using Markov Chain Method

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Modeling Replicator Dynamics in Stochastic Games Using Markov Chain Method Main Track AAMAS 2021, May 3-7, 2021, Online Modeling Replicator Dynamics in Stochastic Games Using Markov Chain Method Chuang Deng Zhihai Rong Department of Automation School of Computer Science and Engineering Shanghai Jiao Tong University University of Electronic Science and Technology of China [email protected] [email protected] Lin Wang Xiaofan Wang Department of Automation School of Mechatronic Engineering and Automation Shanghai Jiao Tong University Shanghai University [email protected] [email protected] ABSTRACT For gaining qualitative and detailed insights into the learning In stochastic games, individuals need to make decisions in multi- process in repeated games, many algorithms and models have been ple states and transitions between states influence the dynamics proposed to build the bridge between evolutionary game theory of strategies significantly. In this work, by describing the dynamic and multi-agent reinforcement learning algorithms, such as FAQ- process in stochastic game as a Markov chain and utilizing the tran- learning [10], regret minimization [11], continuous strategy repli- sition matrix, we introduce a new method, named state-transition cator dynamics [5], IGA [21], IGA-WOLF [2] and so on. Besides replicator dynamics, to obtain the replicator dynamics of a stochas- the assumption that individuals are in the same state, how to de- tic game. Based on our proposed model, we can gain qualitative scribe strategy dynamics with the reciprocity of multiple states and and detailed insights into the influence of transition probabilities separate strategies also raises the attention of researchers. Vrancx on the dynamics of strategies. We illustrate that a set of unbalanced et al. first investigate the problem and they combine replicator transition probabilities can help players to overcome the social dynamics and piecewise dynamics to model the learning behav- dilemmas and lead to mutual cooperation in a cooperation back ior of agents in stochastic games [25]. Hennes et al. propose a state, even if the stochastic game has the same social dilemmas in method called state-coupled replicator dynamics which couples each state. Moreover, we also present that a set of specifically de- the multiple states directly [8]. The derived algorithms from the signed transition probabilities can fix the expected payoffs of one combination of evolutionary game theory and multi-agent rein- player and make him lose the motivation to update his strategies forcement learning algorithms have been applied in multiple re- in the stochastic game. search fields, including femtocell power allocation [27], interdo- main routing price setting [24], design of social agents [6] and de- KEYWORDS sign of new multi-agent reinforcement learning algorithms [7]. In this paper, by describing the dynamic process in stochastic Multi-agent Learning; Evolutionary Game Theory; Replicator Dy- game as a Markov chain [9, 22], we propose a new method, named namics; Stochastic Games state-transition replicator dynamics, to derive a set of ordinary dif- ACM Reference Format: ferential equations to model the learning behavior of players in sto- Chuang Deng, Zhihai Rong, Lin Wang, and Xiaofan Wang. 2021. Modeling chastic games. Moreover, based on the transition matrix of Markov Replicator Dynamics in Stochastic Games Using Markov Chain Method. In chain and the derived replicator dynamics system, we demonstrate Proc. of the 20th International Conference on Autonomous Agents and Multi- that, though players face the same social dilemmas in all states, a agent Systems (AAMAS 2021), Online, May 3–7, 2021, IFAAMAS, 9 pages. set of unbalanced transition probabilities between states can lead 1 INTRODUCTION to cooperation in the cooperation back state where players have a high probability to be in after mutual cooperation. Besides, we also The tragedy of the commons leads to the questions how todrive point out that in stochastic games, there exist several sets of spe- and reinforce cooperative behaviors in social and economic sys- cific transition probabilities which can control the expected pay- tems [4, 17]. These questions have been extensively explored by offs of players. For a stateless matrix game, Press et al. [15] prove analyzing stylized uncooperative game theory models with some that if players’ decisions depend on the previous actions of players feedback mechanisms using evolutionary game theory [14, 16, 18– participating in games, one player can unilaterally sets the payoff 20]. Besides the evolutionary game theory, another method used in of the other player. In stochastic games, transition probabilities can studying the dynamics of strategies is multi-agent reinforcement take this work. Utilizing the analyzing method in stateless matrix learning [1, 3, 12]. These works mainly derive and design multi- game [13, 15, 23] and state-transition replicator dynamics model, agent reinforcement learning algorithms which can lead strategies we present that a set of specifically derived transition probabilities of players to the Nash equilibrium existing in the game. can fix the expected payoffs of one player and this player losesthe Proc. of the 20th International Conference on Autonomous Agents and Multiagent Sys- motivation to update his strategies in this stochastic game. tems (AAMAS 2021), U. Endriss, A. Nowé, F. Dignum, A. Lomuscio (eds.), May 3–7, 2021, Online. © 2021 International Foundation for Autonomous Agents and Multiagent Sys- tems (www.ifaamas.org). All rights reserved. 420 Main Track AAMAS 2021, May 3-7, 2021, Online " �!! (� ,�) 2 BACKGROUND state 1 Agent Q state 2 Agent Q coop. defect defect 2.1 Stochastic games " coop. �!" (� ,�) " " P " " $ $ Stochastic games describe how much rewards players obtain based coop. � , � � ,� P $ $ !! !! !# !# # coop. �!!, �!! �!# ,�!# �!" (� ,�) on their strategies in multiple states. The concept of state in sto- gent " " " " gent $ $ $ $ A defect � ,� � ,� A chastic games refers to the environment where players interact #! #! ## ## defect �#!,�#! �##,�## # with each other. The current state players being in and the joint �!! (� ,�) action players taking not only determine the immediate payoffs players can receive in this round, but also the state they will stay Figure 1: The framework of a two-state two-player two- in the next round. To define a stochastic game, we need to spec- action stochastic game. There are two players, % and &. In ify the set of players, the set of states, the set of actions that each every round, they can be in one state, B1 or B2. In each state, player can take in each state, a transition function that describes each player has a corresponding strategy and payoff matrix. how states change over time and the payoff function which de- The state in the next round depends on the current state, on scribes how players’ actions in each state affect the players’ pay- the players’ joint action and on chance, IB0 ¹B, aº. offs. We follow the definition in [8] to give a definition of stochastic games. The game퐺 = h=, Y, 훀, z, g, c1, ··· , c=i is a stochastic game with : B B = players and : states. In each state B 2 Y = ¹B1, ··· ,B º, every 퐴 (퐵 ) determines the payoffs for player % (&) in state B. Player % g ¹B, aº 0B & g ¹B, aº 1B player 8 has an action set 훀8 ¹Bº and strategy c8 ¹Bº. In each round, receives % = 0% ,0& while player gets & = 0% ,0& every player 8 stays in one sate B and chooses an action 08 from for a given joint action a = ¹0% , 0& º. Player % has strategies p1 = action set 훀8 ¹Bº according to strategy c8 ¹Bº. The payoff function ¹?1, − ?1ºT p2 ¹?2, − ?2ºT B1 B2 Î= = 1 and = 1 in state and , respec- g ¹B, aº 훀 ¹Bº 7! ' a ¹0 , ··· , 0 º 1 2 : 8=1 8 maps the joint action = 1 = tively. Player & also has two strategies, q and q . Since there are to an immediateÎ payoff value for each player. The transition func- only two states, the transition probabilities have the relationship, I¹B, aº = 훀 ¹Bº 7! Δ:−1 1 1 2 2 tion : 8=1 8 determines the probabilistic IB1 ¹B , aº = 1 − IB2 ¹B , aº and IB2 ¹B , aº = 1 − IB1 ¹B , aº. The :− state transition, where Δ 1 is the ¹: − 1º-simplex and IB0 ¹B, aº is framework of a two-state two-player two-action stochastic game 0 the transition probability from state B to B under joint action a. is shown in Figure 1. In this work, we also follow the restriction proposed in [8] that all states B 2 Y are in an ergodic set. This restriction ensures that 2.3 Learning automata the game has no absorbing states. In this section, we give a description on how player can update his strategy with multi-agent reinforcement learning method [26]. A 2.2 Two-state two-player two-action stochastic player can be seen as a learning automata, who updates his strat- games egy by monitoring the reinforcement signal resulted from the ac- Here, we introduce the simplest version of stochastic games: two- tion he has chosen in this round. An automata wants to learn the state two-player two-action stochastic game. optimal strategy and maximizes the expected payoffs. In games containing two players, player % (&) has a strategy p In this paper, we focus on finite action-set learning automata (q). Each player chooses between only two actions, cooperation (2) (FALA) [8]. At the beginning of a round, every player draws a and defection (3). Thus, we can use a probability to fully define one random action 0¹Cº from his action set according to his strategy player’s strategy. For player %, the parameter ? means the proba- 0 ¹Cº. Based on the action 0¹Cº, the environment responds with a re- bility of cooperation and p = ¹?, 1 − ?ºT. Similarly, for player &, ward g ¹Cº. The automata uses this reward to update his strategy to q = ¹@, 1 − @ºT. The payoff function can be represented byabi- 0 ¹C ¸1º.
Load more

Recommended publications
  • Stochastic Game Theory Applications for Power Management in Cognitive Networks
    STOCHASTIC GAME THEORY APPLICATIONS FOR POWER MANAGEMENT IN COGNITIVE NETWORKS A thesis submitted to Kent State University in partial fulfillment of the requirements for the degree of Master of Digital Science by Sham Fung Feb 2014 Thesis written by Sham Fung M.D.S, Kent State University, 2014 Approved by Advisor , Director, School of Digital Science ii TABLE OF CONTENTS LIST OF FIGURES . vi LIST OF TABLES . vii Acknowledgments . viii Dedication . ix 1 BACKGROUND OF COGNITIVE NETWORK . 1 1.1 Motivation and Requirements . 1 1.2 Definition of Cognitive Networks . 2 1.3 Key Elements of Cognitive Networks . 3 1.3.1 Learning and Reasoning . 3 1.3.2 Cognitive Network as a Multi-agent System . 4 2 POWER ALLOCATION IN COGNITIVE NETWORKS . 5 2.1 Overview . 5 2.2 Two Communication Paradigms . 6 2.3 Power Allocation Scheme . 8 2.3.1 Interference Constraints for Primary Users . 8 2.3.2 QoS Constraints for Secondary Users . 9 2.4 Related Works on Power Management in Cognitive Networks . 10 iii 3 GAME THEORY|PRELIMINARIES AND RELATED WORKS . 12 3.1 Non-cooperative Game . 14 3.1.1 Nash Equilibrium . 14 3.1.2 Other Equilibriums . 16 3.2 Cooperative Game . 16 3.2.1 Bargaining Game . 17 3.2.2 Coalition Game . 17 3.2.3 Solution Concept . 18 3.3 Stochastic Game . 19 3.4 Other Types of Games . 20 4 GAME THEORETICAL APPROACH ON POWER ALLOCATION . 23 4.1 Two Fundamental Types of Utility Functions . 23 4.1.1 QoS-Based Game . 23 4.1.2 Linear Pricing Game .
    [Show full text]
  • Deterministic and Stochastic Prisoner's Dilemma Games: Experiments in Interdependent Security
    NBER TECHNICAL WORKING PAPER SERIES DETERMINISTIC AND STOCHASTIC PRISONER'S DILEMMA GAMES: EXPERIMENTS IN INTERDEPENDENT SECURITY Howard Kunreuther Gabriel Silvasi Eric T. Bradlow Dylan Small Technical Working Paper 341 http://www.nber.org/papers/t0341 NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambridge, MA 02138 August 2007 We appreciate helpful discussions in designing the experiments and comments on earlier drafts of this paper by Colin Camerer, Vince Crawford, Rachel Croson, Robyn Dawes, Aureo DePaula, Geoff Heal, Charles Holt, David Krantz, Jack Ochs, Al Roth and Christian Schade. We also benefited from helpful comments from participants at the Workshop on Interdependent Security at the University of Pennsylvania (May 31-June 1 2006) and at the Workshop on Innovation and Coordination at Humboldt University (December 18-20, 2006). We thank George Abraham and Usmann Hassan for their help in organizing the data from the experiments. Support from NSF Grant CMS-0527598 and the Wharton Risk Management and Decision Processes Center is gratefully acknowledged. The views expressed herein are those of the author(s) and do not necessarily reflect the views of the National Bureau of Economic Research. © 2007 by Howard Kunreuther, Gabriel Silvasi, Eric T. Bradlow, and Dylan Small. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source. Deterministic and Stochastic Prisoner's Dilemma Games: Experiments in Interdependent Security Howard Kunreuther, Gabriel Silvasi, Eric T. Bradlow, and Dylan Small NBER Technical Working Paper No. 341 August 2007 JEL No. C11,C12,C22,C23,C73,C91 ABSTRACT This paper examines experiments on interdependent security prisoner's dilemma games with repeated play.
    [Show full text]
  • Stochastic Games with Hidden States∗
    Stochastic Games with Hidden States¤ Yuichi Yamamoto† First Draft: March 29, 2014 This Version: January 14, 2015 Abstract This paper studies infinite-horizon stochastic games in which players ob- serve noisy public information about a hidden state each period. We find that if the game is connected, the limit feasible payoff set exists and is invariant to the initial prior about the state. Building on this invariance result, we pro- vide a recursive characterization of the equilibrium payoff set and establish the folk theorem. We also show that connectedness can be replaced with an even weaker condition, called asymptotic connectedness. Asymptotic con- nectedness is satisfied for generic signal distributions, if the state evolution is irreducible. Journal of Economic Literature Classification Numbers: C72, C73. Keywords: stochastic game, hidden state, connectedness, stochastic self- generation, folk theorem. ¤The author thanks Naoki Aizawa, Drew Fudenberg, Johannes Horner,¨ Atsushi Iwasaki, Michi- hiro Kandori, George Mailath, Takeaki Sunada, and Masatoshi Tsumagari for helpful conversa- tions, and seminar participants at various places. †Department of Economics, University of Pennsylvania. Email: [email protected] 1 1 Introduction When agents have a long-run relationship, underlying economic conditions may change over time. A leading example is a repeated Bertrand competition with stochastic demand shocks. Rotemberg and Saloner (1986) explore optimal col- lusive pricing when random demand shocks are i.i.d. each period. Haltiwanger and Harrington (1991), Kandori (1991), and Bagwell and Staiger (1997) further extend the analysis to the case in which demand fluctuations are cyclic or persis- tent. One of the crucial assumptions of these papers is that demand shocks are publicly observable before firms make their decisions in each period.
    [Show full text]
  • 570: Minimax Sample Complexity for Turn-Based Stochastic Game
    Minimax Sample Complexity for Turn-based Stochastic Game Qiwen Cui1 Lin F. Yang2 1School of Mathematical Sciences, Peking University 2Electrical and Computer Engineering Department, University of California, Los Angeles Abstract guarantees are rather rare due to complex interaction be- tween agents that makes the problem considerably harder than single agent reinforcement learning. This is also known The empirical success of multi-agent reinforce- as non-stationarity in MARL, which means when multi- ment learning is encouraging, while few theoret- ple agents alter their strategies based on samples collected ical guarantees have been revealed. In this work, from previous strategy, the system becomes non-stationary we prove that the plug-in solver approach, proba- for each agent and the improvement can not be guaranteed. bly the most natural reinforcement learning algo- One fundamental question in MBRL is that how to design rithm, achieves minimax sample complexity for efficient algorithms to overcome non-stationarity. turn-based stochastic game (TBSG). Specifically, we perform planning in an empirical TBSG by Two-players turn-based stochastic game (TBSG) is a two- utilizing a ‘simulator’ that allows sampling from agents generalization of Markov decision process (MDP), arbitrary state-action pair. We show that the em- where two agents choose actions in turn and one agent wants pirical Nash equilibrium strategy is an approxi- to maximize the total reward while the other wants to min- mate Nash equilibrium strategy in the true TBSG imize it. As a zero-sum game, TBSG is known to have and give both problem-dependent and problem- Nash equilibrium strategy [Shapley, 1953], which means independent bound.
    [Show full text]
  • Discovery and Equilibrium in Games with Unawareness∗
    Discovery and Equilibrium in Games with Unawareness∗ Burkhard C. Schippery April 20, 2018 Abstract Equilibrium notions for games with unawareness in the literature cannot be interpreted as steady-states of a learning process because players may discover novel actions during play. In this sense, many games with unawareness are \self-destroying" as a player's representa- tion of the game may change after playing it once. We define discovery processes where at each state there is an extensive-form game with unawareness that together with the players' play determines the transition to possibly another extensive-form game with unawareness in which players are now aware of actions that they have discovered. A discovery process is rationalizable if players play extensive-form rationalizable strategies in each game with un- awareness. We show that for any game with unawareness there is a rationalizable discovery process that leads to a self-confirming game that possesses a self-confirming equilibrium in extensive-form rationalizable strategies. This notion of equilibrium can be interpreted as steady-state of both a discovery and learning process. Keywords: Self-confirming equilibrium, conjectural equilibrium, extensive-form rational- izability, unawareness, extensive-form games, equilibrium, learning, discovery. JEL-Classifications: C72, D83. ∗I thank Aviad Heifetz, Byung Soo Lee, and seminar participants at UC Berkeley, the University of Toronto, the Barcelona workshop on Limited Reasoning and Cognition, SAET 2015, CSLI 2016, TARK 2017, and the Virginia Tech Workshop on Advances in Decision Theory 2018 for helpful discussions. An abbreviated earlier version appeared in the online proceedings of TARK 2017 under the title “Self-Confirming Games: Unawareness, Discovery, and Equilibrium".
    [Show full text]
  • Efficient Exploration of Zero-Sum Stochastic Games
    Efficient exploration of zero-sum stochastic games Carlos Martin1 Tuomas Sandholm1,2,3,4 [email protected] [email protected] 1Computer Science Department, Carnegie Mellon University 2Optimized Markets, Inc. 3Strategic Machine, Inc. 4Strategy Robot, Inc. Abstract We investigate the increasingly important and common game-solving setting where we do not have an explicit description of the game but only oracle access to it through gameplay, such as in financial or military simulations and computer games. During a limited-duration learning phase, the algorithm can control the actions of both players in order to try to learn the game and how to play it well. After that, the algorithm has to produce a strategy that has low exploitability. Our motivation is to quickly learn strategies that have low exploitability in situations where evaluating the payoffs of a queried strategy profile is costly. For the stochastic game setting, we propose using the distribution of state-action value functions induced by a belief distribution over possible environments. We compare the performance of various explo- ration strategies for this task, including generalizations of Thompson sampling and Bayes-UCB to this new setting. These two consistently outperform other strategies. 1 Introduction We study the problem of how to efficiently explore zero-sum games whose payoffs and dynamics are initially unknown. The agent is given a certain number of episodes to learn as much useful information about the game as possible. During this learning, the rewards obtained in the game are fictional and thus do not count toward the evaluation of the final strategy.
    [Show full text]
  • Nash Q-Learning for General-Sum Stochastic Games
    Journal of Machine Learning Research 4 (2003) 1039-1069 Submitted 11/01; Revised 10/02; Published 11/03 Nash Q-Learning for General-Sum Stochastic Games Junling Hu [email protected] Talkai Research 843 Roble Ave., 2 Menlo Park, CA 94025, USA Michael P. Wellman [email protected] Artificial Intelligence Laboratory University of Michigan Ann Arbor, MI 48109-2110, USA Editor: Craig Boutilier Abstract We extend Q-learning to a noncooperative multiagent context, using the framework of general- sum stochastic games. A learning agent maintains Q-functions over joint actions, and performs updates based on assuming Nash equilibrium behavior over the current Q-values. This learning protocol provably converges given certain restrictions on the stage games (defined by Q-values) that arise during learning. Experiments with a pair of two-player grid games suggest that such restric- tions on the game structure are not necessarily required. Stage games encountered during learning in both grid environments violate the conditions. However, learning consistently converges in the first grid game, which has a unique equilibrium Q-function, but sometimes fails to converge in the second, which has three different equilibrium Q-functions. In a comparison of offline learn- ing performance in both games, we find agents are more likely to reach a joint optimal path with Nash Q-learning than with a single-agent Q-learning method. When at least one agent adopts Nash Q-learning, the performance of both agents is better than using single-agent Q-learning. We have also implemented an online version of Nash Q-learning that balances exploration with exploitation, yielding improved performance.
    [Show full text]
  • Bayesian Analysis of Deterministic and Stochastic Prisoner's Dilemma Games
    University of Pennsylvania ScholarlyCommons Operations, Information and Decisions Papers Wharton Faculty Research 8-2009 Bayesian Analysis of Deterministic and Stochastic Prisoner's Dilemma Games Howard Kunreuther University of Pennsylvania Gabriel Silvasi Eric. T. Bradlow University of Pennsylvania Dylan. S. Small University of Pennsylvania Follow this and additional works at: https://repository.upenn.edu/oid_papers Part of the Probability Commons, and the Statistical Theory Commons Recommended Citation Kunreuther, H., Silvasi, G., Bradlow, E. T., & Small, D. S. (2009). Bayesian Analysis of Deterministic and Stochastic Prisoner's Dilemma Games. Judgment and Decision Making, 4 (5), 363-384. Retrieved from https://repository.upenn.edu/oid_papers/254 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/oid_papers/254 For more information, please contact [email protected]. Bayesian Analysis of Deterministic and Stochastic Prisoner's Dilemma Games Abstract This paper compares the behavior of individuals playing a classic two-person deterministic prisoner’s dilemma (PD) game with choice data obtained from repeated interdependent security prisoner’s dilemma games with varying proba- bilities of loss and the ability to learn (or not learn) about the actions of one’s counterpart, an area of recent interest in experimental economics. This novel data set, from a series of controlled laboratory experiments, is analyzed using Bayesian hierarchical methods, the first application of such methods in this research domain. We find that individuals are much more likely to be cooperative when payoffs are deterministic than when the out-comes are probabilistic. A key factor explaining this difference is that subjects in a stochastic PD game respond not just to what their counterparts did but also to whether or not they suffered a loss.
    [Show full text]
  • Stochastic Constraint Programming for General Game Playing with Imperfect Information
    Stochastic Constraint Programming for General Game Playing with Imperfect Information Frédéric Koriche, Sylvain Lagrue, Éric Piette, and Sébastien Tabary CRIL Univ. Artois - CNRS UMR 8188, France-62307 Lens {koriche,lagrue,epiette,tabary}@cril.fr Abstract current game state, and the effects of their joint actions are uncer- tain. As such, GDL-II is expressive enough to capture partially The game description language with incomplete observable stochastic games (POSGs), which cover a wide variety GDL-II information ( ) is expressive enough to of multi-agent sequential decision problems. However, such ex- capture partially observable stochastic multi-agent pressiveness does not come without a price: from a computational games. Unfortunately, such expressiveness does viewpoint, the problem of finding a winning strategy in a POSG not come without a price: the problem of finding a is NEXPNP-complete [Goldsmith and Mundhenk, 2007], a com- winning strategy is NEXPNP-hard, a complexity class plexity class which is far beyond the standard PSPACE complexity which is far beyond the reach of modern constraint class of games with complete information. Although several algo- solvers. In this paper, we identify a PSPACE-complete rithms have been devised for tackling specific instances of POSGs, GDL-II fragment of , where agents share the same such as Contract Bridge [Ginsberg, 2001] and Poker [Bowling (partial) observations. We show that this fragment et al., 2015], they are dedicated programs which heavily rely on can be cast as a decomposable stochastic constraint human knowledge about game rules and evaluation functions. By SCSP satisfaction problem ( ) which, in turn, can be contrast, the task of developing general game players for POSGs solved using general-purpose constraint programming appears extremely challenging due to their complexity barrier.
    [Show full text]
  • Perfect Conditional E-Equilibria of Multi-Stage Games with Infinite Sets
    Perfect Conditional -Equilibria of Multi-Stage Games with Infinite Sets of Signals and Actions∗ Roger B. Myerson∗ and Philip J. Reny∗∗ *Department of Economics and Harris School of Public Policy **Department of Economics University of Chicago Abstract Abstract: We extend Kreps and Wilson’s concept of sequential equilibrium to games with infinite sets of signals and actions. A strategy profile is a conditional -equilibrium if, for any of a player’s positive probability signal events, his conditional expected util- ity is within of the best that he can achieve by deviating. With topologies on action sets, a conditional -equilibrium is full if strategies give every open set of actions pos- itive probability. Such full conditional -equilibria need not be subgame perfect, so we consider a non-topological approach. Perfect conditional -equilibria are defined by testing conditional -rationality along nets of small perturbations of the players’ strate- gies and of nature’s probability function that, for any action and for almost any state, make this action and state eventually (in the net) always have positive probability. Every perfect conditional -equilibrium is a subgame perfect -equilibrium, and, in fi- nite games, limits of perfect conditional -equilibria as 0 are sequential equilibrium strategy profiles. But limit strategies need not exist in→ infinite games so we consider instead the limit distributions over outcomes. We call such outcome distributions per- fect conditional equilibrium distributions and establish their existence for a large class of regular projective games. Nature’s perturbations can produce equilibria that seem unintuitive and so we augment the game with a net of permissible perturbations.
    [Show full text]
  • Monte-Carlo Tree Reductions for Stochastic Games Nicolas Jouandeau, Tristan Cazenave
    Monte-Carlo Tree Reductions for Stochastic Games Nicolas Jouandeau, Tristan Cazenave To cite this version: Nicolas Jouandeau, Tristan Cazenave. Monte-Carlo Tree Reductions for Stochastic Games. 19th an- nual Conference on Technologies and Applications of Artificial Intelligence, Nov 2014, Taipei, Taiwan. 10.1007/978-3-319-13987-6_22. hal-02317159 HAL Id: hal-02317159 https://hal.archives-ouvertes.fr/hal-02317159 Submitted on 15 Oct 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Monte-Carlo Tree Reductions for Stochastic Games Nicolas Jouandeau1 and Tristan Cazenave2 1 LIASD, Universit´ede Paris 8, France [email protected] 2 LAMSADE, Universit´eParis-Dauphine, France [email protected] Abstract. Monte-Carlo Tree Search (MCTS) is a powerful paradigm for perfect information games. When considering stochastic games, the tree model that represents the game has to take chance and a huge branching factor into account. As effectiveness of MCTS may decrease in such a setting, tree reductions may be useful. Chance-nodes are a way to deal with random events. Move-groups are another way to deal efficiently with a large branching factor by regrouping nodes.
    [Show full text]
  • Quantum Games: a Review of the History, Current State, and Interpretation
    Quantum games: a review of the history, current state, and interpretation Faisal Shah Khan,∗ Neal Solmeyer & Radhakrishnan Balu,y Travis S. Humblez October 1, 2018 Abstract We review both theoretical and experimental developments in the area of quantum games since the inception of the subject circa 1999. We will also offer a narrative on the controversy that surrounded the subject in its early days, and how this controversy has affected the development of the subject. 1 Introduction Roots of the theory of quantum information lie in the ideas of Wisener and Ingarden [1, 2]. These authors proposed ways to incorporate the uncertainty or entropy of information within the uncertainty inherent in quantum mechanical processes. Uncertainty is a fundamental concept in both physics and the theory of information, hence serving as the natural link between the two disciplines. Uncertainty itself is defined in terms of probability distributions. Every quantum physical object produces a probability distribution when its state is measured. More precisely, a general state of an m-state quantum object is an element of the m m-dimensional complex projective Hilbert space CP , say v = (v1; : : : ; vm) : (1) Upon measurement with respect to the observable states of the quantum object (which are the elements of m an orthogonal basis of CP ), v will produce the probability distribution 1 2 2 Pm 2 jv1j ;:::; jvmj (2) k=1 jvkj arXiv:1803.07919v2 [quant-ph] 27 Sep 2018 over the observable states. Hence, a notion of quantum entropy or uncertainty may be defined that coincides with the corresponding notion in classical information theory [3].
    [Show full text]