A Constructive Study of Stochastic Games with Strategic
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Lecture 4 Rationalizability & Nash Equilibrium Road
Lecture 4 Rationalizability & Nash Equilibrium 14.12 Game Theory Muhamet Yildiz Road Map 1. Strategies – completed 2. Quiz 3. Dominance 4. Dominant-strategy equilibrium 5. Rationalizability 6. Nash Equilibrium 1 Strategy A strategy of a player is a complete contingent-plan, determining which action he will take at each information set he is to move (including the information sets that will not be reached according to this strategy). Matching pennies with perfect information 2’s Strategies: HH = Head if 1 plays Head, 1 Head if 1 plays Tail; HT = Head if 1 plays Head, Head Tail Tail if 1 plays Tail; 2 TH = Tail if 1 plays Head, 2 Head if 1 plays Tail; head tail head tail TT = Tail if 1 plays Head, Tail if 1 plays Tail. (-1,1) (1,-1) (1,-1) (-1,1) 2 Matching pennies with perfect information 2 1 HH HT TH TT Head Tail Matching pennies with Imperfect information 1 2 1 Head Tail Head Tail 2 Head (-1,1) (1,-1) head tail head tail Tail (1,-1) (-1,1) (-1,1) (1,-1) (1,-1) (-1,1) 3 A game with nature Left (5, 0) 1 Head 1/2 Right (2, 2) Nature (3, 3) 1/2 Left Tail 2 Right (0, -5) Mixed Strategy Definition: A mixed strategy of a player is a probability distribution over the set of his strategies. Pure strategies: Si = {si1,si2,…,sik} σ → A mixed strategy: i: S [0,1] s.t. σ σ σ i(si1) + i(si2) + … + i(sik) = 1. If the other players play s-i =(s1,…, si-1,si+1,…,sn), then σ the expected utility of playing i is σ σ σ i(si1)ui(si1,s-i) + i(si2)ui(si2,s-i) + … + i(sik)ui(sik,s-i). -
Learning and Equilibrium
Learning and Equilibrium Drew Fudenberg1 and David K. Levine2 1Department of Economics, Harvard University, Cambridge, Massachusetts; email: [email protected] 2Department of Economics, Washington University of St. Louis, St. Louis, Missouri; email: [email protected] Annu. Rev. Econ. 2009. 1:385–419 Key Words First published online as a Review in Advance on nonequilibrium dynamics, bounded rationality, Nash equilibrium, June 11, 2009 self-confirming equilibrium The Annual Review of Economics is online at by 140.247.212.190 on 09/04/09. For personal use only. econ.annualreviews.org Abstract This article’s doi: The theory of learning in games explores how, which, and what 10.1146/annurev.economics.050708.142930 kind of equilibria might arise as a consequence of a long-run non- Annu. Rev. Econ. 2009.1:385-420. Downloaded from arjournals.annualreviews.org Copyright © 2009 by Annual Reviews. equilibrium process of learning, adaptation, and/or imitation. If All rights reserved agents’ strategies are completely observed at the end of each round 1941-1383/09/0904-0385$20.00 (and agents are randomly matched with a series of anonymous opponents), fairly simple rules perform well in terms of the agent’s worst-case payoffs, and also guarantee that any steady state of the system must correspond to an equilibrium. If players do not ob- serve the strategies chosen by their opponents (as in extensive-form games), then learning is consistent with steady states that are not Nash equilibria because players can maintain incorrect beliefs about off-path play. Beliefs can also be incorrect because of cogni- tive limitations and systematic inferential errors. -
Lecture Notes
GRADUATE GAME THEORY LECTURE NOTES BY OMER TAMUZ California Institute of Technology 2018 Acknowledgments These lecture notes are partially adapted from Osborne and Rubinstein [29], Maschler, Solan and Zamir [23], lecture notes by Federico Echenique, and slides by Daron Acemoglu and Asu Ozdaglar. I am indebted to Seo Young (Silvia) Kim and Zhuofang Li for their help in finding and correcting many errors. Any comments or suggestions are welcome. 2 Contents 1 Extensive form games with perfect information 7 1.1 Tic-Tac-Toe ........................................ 7 1.2 The Sweet Fifteen Game ................................ 7 1.3 Chess ............................................ 7 1.4 Definition of extensive form games with perfect information ........... 10 1.5 The ultimatum game .................................. 10 1.6 Equilibria ......................................... 11 1.7 The centipede game ................................... 11 1.8 Subgames and subgame perfect equilibria ...................... 13 1.9 The dollar auction .................................... 14 1.10 Backward induction, Kuhn’s Theorem and a proof of Zermelo’s Theorem ... 15 2 Strategic form games 17 2.1 Definition ......................................... 17 2.2 Nash equilibria ...................................... 17 2.3 Classical examples .................................... 17 2.4 Dominated strategies .................................. 22 2.5 Repeated elimination of dominated strategies ................... 22 2.6 Dominant strategies .................................. -
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 . -
Economics 201B Economic Theory (Spring 2021) Strategic Games
Economics 201B Economic Theory (Spring 2021) Strategic Games Topics: terminology and notations (OR 1.7), games and solutions (OR 1.1-1.3), rationality and bounded rationality (OR 1.4-1.6), formalities (OR 2.1), best-response (OR 2.2), Nash equilibrium (OR 2.2), 2 2 examples × (OR 2.3), existence of Nash equilibrium (OR 2.4), mixed strategy Nash equilibrium (OR 3.1, 3.2), strictly competitive games (OR 2.5), evolution- ary stability (OR 3.4), rationalizability (OR 4.1), dominance (OR 4.2, 4.3), trembling hand perfection (OR 12.5). Terminology and notations (OR 1.7) Sets For R, ∈ ≥ ⇐⇒ ≥ for all . and ⇐⇒ ≥ for all and some . ⇐⇒ for all . Preferences is a binary relation on some set of alternatives R. % ⊆ From % we derive two other relations on : — strict performance relation and not  ⇐⇒ % % — indifference relation and ∼ ⇐⇒ % % Utility representation % is said to be — complete if , or . ∀ ∈ % % — transitive if , and then . ∀ ∈ % % % % can be presented by a utility function only if it is complete and transitive (rational). A function : R is a utility function representing if → % ∀ ∈ () () % ⇐⇒ ≥ % is said to be — continuous (preferences cannot jump...) if for any sequence of pairs () with ,and and , . { }∞=1 % → → % — (strictly) quasi-concave if for any the upper counter set ∈ { ∈ : is (strictly) convex. % } These guarantee the existence of continuous well-behaved utility function representation. Profiles Let be a the set of players. — () or simply () is a profile - a collection of values of some variable,∈ one for each player. — () or simply is the list of elements of the profile = ∈ { } − () for all players except . ∈ — ( ) is a list and an element ,whichistheprofile () . -
Interim Correlated Rationalizability
Interim Correlated Rationalizability The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Dekel, Eddie, Drew Fudenberg, and Stephen Morris. 2007. Interim correlated rationalizability. Theoretical Economics 2, no. 1: 15-40. Published Version http://econtheory.org/ojs/index.php/te/article/view/20070015 Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:3196333 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA Theoretical Economics 2 (2007), 15–40 1555-7561/20070015 Interim correlated rationalizability EDDIE DEKEL Department of Economics, Northwestern University, and School of Economics, Tel Aviv University DREW FUDENBERG Department of Economics, Harvard University STEPHEN MORRIS Department of Economics, Princeton University This paper proposes the solution concept of interim correlated rationalizability, and shows that all types that have the same hierarchies of beliefs have the same set of interim-correlated-rationalizable outcomes. This solution concept charac- terizes common certainty of rationality in the universal type space. KEYWORDS. Rationalizability, incomplete information, common certainty, com- mon knowledge, universal type space. JEL CLASSIFICATION. C70, C72. 1. INTRODUCTION Harsanyi (1967–68) proposes solving games of incomplete -
Tacit Collusion in Oligopoly"
Tacit Collusion in Oligopoly Edward J. Green,y Robert C. Marshall,z and Leslie M. Marxx February 12, 2013 Abstract We examine the economics literature on tacit collusion in oligopoly markets and take steps toward clarifying the relation between econo- mists’analysis of tacit collusion and those in the legal literature. We provide examples of when the economic environment might be such that collusive pro…ts can be achieved without communication and, thus, when tacit coordination is su¢ cient to elevate pro…ts versus when com- munication would be required. The authors thank the Human Capital Foundation (http://www.hcfoundation.ru/en/), and especially Andrey Vavilov, for …nancial support. The paper bene…tted from discussions with Roger Blair, Louis Kaplow, Bill Kovacic, Vijay Krishna, Steven Schulenberg, and Danny Sokol. We thank Gustavo Gudiño for valuable research assistance. [email protected], Department of Economics, Penn State University [email protected], Department of Economics, Penn State University [email protected], Fuqua School of Business, Duke University 1 Introduction In this chapter, we examine the economics literature on tacit collusion in oligopoly markets and take steps toward clarifying the relation between tacit collusion in the economics and legal literature. Economists distinguish between tacit and explicit collusion. Lawyers, using a slightly di¤erent vocabulary, distinguish between tacit coordination, tacit agreement, and explicit collusion. In hopes of facilitating clearer communication between economists and lawyers, in this chapter, we attempt to provide a coherent resolution of the vernaculars used in the economics and legal literature regarding collusion.1 Perhaps the easiest place to begin is to de…ne explicit collusion. -
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. -
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. -
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. -
Matsui 2005 IER-23Wvu3z
INTERNATIONAL ECONOMIC REVIEW Vol. 46, No. 1, February 2005 ∗ A THEORY OF MONEY AND MARKETPLACES BY AKIHIKO MATSUI AND TAKASHI SHIMIZU1 University of Tokyo, Japan; Kansai University, Japan This article considers an infinitely repeated economy with divisible fiat money. The economy has many marketplaces that agents choose to visit. In each mar- ketplace, agents are randomly matched to trade goods. There exist a variety of stationary equilibria. In some equilibrium, each good is traded at a single price, whereas in another, every good is traded at two different prices. There is a contin- uum of such equilibria, which differ from each other in price and welfare levels. However, it is shown that only the efficient single-price equilibrium is evolution- arily stable. 1. INTRODUCTION In a transaction, one needs to have what his trade partner wants. However, it is hard for, say, an economist who wants to have her hair cut to find a hairdresser who wishes to learn economics. In order to mitigate this problem of a lack of double coincidence of wants, money is often used as a medium of exchange. If there is a generally acceptable good called money, then the economist can divide the trading process into two: first, she teaches economics students to obtain money, and then finds a hairdresser to exchange money for a haircut. The hairdresser accepts the money since he can use it to obtain what he wants. Money is accepted by many people as it is believed to be accepted by many. Focusing on this function of money, Kiyotaki and Wright (1989) formalized the process of monetary exchange. -
SUPPLEMENT to “VERY SIMPLE MARKOV-PERFECT INDUSTRY DYNAMICS: THEORY” (Econometrica, Vol
Econometrica Supplementary Material SUPPLEMENT TO “VERY SIMPLE MARKOV-PERFECT INDUSTRY DYNAMICS: THEORY” (Econometrica, Vol. 86, No. 2, March 2018, 721–735) JAAP H. ABBRING CentER, Department of Econometrics & OR, Tilburg University and CEPR JEFFREY R. CAMPBELL Economic Research, Federal Reserve Bank of Chicago and CentER, Tilburg University JAN TILLY Department of Economics, University of Pennsylvania NAN YANG Department of Strategy & Policy and Department of Marketing, National University of Singapore KEYWORDS: Demand uncertainty, dynamic oligopoly, firm entry and exit, sunk costs, toughness of competition. THIS SUPPLEMENT to Abbring, Campbell, Tilly, and Yang (2018) (hereafter referred to as the “main text”) (i) proves a theorem we rely upon for the characterization of equilibria and (ii) develops results for an alternative specification of the model. Section S1’s Theorem S1 establishes that a strategy profile is subgame perfect if no player can benefit from deviating from it in one stage of the game and following it faith- fully thereafter. Our proof very closely follows that of the analogous Theorem 4.2 in Fudenberg and Tirole (1991). That theorem only applies to games that are “continuous at infinity” (Fudenberg and Tirole, p. 110), which our game is not. In particular, we only bound payoffs from above (Assumption A1 in the main text) and not from below, be- cause we want the model to encompass econometric specifications like Abbring, Camp- bell, Tilly, and Yang’s (2017) that feature arbitrarily large cost shocks. Instead, our proof leverages the presence of a repeatedly-available outside option with a fixed and bounded payoff, exit. Section S2 presents the primitive assumptions and analysis for an alternative model of Markov-perfect industry dynamics in which one potential entrant makes its entry decision at the same time as incumbents choose between continuation and exit.