DOCSLIB.ORG

Two Player Zero Sum Multi-Stage Game Analysis Using Coevolutionary Algorithm

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

TWO PLAYER ZERO SUM MULTI-STAGE GAME ANALYSIS USING COEVOLUTIONARY ALGORITHM. A Thesis submitted in partial fulfilment of the requirements for the degree of Master of Science in Electrical Engineering By SUMEDH SOPAN NAGRALE B.E.,University of Mumbai, 2012 2019 Wright State University WRIGHT STATE UNIVERSITY GRADUATE SCHOOL April 24, 2019 I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY SUPERVISION BY Sumedh Sopan Nagrale ENTITLED Two Player Zero Sum Multi-Stage Game Analysis Using Coevolutionary Algorithm BE ACCEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DE- GREE OF Master of Science in Electrical Engineering. Luther Palmer, III, Ph.D. Thesis Director. Fred D. Garber, Ph.D. Chair, Department of Electrical Engineering Committee on Final Examination: Luther Palmer, III, Ph.D. Pradeep Misra, Ph.D. Xiaodong Zhang, Ph.D. Barry Milligan, Ph.D. Interim Dean of the Graduate School ABSTRACT Nagrale, Sumedh Sopan. M.S.E.E, Department of Electrical Engineering, Wright State University, 2019. Two Player Zero Sum Multi-Stage Game Analysis Using Coevolutionary Algorithm. A New Two player zero sum multistage simultaneous Game has been developed from a real-life sit- uation of dispute between two individual. Research identifies a multistage game as a multi-objective optimization problem and solves it using Coevolutionary algorithm which converges to a solution from pareto optimal solution. A comparison is done between individual stage behaviour and multistage be- haviour. Further, simulations over a range for Crossover rate, Mutation rate and Number of interaction is done to narrow down the range for a range with optimal computation speed. A relationship has been observed which identifies a relationship between population size, number of interactions, crossover rate, mutation rate and computational time. A point from the obtained range is then selected and applied to a new game to see if the point from the narrowed range works. iii Contents 1 Background 1 1.1 Game Theory . .1 1.2 Solution Concept . .7 1.2.1 Minimax Theorem: . .7 1.2.2 Nash Equilibrium . .7 1.2.3 Rationalization . .7 1.3 Contemporary . .9 2 Introduction 10 3 Game 1: Attacker’s Game or Horizontal Movement Game 12 3.0.1 Players . 12 3.0.2 Environment . 12 3.0.3 Actions . 12 3.0.4 States . 13 3.0.5 Representation choice . 13 3.0.6 Utility . 13 3.0.7 Normal form representation of the states . 13 3.0.8 Transition of states based on actions taken . 14 3.1 Game theoretic solution for the Attack Game . 15 4 Methodology 19 4.1 Co-Evolutionary Algorithm . 20 4.2 Solving s0 as an example by Hand . 21 4.2.1 Population and Gene . 21 4.2.2 Fitness Function . 22 4.2.3 Selection: . 24 4.2.4 Crossover: . 25 4.2.5 Mutation: . 26 4.2.6 Fitness Evaluation: . 26 iv 5 Simulation 27 5.1 Simulation for Individual States using CoEv . 27 5.1.1 Individual Stage Game S0 simulation . 27 5.1.2 Individual Stage Game S1 simulation . 28 5.1.3 Individual Stage Game S2 simulation . 29 5.1.4 Observation . 30 5.1.5 Analysis . 31 5.1.6 Conclusion for the simulation . 31 5.2 Simulations for Multistage Game using CoEv Algorithm . 32 5.2.1 Simulation . 32 5.2.2 Observations . 32 5.2.3 Analysis . 33 5.2.4 Conclusion for the simulation . 33 5.3 Simulation for Parameters with optimal Computation time . 34 5.3.1 Simulation . 34 5.3.2 Observation . 36 5.3.3 Analysis . 36 5.4 Derivation for Computational time . 36 5.4.1 Relation between Generation, Interaction and Time taken . 38 5.4.2 Comparison between values Actual time and real time taken for Interaction . 41 5.5 Conclusion . 42 6 Game 2: Movement Game 43 6.0.1 Players . 43 6.0.2 Environment . 43 6.0.3 Actions . 43 6.0.4 States . 44 6.0.5 Utility . 44 6.0.6 Representation choice . 44 6.0.7 Normal form representation of the states . 44 6.1 Simulation . 46 6.2 Observation and Analysis . 46 6.3 Conclusion . 46 7 Discussion 47 7.1 Future Work . 47 7.1.1 Multiple task Learning . 47 7.1.2 Thesis Extension . 47 7.1.3 Application of research . 48 v 8 Conclusion 49 9 Summary 50 Bibliography 51 vi List of Figures 3.1 Game I States and their equivalent representation s0,s1,s2 ............... 13 3.2 There are multiple Nash equilibriums (R,R) , (R,L), (L,R),(L,L) . 15 3.3 pure stage game S1 equilibrium (L,R) . 15 3.4 pure stage game S2 equilibrium (R,L) . 16 4.1 Genetic Algorithm Outline . 20 4.2 Coevolutionary Algorithm Outline . 21 5.1 State S0 simulation I . 28 5.2 State S0 simulation II . 28 5.3 State S1 simulation I . 29 5.4 State S1 simulation II . 29 5.5 State S2 simulation I . 30 5.6 State S2 simulation II . 30 5.7 State S0 + S1 + S2 simulation I . 32 5.8 Interactions 24 and 48 . 34 5.9 Interactions 72 and 96 . 34 5.10 Interactions 120 and 144 . 35 5.11 Interactions 168 and 192 . 35 5.12 Interactions 216 and 240 . 35 5.13 Plot for G Vs I, Computation time Vs I and Total time Vs I . 39 5.14 Plot for G Vs I, Computation time Vs I and Total time Vs I . 39 5.15 Plot for G Vs I, Computation time Vs I and Total time Vs I . 39 5.16 Plot for G Vs I, Computation time Vs I and Total time Vs I . 39 5.17 PS vsI.......................................... 41 5.18 Game I States and their equivalent representation s0,s1,s2 ............... 42 5.19 Game I States and their equivalent representation s0,s1,s2 ............... 42 6.1 Game II States s0,s1,s2,s3 ............................... 44 6.2 State S0 + S1 + S2 + S3 simulation I . 46 vii List of Tables 1.1 3x3 Matrix: Rock-Paper-Scissor Normal-Form Game . .5 3.1 4x4 Matrix: s0 Normal-Form Game . 14 3.2 4x4 Matrix: s1 Normal-Form Game . 14 3.3 4x4 Matrix: s2 Normal-Form Game . 14 3.4 4x4 Matrix: s0 Reduced Normal-Form Game by Iterative deletion of dominant strategy 16 3.5 4x4 Matrix: s1 Reduced Normal-Form Game by Iterative deletion of dominant strategy 17 3.6 Expected payoff for P5 for mixing different probabilities . 18 4.1 solution for P4 s0 ................................... 19 4.2 Gene example for P4 s0 population . 22 4.3 P4 s0 example and corresponding binary encoding . 23 4.4 Game Matrix in state s0 ................................. 23 4.5 fitness function value P4 s0 population . 23 4.6 P4 s0 population with binary encoded values . 24 4.7 P4 s0 binary encoded population . 24 4.8 P4 s0 population with fitness and noramlised values . 24 4.9 P4 s0 population . 24 4.10 P4 s0 sorted population with respect to fitness . ..
Recommended publications
  • Labsi Working Papers

    Labsi Working Papers

    UNIVERSITY OF SIENA S.N. O’ H iggins Arturo Palomba Patrizia Sbriglia Second Mover Advantage and Bertrand Dynamic Competition: An Experiment May 2010 LABSI WORKING PAPERS N. 28/2010 SECOND MOVER ADVANTAGE AND BERTRAND DYNAMIC COMPETITION: AN EXPERIMENT § S.N. O’Higgins University of Salerno [email protected] Arturo Palomba University of Naples II [email protected] Patrizia Sbriglia §§ University of Naples II [email protected] Abstract In this paper we provide an experimental test of a dynamic Bertrand duopolistic model, where firms move sequentially and their informational setting varies across different designs. Our experiment is composed of three treatments. In the first treatment, subjects receive information only on the costs and demand parameters and on the price’ choices of their opponent in the market in which they are positioned (matching is fixed); in the second and third treatments, subjects are also informed on the behaviour of players who are not directly operating in their market. Our aim is to study whether the individual behaviour and the process of equilibrium convergence are affected by the specific informational setting adopted. In all treatments we selected students who had previously studied market games and industrial organization, conjecturing that the specific participants’ expertise decreased the chances of imitation in treatment II and III. However, our results prove the opposite: the extra information provided in treatment II and III strongly affects the long run convergence to the market equilibrium. In fact, whilst in the first session, a high proportion of markets converge to the Nash-Bertrand symmetric solution, we observe that a high proportion of markets converge to more collusive outcomes in treatment II and more competitive outcomes in treatment III.
  • 1 Sequential Games

    1 Sequential Games

    1 Sequential Games We call games where players take turns moving “sequential games”. Sequential games consist of the same elements as normal form games –there are players, rules, outcomes, and payo¤s. However, sequential games have the added element that history of play is now important as players can make decisions conditional on what other players have done. Thus, if two people are playing a game of Chess the second mover is able to observe the …rst mover’s initial move prior to making his initial move. While it is possible to represent sequential games using the strategic (or matrix) form representation of the game it is more instructive at …rst to represent sequential games using a game tree. In addition to the players, actions, outcomes, and payo¤s, the game tree will provide a history of play or a path of play. A very basic example of a sequential game is the Entrant-Incumbent game. The game is described as follows: Consider a game where there is an entrant and an incumbent. The entrant moves …rst and the incumbent observes the entrant’sdecision. The entrant can choose to either enter the market or remain out of the market. If the entrant remains out of the market then the game ends and the entrant receives a payo¤ of 0 while the incumbent receives a payo¤ of 2. If the entrant chooses to enter the market then the incumbent gets to make a choice. The incumbent chooses between …ghting entry or accommodating entry. If the incumbent …ghts the entrant receives a payo¤ of 3 while the incumbent receives a payo¤ of 1.
  • Notes on Sequential and Repeated Games

    Notes on Sequential and Repeated Games

    Notes on sequential and repeated games 1 Sequential Move Games Thus far we have examined games in which players make moves simultaneously (or without observing what the other player has done). Using the normal (strategic) form representation of a game we can identify sets of strategies that are best responses to each other (Nash Equilibria). We now focus on sequential games of complete information. We can still use the normal form representation to identify NE but sequential games are richer than that because some players observe other players’decisions before they take action. The fact that some actions are observable may cause some NE of the normal form representation to be inconsistent with what one might think a player would do. Here’sa simple game between an Entrant and an Incumbent. The Entrant moves …rst and the Incumbent observes the Entrant’s action and then gets to make a choice. The Entrant has to decide whether or not he will enter a market or not. Thus, the Entrant’s two strategies are “Enter” or “Stay Out”. If the Entrant chooses “Stay Out” then the game ends. The payo¤s for the Entrant and Incumbent will be 0 and 2 respectively. If the Entrant chooses “Enter” then the Incumbent gets to choose whether or not he will “Fight”or “Accommodate”entry. If the Incumbent chooses “Fight”then the Entrant receives 3 and the Incumbent receives 1. If the Incumbent chooses “Accommodate”then the Entrant receives 2 and the Incumbent receives 1. This game in normal form is Incumbent Fight if Enter Accommodate if Enter .
  • SEQUENTIAL GAMES with PERFECT INFORMATION Example

    SEQUENTIAL GAMES with PERFECT INFORMATION Example

    SEQUENTIAL GAMES WITH PERFECT INFORMATION Example 4.9 (page 105) Consider the sequential game given in Figure 4.9. We want to apply backward induction to the tree. 0 Vertex B is owned by player two, P2. The payoffs for P2 are 1 and 3, with 3 > 1, so the player picks R . Thus, the payoffs at B become (0, 3). 00 Next, vertex C is also owned by P2 with payoffs 1 and 0. Since 1 > 0, P2 picks L , and the payoffs are (4, 1). Player one, P1, owns A; the choice of L gives a payoff of 0 and R gives a payoff of 4; 4 > 0, so P1 chooses R. The final payoffs are (4, 1). 0 00 We claim that this strategy profile, { R } for P1 and { R ,L } is a Nash equilibrium. Notice that the 0 00 strategy profile gives a choice at each vertex. For the strategy { R ,L } fixed for P2, P1 has a maximal payoff by choosing { R }, ( 0 00 0 00 π1(R, { R ,L }) = 4 π1(R, { R ,L }) = 4 ≥ 0 00 π1(L, { R ,L }) = 0. 0 00 In the same way, for the strategy { R } fixed for P1, P2 has a maximal payoff by choosing { R ,L }, ( 00 0 00 π2(R, {∗,L }) = 1 π2(R, { R ,L }) = 1 ≥ 00 π2(R, {∗,R }) = 0, where ∗ means choose either L0 or R0. Since no change of choice by a player can increase that players own payoff, the strategy profile is called a Nash equilibrium. Notice that the above strategy profile is also a Nash equilibrium on each branch of the game tree, mainly starting at either B or starting at C.
  • Finitely Repeated Games

    Finitely Repeated Games

    Repeated games 1: Finite repetition Universidad Carlos III de Madrid 1 Finitely repeated games • A finitely repeated game is a dynamic game in which a simultaneous game (the stage game) is played finitely many times, and the result of each stage is observed before the next one is played. • Example: Play the prisoners’ dilemma several times. The stage game is the simultaneous prisoners’ dilemma game. 2 Results • If the stage game (the simultaneous game) has only one NE the repeated game has only one SPNE: In the SPNE players’ play the strategies in the NE in each stage. • If the stage game has 2 or more NE, one can find a SPNE where, at some stage, players play a strategy that is not part of a NE of the stage game. 3 The prisoners’ dilemma repeated twice • Two players play the same simultaneous game twice, at ! = 1 and at ! = 2. • After the first time the game is played (after ! = 1) the result is observed before playing the second time. • The payoff in the repeated game is the sum of the payoffs in each stage (! = 1, ! = 2) • Which is the SPNE? Player 2 D C D 1 , 1 5 , 0 Player 1 C 0 , 5 4 , 4 4 The prisoners’ dilemma repeated twice Information sets? Strategies? 1 .1 5 for each player 2" for each player D C E.g.: (C, D, D, C, C) Subgames? 2.1 5 D C D C .2 1.3 1.5 1 1.4 D C D C D C D C 2.2 2.3 2 .4 2.5 D C D C D C D C D C D C D C D C 1+1 1+5 1+0 1+4 5+1 5+5 5+0 5+4 0+1 0+5 0+0 0+4 4+1 4+5 4+0 4+4 1+1 1+0 1+5 1+4 0+1 0+0 0+5 0+4 5+1 5+0 5+5 5+4 4+1 4+0 4+5 4+4 The prisoners’ dilemma repeated twice Let’s find the NE in the subgames.
  • Stochastic Game Theory Applications for Power Management in Cognitive Networks

    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 .
  • Implementation Theory*

    Implementation Theory*

    Chapter 5 IMPLEMENTATION THEORY* ERIC MASKIN Institute for Advanced Study, Princeton, NJ, USA TOMAS SJOSTROM Department of Economics, Pennsylvania State University, University Park, PA, USA Contents Abstract 238 Keywords 238 1. Introduction 239 2. Definitions 245 3. Nash implementation 247 3.1. Definitions 248 3.2. Monotonicity and no veto power 248 3.3. Necessary and sufficient conditions 250 3.4. Weak implementation 254 3.5. Strategy-proofness and rich domains of preferences 254 3.6. Unrestricted domain of strict preferences 256 3.7. Economic environments 257 3.8. Two agent implementation 259 4. Implementation with complete information: further topics 260 4.1. Refinements of Nash equilibrium 260 4.2. Virtual implementation 264 4.3. Mixed strategies 265 4.4. Extensive form mechanisms 267 4.5. Renegotiation 269 4.6. The planner as a player 275 5. Bayesian implementation 276 5.1. Definitions 276 5.2. Closure 277 5.3. Incentive compatibility 278 5.4. Bayesian monotonicity 279 * We are grateful to Sandeep Baliga, Luis Corch6n, Matt Jackson, Byungchae Rhee, Ariel Rubinstein, Ilya Segal, Hannu Vartiainen, Masahiro Watabe, and two referees, for helpful comments. Handbook of Social Choice and Welfare, Volume 1, Edited by K.J Arrow, A.K. Sen and K. Suzumura ( 2002 Elsevier Science B. V All rights reserved 238 E. Maskin and T: Sj'str6m 5.5. Non-parametric, robust and fault tolerant implementation 281 6. Concluding remarks 281 References 282 Abstract The implementation problem is the problem of designing a mechanism (game form) such that the equilibrium outcomes satisfy a criterion of social optimality embodied in a social choice rule.
  • Collusion Constrained Equilibrium

    Collusion Constrained Equilibrium

    Theoretical Economics 13 (2018), 307–340 1555-7561/20180307 Collusion constrained equilibrium Rohan Dutta Department of Economics, McGill University David K. Levine Department of Economics, European University Institute and Department of Economics, Washington University in Saint Louis Salvatore Modica Department of Economics, Università di Palermo We study collusion within groups in noncooperative games. The primitives are the preferences of the players, their assignment to nonoverlapping groups, and the goals of the groups. Our notion of collusion is that a group coordinates the play of its members among different incentive compatible plans to best achieve its goals. Unfortunately, equilibria that meet this requirement need not exist. We instead introduce the weaker notion of collusion constrained equilibrium. This al- lows groups to put positive probability on alternatives that are suboptimal for the group in certain razor’s edge cases where the set of incentive compatible plans changes discontinuously. These collusion constrained equilibria exist and are a subset of the correlated equilibria of the underlying game. We examine four per- turbations of the underlying game. In each case,we show that equilibria in which groups choose the best alternative exist and that limits of these equilibria lead to collusion constrained equilibria. We also show that for a sufficiently broad class of perturbations, every collusion constrained equilibrium arises as such a limit. We give an application to a voter participation game that shows how collusion constraints may be socially costly. Keywords. Collusion, organization, group. JEL classification. C72, D70. 1. Introduction As the literature on collective action (for example, Olson 1965) emphasizes, groups often behave collusively while the preferences of individual group members limit the possi- Rohan Dutta: [email protected] David K.
  • Chapter 16 Oligopoly and Game Theory Oligopoly Oligopoly

    Chapter 16 Oligopoly and Game Theory Oligopoly Oligopoly

    Chapter 16 “Game theory is the study of how people Oligopoly behave in strategic situations. By ‘strategic’ we mean a situation in which each person, when deciding what actions to take, must and consider how others might respond to that action.” Game Theory Oligopoly Oligopoly • “Oligopoly is a market structure in which only a few • “Figuring out the environment” when there are sellers offer similar or identical products.” rival firms in your market, means guessing (or • As we saw last time, oligopoly differs from the two ‘ideal’ inferring) what the rivals are doing and then cases, perfect competition and monopoly. choosing a “best response” • In the ‘ideal’ cases, the firm just has to figure out the environment (prices for the perfectly competitive firm, • This means that firms in oligopoly markets are demand curve for the monopolist) and select output to playing a ‘game’ against each other. maximize profits • To understand how they might act, we need to • An oligopolist, on the other hand, also has to figure out the understand how players play games. environment before computing the best output. • This is the role of Game Theory. Some Concepts We Will Use Strategies • Strategies • Strategies are the choices that a player is allowed • Payoffs to make. • Sequential Games •Examples: • Simultaneous Games – In game trees (sequential games), the players choose paths or branches from roots or nodes. • Best Responses – In matrix games players choose rows or columns • Equilibrium – In market games, players choose prices, or quantities, • Dominated strategies or R and D levels. • Dominant Strategies. – In Blackjack, players choose whether to stay or draw.
  • Implementation and Strong Nash Equilibrium

    Implementation and Strong Nash Equilibrium

    LIBRARY OF THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY Digitized by the Internet Archive in 2011 with funding from Boston Library Consortium Member Libraries http://www.archive.org/details/implementationstOOmask : } ^mm working paper department of economics IMPLEMENTATION AND STRONG NASH EQUILIBRIUM Eric Maskin Number 216 January 1978 massachusetts institute of technology 50 memorial drive Cambridge, mass. 021 39 IMPLEMENTATION AND STRONG NASH EQUILIBRIUM Eric Maskin Number 216 January 1978 I am grateful for the financial support of the National Science Foundation. A social choice correspondence (SCC) is a mapping which assoc- iates each possible profile of individuals' preferences with a set of feasible alternatives (the set of f-optima) . To implement *n SCC, f, is to construct a game form g such that, for all preference profiles the equilibrium set of g (with respect to some solution concept) coincides with the f-optimal set. In a recent study [1], I examined the general question of implementing social choice correspondences when Nash equilibrium is the solution concept. Nash equilibrium, of course, is a strictly noncooperative notion, and so it is natural to consider the extent to which the results carry over when coalitions can form. The cooperative counterpart of Nash is the strong equilibrium due to Aumann. Whereas Nash defines equilibrium in terms of deviations only by single individuals, Aumann 's equilibrium incorporates deviations by every conceivable coalition. This paper considers implementation for strong equilibrium. The results of my previous paper were positive. If an SCC sat- isfies a monotonicity property and a much weaker requirement called no veto power, it can be implemented by Nash equilibrium.
  • Deterministic and Stochastic Prisoner's Dilemma Games: Experiments in Interdependent Security

    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∗

    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.