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

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