Akio Matsumoto and Ferenc Szidarovszky Game Theory and Its Applications 1st ed 2016 Akio Matsumoto Department of Economic, Chuo University, Hachioji, Tokyo, Japan Ferenc Szidarovszky Department of Applied Mathematics, University of Pécs, Pécs, Hungary ISBN 978-4-431-54785-3 e-ISBN 978-4-431-54786-0 DOI 10.1007/978-4-431-54786-0 Springer Tokyo Heidelberg New York Dordrecht London Library of Congress Control Number: 2015947124 © Springer Japan 2016 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Printed on acid-free paper Springer Japan KK is part of Springer Science+Business Media (www.springer.com) Preface The authors of this book had several decades of research in different areas of game theory until the mid 1990s, when they met in a conference in Odense, Denmark Since then they work together on oligopolies and different dynamic economic systems, and meet at least once every year in Tokyo and either in Tucson, Arizona or in Pécs, Hungary This book has two origins First it is based on game theory short courses presented in several countries including Japan, Hungary, China, and Taiwan among others The second author introduced and taught for several years a one-semester graduate-level game theory course at the University of Arizona for students in engineering and management The class notes of that course is the second origin of this book The objective of this book is to introduce the readers into the main concepts, methods, and applications of game theory, the subject, which has continuously increasing importance in applications in many fields of quantitative sciences including economics, social science, engineering, biology etc The wide variety of applications are illustrated with the particular examples introduced in the second and third chapters as well as with the case studies of the last chapter We strongly recommend this book to undergraduate and graduate students, researchers, and practitioners in all fields of quantitative science where decision problems might arise involving more than one decision makers, stake holders, or interest groups As we will see later in the different chapters, the most appropriate solution concept and the corresponding solution methodology for any problem is a function of the behavior of the decision makers and their interrelationships, and the available information So before applying any method from this book, these conditions have to be examined Then the most appropriate method has to be selected and applied to get the solution, which has to be then interpreted and applied in practice We sincerely hope that this book will help the readers to understand the main concepts and methodology of game theory and it will help to select the most appropriate model, solution concept and method, and to use the obtained result in applying it in their practical problems The authors are thankful to the Department of Economics of Chuo University, Tokyo as well as the Systems and Industrial Engineering Department of the University of Arizona, Tucson for their hospitality during the joint works of the authors The more recent support of the Applied Mathematics Department of the University of Pécs, Hungary is also appreciated In addition, the authors wish to express their special thanks to Dr Taisuke Matsubae for his assistance in preparing the manuscript and the final edited version of this book Contents Introduction Part I Noncooperative Games Discrete Static Games 2.1 Examples of Two-Person Finite Games 2.2 General Description of Two-Person Finite Games 2.3 -person Finite Games Continuous Static Games 3.1 Examples of Two-Person Continuous Games 3.2 Examples of -Person Continuous Games Relation to Other Mathematical Problems 4.1 Nonlinear Optimization 4.2 Fixed Point Problems Existence of Equilibria 5.1 General Existence Conditions 5.2 Bimatrix and Matrix Games 5.3 Mixed Extensions of N-person Finite Games 5.4 Multiproduct Oligopolies Computation of Equilibria 6.1 Application of the Kuhn–Tucker Conditions 6.2 Reduction to an Optimization Problem 6.3 Solution of Bimatrix Games 6.4 Solution of Matrix Games 6.5 Solution of Oligopolies Special Matrix Games 7.1 Matrix with Identical Elements 7.2 The Case of Diagonal Matrix 7.3 Symmetric Matrix Games 7.4 Relation Between Matrix Games and Linear Programming 7.5 Method of Fictitious Play 7.6 Method of von Neumann Uniqueness of Equilibria Repeated and Dynamic Games 9.1 Leader-Follower Games 9.2 Dynamic Games with Simultaneous Moves 9.3 Dynamic Games with Sequential Moves 9.4 Extensive Forms of Dynamic Games 9.5 Subgames and Subgame-Perfect Nash Equilibria 10 Games Under Uncertainty 10.1 Static Bayesian Games 10.2 Dynamic Bayesian Games Part II Cooperative Games 11 Solutions Based on Characteristic Functions 11.1 The Core 11.2 Stable Sets 11.3 The Nucleolus 11.4 The Shapley Values 11.5 The Kernel and the Bargaining Set 12 Conflict Resolution 12.1 The Nash Bargaining Solution 12.2 Alternative Solution Concepts 12.3 -person Conflicts 13 Multiobjective Optimization 13.1 Lexicographic Method 13.2 The -constraint Method 13.3 The Weighting Method 13.4 Distance-Based Methods 13.5 Direction-Based Methods 14 Social Choice 14.1 Methods with Symmetric Players 14.2 Methods with Powers of Players 15 Case Studies and Applications 15.1 A Salesman’s Dilemma 15.2 Oligopoly in Water Management 15.3 A Forestry Management Problem 15.4 International Fishing 15.5 A Water Distribution Problem 15.6 Control in Oligopolies 15.7 Effect of Information Lag in Oligopoly Appendix A: Vector and Matrix Norms Appendix B: Convexity, Concavity Appendix C: Optimum Conditions Appendix D: Fixed Point Theorems Appendix E: Monotonic Mappings Appendix F: Duality in Linear Programming Appendix G: Multiobjective Optimization Appendix H: Stability and Controllability References Index List of Figures Figure 2.1 Equilibria in Example 2.8 Figure 2.2 Structure of the city Figure 3.1 Best responses in Example 3.1 Figure 3.2 Best responses in Example 3.2 Figure 3.3 Best responses in Case Figure 3.4 Best responses in Case Figure 3.5 Best responses in Case Figure 3.6 Best responses in Case Figure 3.7 Payoff function of player in Example 3.4 Figure 3.8 Illustration of Figure 3.9 Best responses in Example 3.4 Figure 3.10 Payoff function of player in Example 3.5 Figure 3.11 Best responses in Example 3.5 Figure 3.12 Payoff of player in Example 3.6 Figure 3.13 Best responses in Example 3.7 Figure 3.14 Payoff in Example 3.8 Figure 3.15 Best responses in Example 3.8 Figure 3.16 Payoff in Example 3.9 Figure 3.17 Best responses in Example 3.9 Figure 3.18 Best responses in Example 3.10 Figure 3.19 Payoff in Example 3.11 Figure 3.20 Payoff in Example 3.11 Figure 3.21 Best responses in Example 3.11 Figure 3.22 Payoff in Example 3.12 Figure 3.23 Payoff in Example 3.12 Figure 3.24 Best responses in Example 3.12 Figure 5.1 Payoff functions of Example 5.4 Figure 5.2 Best responses in Example 5.5 properties (a), (b) and (c) Instead of looking for a usually nonexisting optimal solution, we relax the above conditions by looking for nondominated solutions A feasible is called weakly nondominated if there is no such that for all That is, we cannot improve all objectives simultaneously on the feasible set A feasible is called strongly nondominated if no objective can be improved without worthening at least one other objective That is, there is no such that Figure G.1 shows the difference between weakly and strongly nondominated solutions, where Notice that point A is the only strongly nondominated solution, but there are infinitely many weakly nondominated solutions: the linear segments and Clearly, every strongly nondominated solution is also weakly nondominated, but the weakly nondominated solutions are not necessarily strongly nondominated Fig G.1 Weakly and strongly nondominated solutions The feasible set X shows the possible decision alternatives giving us all possibilities what we can In addition to X , the objective space is usually considered: This set gives all possible simultaneous objective values, that is, it represents what we can get Similarly to set X , a vector is weakly nondominated, if there is no other point such that in all components A point such that is strongly nondominated if there is no other point in F in every component Clearly, all weakly or strongly nondominated points of F are boundary points In the economic literature nondominated solutions are often called Pareto optimal There is no guarantee in general that a problem ( G.2 ) has nondominated solution, and even if it has, the solution is not necessarily unique For example, the problem has no nondominated solution, and problem has infinitely many nondominated solutions: = arbitrary, The existence of nondominated solutions is guaranteed by the following general result Assume F is nonempty, closed and for each there exists a value such that for all Then problem ( G.2 ) has at least one strongly nondominated solution In the case of multiple nondominated solutions the choice of the most appropriate solution needs additional preference information since they are not satisfying properties (a) and (c) of the optima of single-objective optimum problems Different nondominated solutions usually give different objective function values, and in order to compare them we need additional preference information about n element vectors Depending on the different formulations of such preferences different solution concepts and methods are developed A comprehensive summary of the most commonly used methods is given, for example, in Szidarovszky et al (1986) Appendix H Stability and Controllability A time-invariant nonlinear system is given by the difference equation (H.1) with discrete timescales and as (H.2) in continuous timescales where with It is usually assumed that is continuous on D and starting from arbitrary initial value , Eqs ( H.1 ) and ( H.2 ) have unique trajectories in D Vector is called the state of the system at time t The equilibrium or steady state of system ( H.1 ) is an such that , and that of system ( H.2 ) is an will remain such that That is, if the state becomes at any time, then the state for all future times The stability theory of dynamic systems tries to answer the question of what the asymptotic behavior of the state trajectory is if differs from the equilibrium; and under what condition approaches the equilibrium in the long run The equilibrium is called locally asymptotically stable , if there is an such that as if That is, if the initial state is selected sufficiently close to the equilibrium, then the state trajectory converges back to the equilibrium as An equilibrium is called globally asymptotically stable if as regardless of the selection of the initial state We can call a system asymptotically stable if its equilibrium is asymptotically stable One of the most frequently applied methods of checking stability of nonlinear systems is the local linearization , when the right-hand sides of Eqs ( H.1 ) and ( H.2 ) are replaced by their linear Taylor polynomials centered at : (H.3) where denotes the Jacobian matrix of at So the linearized version of Eq ( H.1 ) has the form and by introducing the notation , it can be rewritten as (H.4) The linearized version of Eq ( H.2 ) can be written as that is, (H.5) since is a constant with zero derivative In both cases, the linearized equation becomes a homogeneous linear equation and the asymptotic stability of in the linearized equations is equivalent to the asymptotic stability of the zero equilibrium of equations ( H.4 ) and ( H.5 ) It is easy to prove that in the case of linear systems local asymptotic stability implies global asymptotic stability; however, this is not true for nonlinear systems However, the following fact gives a practical method to check stability of nonlinear systems Assume that the homogeneous system ( H.4 ) (or ( H.5 )) is asymptotically stable, then the nonlinear system ( H.1 ) (or ( H.2 )) is locally asymptotically stable So the asymptotic stability of the linearized system implies the local asymptotic stability of the nonlinear system The reverse of this fact is not true, there are asymptotically stable nonlinear systems with linearized system which are not asymptotically stable Such example in the discrete case is system with the unique equilibrium with , and in the continuous case The asymptotical stability of the linear systems ( H.4 ) and ( H.5 ) can be decided based on the eigenvalues of the Jacobian matrix as follows System ( H.4 ) is asymptotically stable if and only if for all eigenvalues of , and system ( H.5 ) is asymptotically stable if and only if Re , that is, the real parts of all eigenvalues are negative The eigenvalues of an matrix are the roots of the n th degree characteristic polynomial, so there are n eigenvalues where multiple eigenvalues are counted with their multiplicities In the case of low degree polynomials the stability conditions can be verified without solving the polynomial equations If , the characteristic polynomial is quadratic: Then both roots have negative real parts if and only if both coefficients and are positive It can be also proved that both roots are inside the unit circle if and only if (H.6) If , then the characteristic polynomial is cubic: All roots have negative real parts if and only if all coefficients are positive and (H.7) In the discrete case, all roots are inside the unit circle if and only if (H.8) A time invariant discrete linear control system can be written as (H.9) where is , is constant matrix, is n -element, is an m -element vector Here, as before, is the state of the system and is the input, or control of the system at time t In the continuous case the system equation has the form: (H.10) Let be a given future time period, and a selected given state vector We say that system ( H.9 ) or ( H.10 ) is controllable to at time T if there is an input function such that the state of the system becomes at This type of control is called the final state control , since we not care about the system for We say that a system is completely controllable at time T , if it is controllable to any final state at time T The controllability of systems ( H.9 ) and ( H.10 ) can be easily verified by the Kalman controllablility conditions Define the Kalman controllability matrix (H.11) which is an constant matrix Then system ( H.9 ) is completely controllable at any time , if and only if matrix has full rank, that is, System ( H.10 ) is completely controllable at any time if and only if Since matrix has n rows, its rank cannot exceed n , so the maximum possible rank of matrix is n For more details the interested reader may consult any textbook on linear systems, for example, Szidarovszky and Bahill (1992) References Ahmadi, A., & Salazar Moreno, R (2013) Game theory applications in a water distribution problem Journal of Water Resources and 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return Chess-game Chicken game Clear cut Coalitions Column norm Competition of gas stations Complete information Computation of equilibria Concavity Conflict resolution Constant-sum game Continuous games Continuous static game Continuous time scales Contraction mapping Contraction mapping theorem Contraction property Control Control in oligopolies Convex game Convexity Cooperative games Core Core allocation rule Cournot duopoly Cournot oligopoly D Decision alternatives Decision makers Decision space Decisions Dictatorship Direction of improvement Direction-based methods Disagreement payoff vector Disagreement point Discrete static games Discrete time scales Distance-based methods Domestic water users Dominant strategy Dual form Duality in Linear Programming Duel with sound Duel without sound Duopoly Duopoly Stackelberg game Dynamic Bayesian games Dynamic games with sequential moves Dynamic games with simultaneous moves Dynamic oligopoly E Efficient payoff vectors Endpoint -constraint method Equal loss method Equal loss principle Equilibrium(a) Equilibrium problems Essential constant-sum game Essential game Euclidean distance Euclidean norm Existence of equilibria Extensive forms of dynamic games F Feasibility Final state control Finite game with complete and perfect information Finite rooted tree game Finite trees First price auction Fixed point Fixed point problem(s) Fixed points Fixed point theorems Forestry management problem Frobenius norm G Game against nature Game of privilege Games under uncertainty Gauss-Seidel iteration process General existence conditions Geometric distance Globally asymptotically stable Good citizens Gradient adjustments Grand coalition H Hare system I Ideal payoff vector Ideal point Ideally worst point Imperfect information Importance weights Imputation Incomplete information Independence from increasing linear transformations Independence from unfavorable alternatives Individually rational payoff configuration Individual monotonicity Industrial users Inessential game Information lag in oligopoly Information set International fishing Irregular strip cut K Kakutani fixed point theorem Kalai-Smorodinsky solution Kalman matrix Kernel Kuhn–Tucker conditions Kuhn–Tucker regularity condition L Lagrange multipliers Lagrangian Leader-follower game Least-square nucleolus Lexicographic method Lexicographic nucleolus Linear programming problems Local linearization Locally asymptotically stable Location game Losing coalition M Marginal worth of player Marginal worth vector Market sharing Matrix game(s) Matrix with identical elements Maximin construction Maximum distance Maximum norm Mediator Method of fictitious play Method of von Neumann Minkowski distances Mixed equilibrium Mixed extension Mixed extensions of N-person finite games Mixed strategy Mixed strategy equilibrium Monotonic game Monotonicity axiom Monotonic mappings Multiobjective optimization Multiproduct Oligopolies N N-person conflicts N-person continuous games N-person finite games Nash axioms Nash bargaining solution Nash equilibrium(a) Nash-product Negotiation process Nikaido–Isoda theorem Nikaido-Isoda theorem Noncooperative game(s) Non-dominated imputations Nondominated solution Nonlinear optimization Nonsymmetric Nash bargaining solution Norm of matrix Norm of vector Normal form Normalized payoff functions Nucleolus O Objective space Oligopoly Oligopoly in water management Optimization problem Optimum conditions Ordinal preferences Output adjustments toward best responses P Pairwise comparisons Pareto frontier Pareto optimality Pareto optimal solution Payoff function(s) Payoff matrix Payoff space Payoff table Perfect Bayesian equilibrium Perfect information Players Plurality voting Point-to-point mapping Point-to-set mapping Position game Powers of players Preference graph Preimputations Primal form Primal–dual pair Prisoner's dilemma Pure strategy equilibrium Pure strategy(-ies) Q Quadratic programming form Quadratic programming problem R Rational game Rationality Reduction to an optimization problem Relation between matrix games and linear programming Relative excesses Repeated and dynamic games Result of Rosen Root Row norm S Saddle points Salesman's Dilemma Satisfaction function Second price auction Sequential bargaining Set valued mapping Shapley values Sharing a pie Signaling games Simple game Simultaneous payoff Simultaneous strategy Single-person decision problem Skew-symmetric matrix Social choice Solution of bimatrix games Solution of matrix games Solution of oligopolies Special matrix games Speed of adjustment Spying game Stability analysis Stability and Controllability Stable sets Stackelberg equilibrium Stackelberg game Static Bayesian games Static expectations Steady state Strategic equivalence Strategies Strategy set Strictly diagonally concave game Strictly monotonic function Strong duality theorem Strongly nondominated solution Subgame Subgame-perfect equilibria Superadditive game Surplus of player Symmetric matrix game(s) Symmetric players T Terminal node Timing game Two-person conflicts Two-person continuous games Two-person finite games Two-person zero-sum game U Unequal players Uniform thinning Uniqueness of equilibria Utility function V Value of the matrix game Vector and Matrix Norms Veto player von Neumann–Morgenstern stable set Voting game W Wages and employment Waste management Water distribution problem Weak duality property Weakly nondominated solution Weakly Pareto optimal solutions Weakly superadditive game Weierstrass theorem Weighting method Winning coalition Worst possible payoff vector Z (0, 1)-normalized game Zero-sum game ... Noncooperative Games © Springer Japan 2016 Akio Matsumoto and Ferenc Szidarovszky, Game Theory and Its Applications, DOI 10.1007/978-4-431-54786-0_2 Discrete Static Games Akio Matsumoto1 and Ferenc... Springer Japan 2016 Akio Matsumoto and Ferenc Szidarovszky, Game Theory and Its Applications, DOI 10.1007/978-4-431-54786-0_1 Introduction Akio Matsumoto1 and Ferenc Szidarovszky2 (1) Department... the theory of games and their applications, so both researches and application-oriented experts can benefit from it and can use the material of this book in their work The solution concepts and