Max Plus Algebra, Optimization and Game Theory
09 Jan 2018-Vol. 302, pp 345-361
TL;DR: This paper discusses various optimization problem using methods based on max-plus algebra, which has maximization and addition as its basic arithmetic operations, and presents some sub-classes of mathematical optimization problems like linear programming, convex quadratic programming problem, fractional programming problem
Abstract: Max-plus algebra has been applied to several fields like matrix algebra, cryptography, transportation, manufacturing, information technology and study of discrete event systems like subway traffic networks, parallel processing systems, telecommunication networks for many years. In this paper, we discuss various optimization problem using methods based on max-plus algebra, which has maximization and addition as its basic arithmetic operations. We present some sub-classes of mathematical optimization problems like linear programming, convex quadratic programming problem, fractional programming problem, bimatrix game problem and some classes of stochastic game problem in max algebraic framework and discuss various connections between max-plus algebra and optimization.
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TL;DR: The algorithm provided provides a constructive proof of the existence of the value and of optimal stationary strategies for both players and the finiteness of the algorithm proves also the ordered field property of the switching control stochastic game.
Abstract: In this paper two-person zero-sum stochastic games are considered with the average payoff as criterion. It is assumed that in each state one of the players governs the transitions. We will establish an algorithm, which yields in a finite number of iterations the solution of the game i.e. the value of the game and optimal stationary strategies for both players. An essential part of our algorithm is formed by the linear programming problem which solves a one player control stochastic game. Furthermore, our algorithm provides a constructive proof of the existence of the value and of optimal stationary strategies for both players. In addition, the finiteness of our algorithm proves also the ordered field property of the switching control stochastic game.
Wir betrachten stochastische Zweipersonen-Nullsummenspiele mit der durchschnittlichen Auszahlung als Kriterium. Wir nehmen an, daβ in jedem Zustand einer der Spieler das Ubergangsgesetz kontrolliert und entwickeln einen Algorithmus, der nach endlichen vielen Iterationsschritten die Losung des Spiels -- d. h. den Spielwert und optimale stationare Strategien fur beide Spieler -- liefert. Ein wesentlicher Teil unseres Algorithmus besteht aus dem linearen Programm, das ein stochastisches Spiel lost, bei dem ein Spieler das Ubergangsgesetz bestimmt. Daruber hinaus geben wir mit unserem Algorithmus einen konstruktiven Beweis der Existenz des Spielwertes und optimaler stationarer Strategien fur beide Spieler. Weiter zeigt die Endlichkeit unseres Algorithmus die "ordered field property" stochastischer Spiele mit wechselnder Kontrolle des Ubergangsgesetzes.
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TL;DR: In a stochastic game the play proceeds by steps from position to position, according to transition probabilities controlled jointly by the two players, and the expected total gain or loss is bounded by M, which depends on N 2 + N matrices.
Abstract: In a stochastic game the play proceeds by steps from position to position, according to transition probabilities controlled jointly by the two players. We shall assume a finite number, N , of positions, and finite numbers m K , n K of choices at each position; nevertheless, the game may not be bounded in length. If, when at position k , the players choose their i th and j th alternatives, respectively, then with probability s i j k > 0 the game stops, while with probability p i j k l the game moves to position l . Define s = min k , i , j s i j k . Since s is positive, the game ends with probability 1 after a finite number of steps, because, for any number t , the probability that it has not stopped after t steps is not more than (1 − s ) t .
Payments accumulate throughout the course of play: the first player takes a i j k from the second whenever the pair i , j is chosen at position k. If we define the bound M: M = max k , i , j | a i j k | , then we see that the expected total gain or loss is bounded by M + ( 1 − s ) M + ( 1 − s ) 2 M + … = M / s . (1)
The process therefore depends on N 2 + N matrices P K l = ( p i j k l | i = 1 , 2 , … , m K ; j = 1 , 2 , … , n K ) A K = ( a i j k | …
2,622 citations
TL;DR: This book proposes a unified mathematical treatment of a class of 'linear' discrete event systems, which contains important subclasses of Petri nets and queuing networks with synchronization constraints, which is shown to parallel the classical linear system theory in several ways.
Abstract: This book proposes a unified mathematical treatment of a class of 'linear' discrete event systems, which contains important subclasses of Petri nets and queuing networks with synchronization constraints. The linearity has to be understood with respect to nonstandard algebraic structures, e.g. the 'max-plus algebra'. A calculus is developed based on such structures, which is followed by tools for computing the time behaviour to such systems. This algebraic vision lays the foundation of a bona fide 'discrete event system theory', which is shown to parallel the classical linear system theory in several ways.
1,424 citations
01 Dec 1996
TL;DR: In this article, the authors present a series of courses and prerequisites for the development of stochastic games with a focus on reducing the complexity of the problem of finding the optimal solution.
Abstract: 1 Introduction.- 1.0 Background.- 1.1 Raison d'Etre and Limitations.- 1.2 A Menu of Courses and Prerequisites.- 1.3 For the Cognoscenti.- 1.4 Style and Nomenclature.- I Mathematical Programming Perspective.- 2 Markov Decision Processes: The Noncompetitive Case.- 2.0 Introduction.- 2.1 The Summable Markov Decision Processes.- 2.2 The Finite Horizon Markov Decision Process.- 2.3 Linear Programming and the Summable Markov Decision Models.- 2.4 The Irreducible Limiting Average Process.- 2.5 Application: The Hamiltonian Cycle Problem.- 2.6 Behavior and Markov Strategies.- 2.7 Policy Improvement and Newton's Method in Summable MDPs.- 2.8 Connection Between the Discounted and the Limiting Average Models.- 2.9 Linear Programming and the Multichain Limiting Average Process.- 2.10 Bibliographic Notes.- 2.11 Problems.- 3 Stochastic Games via Mathematical Programming.- 3.0 Introduction.- 3.1 The Discounted Stochastic Games.- 3.2 Linear Programming and the Discounted Stochastic Games.- 3.3 Modified Newton's Method and the Discounted Stochastic Games.- 3.4 Limiting Average Stochastic Games: The Issues.- 3.5 Zero-Sum Single-Controller Limiting Average Game.- 3.6 Application: The Travelling Inspector Model.- 3.7 Nonlinear Programming and Zero-Sum Stochastic Games.- 3.8 Nonlinear Programming and General-Sum Stochastic Games.- 3.9 Shapley's Theorem via Mathematical Programming.- 3.10 Bibliographic Notes.- 3.11 Problems.- II Existence, Structure and Applications.- 4 Summable Stochastic Games.- 4.0 Introduction.- 4.1 The Stochastic Game Model.- 4.2 Transient Stochastic Games.- 4.2.1 Stationary Strategies.- 4.2.2 Extension to Nonstationary Strategies.- 4.3 Discounted Stochastic Games.- 4.3.1 Introduction.- 4.3.2 Solutions of Discounted Stochastic Games.- 4.3.3 Structural Properties.- 4.3.4 The Limit Discount Equation.- 4.4 Positive Stochastic Games.- 4.5 Total Reward Stochastic Games.- 4.6 Nonzero-Sum Discounted Stochastic Games.- 4.6.1 Existence of Equilibrium Points.- 4.6.2 A Nonlinear Compementarity Problem.- 4.6.3 Perfect Equilibrium Points.- 4.7 Bibliographic Notes.- 4.8 Problems.- 5 Average Reward Stochastic Games.- 5.0 Introduction.- 5.1 Irreducible Stochastic Games.- 5.2 Existence of the Value.- 5.3 Stationary Strategies.- 5.4 Equilibrium Points.- 5.5 Bibliographic Notes.- 5.6 Problems.- 6 Applications and Special Classes of Stochastic Games.- 6.0 Introduction.- 6.1 Economic Competition and Stochastic Games.- 6.2 Inspection Problems and Single-Control Games.- 6.3 The Presidency Game and Switching-Control Games.- 6.4 Fishery Games and AR-AT Games.- 6.5 Applications of SER-SIT Games.- 6.6 Advertisement Models and Myopic Strategies.- 6.7 Spend and Save Games and the Weighted Reward Criterion.- 6.8 Bibliographic Notes.- 6.9 Problems.- Appendix G Matrix and Bimatrix Games and Mathematical Programming.- G.1 Introduction.- G.2 Matrix Game.- G.3 Linear Programming.- G.4 Bimatrix Games.- G.5 Mangasarian-Stone Algorithm for Bimatrix Games.- G.6 Bibliographic Notes.- Appendix H A Theorem of Hardy and Littlewood.- H.1 Introduction.- H.2 Preliminaries, Results and Examples.- H.3 Proof of the Hardy-Littlewood Theorem.- Appendix M Markov Chains.- M.1 Introduction.- M.2 Stochastic Matrix.- M.3 Invariant Distribution.- M.4 Limit Discounting.- M.5 The Fundamental Matrix.- M.6 Bibliographic Notes.- Appendix P Complex Varieties and the Limit Discount Equation.- P.1 Background.- P.2 Limit Discount Equation as a Set of Simultaneous Polynomials.- P.3 Algebraic and Analytic Varieties.- P.4 Solution of the Limit Discount Equation via Analytic Varieties.- References.
1,191 citations
Book•
08 Nov 2010
TL;DR: Max-algebra: Two Special Features.- One-sided Max-linear Systems and Max- algebraic Subspaces.
Abstract: Max-algebra: Two Special Features.- One-sided Max-linear Systems and Max-algebraic Subspaces.- Eigenvalues and Eigenvectors.- Maxpolynomials. The Characteristic Maxpolynomial.- Linear Independence and Rank. The Simple Image Set.- Two-sided Max-linear Systems.- Reachability of Eigenspaces.- Generalized Eigenproblem.- Max-linear Programs.- Conclusions and Open Problems.
488 citations
Book•
27 Nov 2005
389 citations