Completely Mixed Strategies for Generalized Bimatrix and Switching Controller Stochastic Game
TL;DR: This paper revisits a result by Jurg et al. (Linear Algebra Appl 141:61–74, 1990) where the necessary and sufficient condition for a bimatrix game to be weakly completely mixed and presents an alternate proof of this result using linear complementarity approach.
Abstract: In this paper, we revisit a result by Jurg et al. (Linear Algebra Appl 141:61–74, 1990) where the necessary and sufficient condition for a bimatrix game to be weakly completely mixed is given. We present an alternate proof of this result using linear complementarity approach. We extend this result to a generalization of bimatrix game introduced by Gowda and Sznajder (Int J Game Theory 25:1–12, 1996) via a generalization of linear complementarity problem introduced by Cottle and Dantzig (J Comb Theory 8:79–90, 1970). We further study completely mixed switching controller stochastic game (in which transition structure is a natural generalization of the single controller games) and extend the results obtained by Filar (Proc Am Math Soc 95:585–594, 1985) for completely mixed single controller stochastic game to completely mixed switching controller stochastic game. A numerical method is proposed to compute a completely mixed strategy for a switching controller stochastic game.
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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.
4 citations
Proceedings Article•
01 Dec 1985TL;DR: In this article, one-step algorithms are presented for two classes of structured stochastic games, namely, those with additive rewards and transitions and those which have switching controllers, which can be derived from optimal solutions to appropriate bilinear programs.
Abstract: One-step algorithms are presented for two classes of structured stochastic games, namely, those with additive rewards and transitions and those which have switching controllers. Solutions to such classes of games under the average reward criterion can be derived from optimal solutions to appropriate bilinear programs. The validity of using bilinear programming as a solution method follows from two preliminary theorems, the first of which is a complete classification of undiscounted stochastic games with optimal stationary strategies. The second of these preliminary theorems relaxes the conditions of the classification theorem for certain classes of stochastic games and provides the basis for the bilinear programming results. Analogous results hold for the discounted stochastic games with the above special structures.
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18 Feb 1992
TL;DR: In this article, the authors present an overview of existing and multiplicity of degree theory and propose pivoting methods and iterative methods for degree analysis, including sensitivity and stability analysis.
Abstract: Introduction. Background. Existence and Multiplicity. Pivoting Methods. Iterative Methods. Geometry and Degree Theory. Sensitivity and Stability Analysis. Chapter Notes and References. Bibliography. Index.
2,897 citations
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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
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
TL;DR: An algebraic proof of the existence of equilibrium points for two-person non-zero-sum games is given in this paper, leading to an efficient scheme for computing an equilibrium point, which is valid for any ordered field.
Abstract: An algebraic proof is given of the existence of equilibrium points for bimatrix (or two-person, non-zero-sum) games. The proof is constructive, leading to an efficient scheme for computing an equilibrium point. In a nondegenerate case, the number of equilibrium points is finite and odd. The proof is valid for any ordered field.
1,087 citations
"Completely Mixed Strategies for Gen..." refers background in this paper
...Lemke and Howson [11] gave an efficient and constructive procedure for obtaining an equilibrium pair by solving LCP(q, M) where M = [ 0 A BT 0 ] and q = [−em −en ] ....
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...Lemke and Howson [11] gave an efficient and constructive procedure for obtaining an equilibrium pair by solving LCP(q, M) where M = [ 0 A BT 0 ] and q = −em −en ]...
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