Two-person Nonzero-sum Games Research Articles

NASH EQUILIBRIUM STRATEGIES REVISITED IN SOFTWARE RELEASE GAMES

A software release game was formulated by Zeephongsekul and Chiera [Zeephongsekul, P. & Chiera, C. (1995). Optimal software release policy based on a two-person game of timing. Journal of Applied Probability 32: 470–481] and was reconsidered by Dohi et al. [Dohi, T., Teraoka, Y., & Osaki, S. (2000). Software release games. Journal of Optimization Theory and Applications 105(2): 325–346] in a framework of two-person nonzero-sum games. In this paper, we further point out the faults in the above literature and revisit the Nash equilibrium strategies in the software release games from the viewpoints of both silent and noisy type of games. It is shown that the Nash equilibrium strategies in the silent and noisy of software release games exist under some parametric conditions.

Nonzero-Sum Stochastic Games with Probability Criteria

In this paper, we consider two-person nonzero-sum discrete-time stochastic games under the probability criterion. Taking $$\lambda $$ for player 1 and $$\mu $$ for player 2 as their profit goal, the two players are concerned with the probabilities that the rewards they earn before the first passage to some target state set are more than $$\lambda $$ and $$\mu $$, respectively. We firstly give a characterization of the probabilities, and then, under a mild condition, we show that the optimal value function for each player is the unique solution to the corresponding optimality equation by an iterative approximation, and then establish the existence of Nash equilibria. Finally, a queueing system is provided to show the application of our main result.

GRAPHICAL METHOD FOR INTERVAL BIMATRIX GAMES

This paper deals with two-person non-zero sum games with interval payoffs. Graphical method to find a mixed strategy equilibrium is adapted to interval bimatrix games. In addition, interval bimatrix games Nash equilibrium is attained by graphical method. Nu- merical examples are also illustrated.

An Analytıc Method for Interval Bimatrix Games

This paper deals with ݔ two-person non-zero sum games with interval pay-offs. An analytic method for solving such games is given. A pair of Nash Equilibrium is found by using the method. The analytic method is effective to find at least one Nash Equilibrium (N.E) for two-person bimatrix games. Therefore, the analytic method for two-person bimatrix games is adapted to interval bimatrix games.

Stochastic Games for Continuous-Time Jump Processes Under Finite-Horizon Payoff Criterion

In this paper we study two-person nonzero-sum games for continuous-time jump processes with the randomized history-dependent strategies under the finite-horizon payoff criterion. The state space is countable, and the transition rates and payoff functions are allowed to be unbounded from above and from below. Under the suitable conditions, we introduce a new topology for the set of all randomized Markov multi-strategies and establish its compactness and metrizability. Then by constructing the approximating sequences of the transition rates and payoff functions, we show that the optimal value function for each player is a unique solution to the corresponding optimality equation and obtain the existence of a randomized Markov Nash equilibrium. Furthermore, we illustrate the applications of our main results with a controlled birth and death system.

Nonzero-sum games for continuous-time Markov chains with unbounded transition and average payoff rates

This paper attempts to study two-person nonzero-sum games for denumerable continuous-time Markov chains determined by transition rates, with an expected average criterion. The transition rates are allowed to be unbounded, and the payoff functions may be unbounded from above and from below. We give suitable conditions under which the existence of a Nash equilibrium is ensured. More precisely, using the socalled “vanishing discount” approach, a Nash equilibrium for the average criterion is obtained as a limit point of a sequence of equilibrium strategies for the discounted criterion as the discount factors tend to zero. Our results are illustrated with a birth-and-death game.

SIMPLE EQUILIBRIA IN SEMI-INFINITE GAMES

In the paper two-person nonzero-sum semi-infinite games with bounded payoffs are studied — both with countable and uncountable infinite strategy space. Under some concavity/convexity assumptions they are shown to possess ε-equilibria (equilibria) in strategies with supports consisting of at most two points of the players' pure strategy spaces. Further the games without any concavity/convexity properties are studied. It is proved that they possess ε-equilibria (equilibria) in strategies having certain finite supports.

Extra-proximal methods for solving two-person nonzero-sum games

We consider two-person nonzero-sum game, both in the classical form and in the form of a game with coupled variables. An extra-proximal approach for finding the game’s solutions is suggested and justified. We provide our algorithm with an analysis of its convergence.

Nonzero-sum games for continuous-time Markov chains with unbounded discounted payoffs

In this paper, we study two-person nonzero-sum games for continuous-time Markov chains with discounted payoff criteria and Borel action spaces. The transition rates are possibly unbounded, and the payoff functions might have neither upper nor lower bounds. We give conditions that ensure the existence of Nash equilibria in stationary strategies. For the zero-sum case, we prove the existence of the value of the game, and also provide arecursiveway to compute it, or at least to approximate it. Our results are applied to a controlled queueing system. We also show that if the transition rates areuniformly bounded, then a continuous-time game is equivalent, in a suitable sense, to a discrete-time Markov game.

Nonzero-sum games for continuous-time Markov chains with unbounded discounted payoffs

In this paper, we study two-person nonzero-sum games for continuous-time Markov chains with discounted payoff criteria and Borel action spaces. The transition rates are possibly unbounded, and the payoff functions might have neither upper nor lower bounds. We give conditions that ensure the existence of Nash equilibria in stationary strategies. For the zero-sum case, we prove the existence of the value of the game, and also provide arecursiveway to compute it, or at least to approximate it. Our results are applied to a controlled queueing system. We also show that if the transition rates areuniformly bounded, then a continuous-time game is equivalent, in a suitable sense, to a discrete-time Markov game.

Generalized quantization scheme for two-person non-zero sum games

We have proposed a generalized quantization scheme for non-zero sum games which can be reduced to two existing quantization schemes under appropriate set of parameters. Some other important situations are identified which are not apparent in the exiting two quantizations schemes.

Internet pricing with a game theoretical approach: concepts and examples

The basic concepts of three branches of game theory, leader-follower, cooperative, and two-person nonzero sum games, are reviewed and applied to the study of the Internet pricing issue. In particular, we emphasize that the cooperative game (also called the bargaining problem) provides an overall picture for the issue. With a simple model for Internet quality of service (QoS), we demonstrate that the leader-follower game may lead to a solution that is not Pareto optimal and in some cases may be "unfair," and that the cooperative game may provide a better solution for both the Internet service provider (ISP) and the user. The practical implication of the results is that government regulation or arbitration may be helpful. The QoS model is also applied to study the competition between two ISPs, and we find a Nash equilibrium point from which the two ISPs would not move out without cooperation. The proposed approaches can be applied to other Internet pricing problems such as the Paris Metro pricing scheme.

An "Informationally Robust Equilibrium" for Two-Person Nonzero-Sum Games

This paper considers the implications of explicit slight uncertainty about the strategic sequence of moves, that is, about the informational structure, for a two-person nonzero-sum game. It is shown that the set of associated informationally robust equilibria must be nonempty and must be a subset of the set of Nash equilibria. Although the present approach is conceptually reminiscent of that underlying perfectness, the two concepts are shown to be logically unrelated. There may exist no equilibrium which is both perfect and informationally robust. Journal of Economic Literature Classification Number: C72.

Games, Theory and Applications.

Anyone with a knowledge of basic mathematics will find this an accessible and informative introduction to game theory. It opens with the theory of two-person zero-sum games, two-person non-zero sum games, and n-person games, at a level between nonmathematical introductory books and technical mathematical game theory books. Succeeding sections focus on a variety of applications-including introductory explanations of gaming and meta games-that offer nonspecialists information about new areas of game theory at a comprehensible level. Numerous exercises appear with full solutions, in addition to an extensive bibliography, 80 problems with worked solutions, and over 30 illustrations useful for the theory of non-zero and n-person games.

Constrained games as complementary eigenvalue problems

A class of two-person nonzero sum games where the strategy choices are constrained in some form for each player is analyzed here to show the equivalent nonlinear programs which must be solved for the Cournot-Nash equilibrium. This equilibrium solution is shown in appropriate cases to lead to complementary eigenvalue problems, which have applications in normal solutions of stochastic LP models and optimal design problems in linear regression theory.

Stackelberg strategy for linear quadratic partial differential games

Characterizations of open-loop Stackelberg strategy for two-person nonzero-sum dynamic games described by linear operators (including unbounded ones) in Hilbert spaces are presented. The method of Lions is used and it guarantees the existence, uniqueness, as well as necessary and sufficient conditions for the open-loop Stackelberg strategy of such games. Special cases considered include games governed by ordinary differential equations, linear elliptic and parabolic partial differential equations with quadratic cost functions.

Informational properties of the Nash solutions of two stochastic nonzero-sum games

Within the framework of stochastic two-person nonzero-sum games, we deal with two commonly used models in engineering and economics—namely, the LQG (Linear-Quadratic-Gaussian) and the duopoly problems. We investigate how variations in information available to either player affect the equilibrium Nash strategies for these two models, whose existence and uniqueness have been proven in the paper. We show that for the LQG model better information for either player results in lower average Nash costs for both players; whereas for the duopoly model better information for one player helps him alone to achieve a higher average Nash profit, and it hurts the other player in the sense that his average Nash profit decreases. We further relate these properties of the Nash solutions for these two games to some of the distinct features of zero-sum games and team problems.

The existence and the shape of the equilibrium strategies for some two-person non-zero sum games with the choice of the moment

The abstracts (in two languages) can be found in the pdf file of the article.
 Original author name(s) and title in Russian and Lithuanian:
 Д. П. Суджюте. Существование и вид равновесных стратегий некоторых не антагонистических игр двух лиц с выбором момента времени
 D. Sūdžiūtė. Kai kurių neantagonistinių dviejų asmenų lošimų su laiko momento parinkimu pusiausvyros strategijų egzistavimas ir pavidalas

Two-person nonzero-sum games and quadratic programming

It is shown that a necessary and sufficient condition that a point be a Nash equilibrium point of a two-person, nonzero-sum game with a finite number of pure strategies is that the point be a solution of a single programming problem with linear constraints and a quadratic objective function that has a global maximum of zero. Every equilibrium point is a solution of this programming problem. For the case of a zero-sum game, the quadratic programming problem degenerates to the well-known dual linear programs associated with the game.

Equilibrium Points in Bimatrix Games

Previous article Next article Equilibrium Points in Bimatrix GamesN. N. Vorob’evN. N. Vorob’evhttps://doi.org/10.1137/1103024PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractAn algorithm for computing all equilibrium points (situations) for the case of bimatrix (i.e., finite two-person, non-cooperative, non-zero-sum) games is given.[1] John Nash, Non-cooperative games, Ann. of Math. (2), 54 (1951), 286–295 MR0043432 0045.08202 CrossrefGoogle Scholar[2] L. S. Shapley and , R. N. Snow, Basic solutions of discrete gamesContributions to the Theory of Games, Annals of Mathematics Studies, no. 24, Princeton University Press, Princeton, N. J., 1950, 27–35 MR0039216 0041.25403 Google Scholar[3] David Blackwell and , M. A. Girshick, Theory of games and statistical decisions, John Wiley and Sons, Inc., New York, 1954xi+355 MR0070134 0056.36303 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Where Do Mistakes Lead? A Survey of Games with Incompetent Players10 February 2022 | Dynamic Games and Applications, Vol. 97 Cross Ref , Vol. 689 Cross Ref Sets of Nash Equilibria in Polymatrix Mixed-Strategy Games10 March 2018 Cross Ref Approximation of Isomorphic Infinite Two-Person Non-Cooperative Games by Variously Sampling the Players’ Payoff Functions and Reshaping Payoff Matrices into Bimatrix Game23 January 2017 | Applied Computer Systems, Vol. 20, No. 1 Cross Ref Ordered Field Property for Semi-Markov Games when One Player Controls Transition Probabilities and Transition TimesInternational Game Theory Review, Vol. 17, No. 02 Cross Ref ORDERED FIELD PROPERTY IN A SUBCLASS OF FINITE SER-SIT SEMI-MARKOV GAMESInternational Game Theory Review, Vol. 15, No. 04 Cross Ref Incompetence and impact of training in bimatrix gamesAutomatica, Vol. 48, No. 10 Cross Ref Routing of Hazardous Material Shipments Under the Threat of Terrorist Attack12 January 2010 Cross Ref Enumeration of Nash equilibria for two-player games6 March 2009 | Economic Theory, Vol. 42, No. 1 Cross Ref A taxonomy of best-reply multifunctions in 2×2×2 trimatrix games21 September 2007 | TOP, Vol. 15, No. 2 Cross Ref A Taxonomy of Best-Reply Multifunctions in 2x2x2 Trimatrix GamesSSRN Electronic Journal Cross Ref Enumeration of All the Extreme Equilibria in Game Theory: Bimatrix and Polymatrix Games29 November 2006 | Journal of Optimization Theory and Applications, Vol. 129, No. 3 Cross Ref Hard-to-Solve Bimatrix GamesEconometrica, Vol. 74, No. 2 Cross Ref Solving three-player games by the matrix approach with application to an electric power marketIEEE Transactions on Power Systems, Vol. 18, No. 4 Cross Ref Tuning of discretization in bimatrix game approach to power system market analysisIEEE Transactions on Power Systems, Vol. 18, No. 2 Cross Ref Chapter 44 Non-zero-sum two-person games Cross Ref Chapter 45 Computing equilibria for two-person games Cross Ref Game Theory 1950–2000 Cross Ref Cooperation and self-interest: Pareto-inefficiency of Nash equilibria in finite random games18 August 1998 | Proceedings of the National Academy of Sciences, Vol. 95, No. 17 Cross Ref Pareto equilibria for bimatrix gamesComputers & Mathematics with Applications, Vol. 25, No. 10-11 Cross Ref On the structure of the set of perfect equilibria in bimatrix games1 March 1993 | Operations-Research-Spektrum, Vol. 15, No. 1 Cross Ref Un nuevo algoritmo para la resolucion de juegos bimatricialesTrabajos de Investigacion Operativa, Vol. 7, No. 1 Cross Ref Constructing bimatrix games with unique equilibrium pointsMathematical Social Sciences, Vol. 15, No. 1 Cross Ref B Cross Ref Maximal nash subsets for bimatrix gamesNaval Research Logistics Quarterly, Vol. 28, No. 1 Cross Ref Polymatrix Games with Joint ConstraintsB. Curtis Eaves17 February 2012 | SIAM Journal on Applied Mathematics, Vol. 24, No. 3AbstractPDF (489 KB)On the Set of Equilibria of a Bimatrix Game: a Survey Cross Ref Exponentially Many Steps for Finding a Nash Equilibrium in a Bimatrix Game Cross Ref Equilibrium Points of Bimatrix GamesO. L. Mangasarian13 July 2006 | Journal of the Society for Industrial and Applied Mathematics, Vol. 12, No. 4AbstractPDF (301 KB)Two-person nonzero-sum games and quadratic programmingJournal of Mathematical Analysis and Applications, Vol. 9, No. 3 Cross Ref Volume 3, Issue 3| 1958Theory of Probability & Its Applications History Submitted:12 March 1958Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1103024Article page range:pp. 297-309ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics