Two-person Nonzero Sum Research Articles

A development in the existing nonlinear model and to study its impact on the Nash equilibrium in a two-person non-zero sum game

A non-zero sum two-person game can be formulated as a nonlinear program problem for finding the Nash equilibrium. Although intensive research has been done in developing methods for solving such nonlinear problems, very little work has been done in modifying the existing model in order to generate a more refined Nash solution. The study focuses on modifying the nonlinear programming model by incorporating some additional constraints and analysing if the altered model would generate a better result. The utility of the additional constraints is discussed, in terms of the payoff received by either of the players, for different problem situations.

A development in the existing nonlinear model and to study its impact on the Nash equilibrium in a two-person non-zero sum game

A non-zero sum two-person game can be formulated as a nonlinear program problem for finding the Nash equilibrium. Although intensive research has been done in developing methods for solving such nonlinear problems, very little work has been done in modifying the existing model in order to generate a more refined Nash solution. The study focuses on modifying the nonlinear programming model by incorporating some additional constraints and analysing if the altered model would generate a better result. The utility of the additional constraints is discussed, in terms of the payoff received by either of the players, for different problem situations.

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.

The audience effect and the role of deception in the expression of male mating preferences

Males of several species have been shown to alter their mate preference in the presence of an eavesdropping rival. This evasive tactic has been interpreted as an attempt by the courting male to drive the attention of the rival away from the preferred female. The fitness return of this deceptive strategy will depend on the costs of cheating for the actor (the displayer) and the benefits for the rival (the bystander) of copying the choice of the displayer. We developed a two-person nonzero sum game between two males that compete for mating with one of two receptive females. Males could assess female quality with a varying level of uncertainty, which was modelled using a Bayesian statistical decision theory approach. We explored the actor and bystander payoffs under different levels of uncertainty in mate assessment and difference in quality between females. We found that when being eavesdropped on is costly (i.e. when females differ largely in quality), males are expected to cheat to reduce the amount of public information that is available to the unintended audience. However, under these circumstances, the value of the public information is low and the bystander is not expected to copy the choice of the actor. Our model suggests that deceptive male choice may evolve only under relatively restricted conditions and suggest that other explanations, such as, for example, a reduction in the risk of precopulatory male–male competition may be more likely. Future theoretical and empirical work will be necessary to test alternative interpretations of the audience effects in male mate choice.

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.

Distribution Design of Two Rival Decenteralized Supply Chains: a Two-person Nonzero Sum Game Theory Approach

We consider competition between two decentralized supply chains network under demand uncertainty. Each chain consists of one risk-averse manufacturer and a group of risk-averse retailers. These two chains present substitutable products to the geographical dispensed markets. The markets’ demands are contingent upon prices, service levels, and advertising efforts of two supply chains. We formulate the distribution design problem of two rival chains as a two-person nonzero sum game. Since strategic decisions of distribution designs often have priority over tactical ones, we first calculate the equilibrium of tactical decisions for each pair of distribution design scenarios. Then, according to the presented methodology, we find Nash equilibrium solutions of distribution network scenarios for two rival chains. Eventually, the research has concluded with a numerical example in order to illustrate the real applications of the methodology.

A Numerical Construction Algorithm of Nash and Stackelberg Solutions for Two-person Non-zero Sum Linear Positional Differential Games

The report evolves a method, which uses the formalization and results of positional antagonistic differential games theory, developed by N. N. Krasovskii and his scientific school, for constructing solutions of a class of non-antagonistic differential games. The method transforms non-antagonistic game into so-called non-standard optimal control problem. Numerical solutions for Stackelberg games are constructed by an algorithm developed by S. Osipov. Numerical Nash solution construction algorithm based upon auxiliary bimatrix games sequence is presented. Used computational geometry algorithms include convex hull construction, union and intersection of polygons and a Minkowski sum for polygons. Results of numerical experiment for a material point motion in plane are presented. The point is moved by force formed by two players. Every player has his personal target point. Among the obtained results, there is a Nash solution such that along the corresponding trajectory the position of the game is non-antagonistic, at first, and then becomes globally antagonistic starting from some moment of time.

Nondominated Equilibrium Solutions of a Multiobjective Two-Person Nonzero-Sum Game and Corresponding Mathematical Programming Problem

With reference to a multiobjective two-person nonzero-sum game, we define nondominated equilibrium solutions and provide a necessary and sufficient condition for a pair of mixed strategies to be a nondominated equilibrium solution. Using the necessary and sufficient condition, we formulate a mathematical programming problem yielding nondominated equilibrium solutions. We give a numerical example and demonstrate that nondominated equilibrium solutions can be obtained by solving the formulated mathematical programming problem.

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.

Solving Two-Person Nonzero-Sum Games with the Help of Differential Equations

Solving Two-Person Nonzero-Sum Games with the Help of Differential Equations

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.

Markov stopping games with random priority

In the paper a construction of Nash equilibria for a random priority finite horizon two-person non-zero sum game with stopping of Markov process is given. The method is used to solve the two-person non-zero-sum game version of the secretary problem. Each player can choose only one applicant. If both players would like to select the same one, then the lottery chooses the player. The aim of the players is to choose the best candidate. An analysis of the solutions for different lotteries is given. Some lotteries admit equilibria with equal Nash values for the players.

An algorithm for the mixed-integer nonlinear bilevel programming problem

The bilevel programming problem (BLPP) is a two-person nonzero sum game in which play is sequential and cooperation is not permitted. In this paper, we examine a class of BLPPs where the leader controls a set of continuous and discrete variables and tries to minimize a convex nonlinear objective function. The follower's objective function is a convex quadratic in a continuous decision space. All constraints are assumed to be linear. A branch and bound algorithm is developed that finds global optima. The main purpose of this paper is to identify efficient branching rules, and to determine the computational burden of the numeric procedures. Extensive test results are reported. We close by showing that it is not readily possible to extend the algorithm to the more general case involving integer follower variables.

A Game of Timing which Shifts Stochastically from a Noisy Version, to a Silent Version

Abstract We consider a two-person non-zero sum game of timing in which the state of bullets possessed by both players shifts stochastically from noisy to silent at random time T in [0, 1] with cdf K(t). The problem is a continuation of the previous paper where the state of the bullets shifted stochastically from silent to noisy. It is shown that the change of information structure available to both the players yields interesting equilibrium strategies. Furthermore, we give remarks on a case where the switching time from a noisy to silent state is not a random variable but a deterministic time in [0, 1], though we cannot derive an equilibrium point except in special cases.

A Game of Timing in whichthePlayers Improve their Information Patterns

Abstract We consider a two-person nonzero sum game of timing in which the stales of bullets possessed by both players shift stochastically from silent to noisy at random time T in [0, 1] with cdf H(t). The model is evidently an exlension of the classical games of timing to a nonzero sum version under an uncertain information structure. It is shown that the shift of information structure available to the both players yields interesting equilibrium strategies.

Excess energy marketing problem

The excess energy marketing problem (EEMP) is the successful selling of Pacific Northwest (PNW) surplus electrical energy, as constrained by the 1964 Regional Preference Act (PL 88-552) and the available capacity on the Pacific Northwest/Southwest Inertie Network. A favored strategy would simultaneously maximize PNW revenues and Pacific Southwest (PSW) savings from dual-fuel displacement while cultivating other markets. If the price for excess energy (firm/nonfirm) were allowed to float to market conditions, the present (1983) Intertie capacity were increased and new DC interties were planned, and conservation were used to level the magnitude and extend the duration of the present surplus energy, then one solution and the recommended marketing policy is an innovative Stackelberg game. Initially, this is a two-person non-zero sum bimatrix game with the leader Bonneville Power Administration (BPA) and the follower California utilities (CU). The strategic form of the game is a Prisoner's Dilemma in which BPA pursues an iterative tit-for-tat scheme following determination of excess energy magnitude, duration, and equilibrium price. Two lower bounds for revenue estimates from sans-strategic policies are theorized by Fourier series and vector autoregressive models. An upper bound for revenues, which might be obtained by pursuing this Stackelberg game, is theorized by assuming BPA adopts the game as policy and CU cooperates at each stage over a game horizon determined by accurate supply/demand forecasts. This dynamic stochastic Stackelberg game is an EEMP solution that will benefit both players, provide Regional economic stability, and spur National energy independence.

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.

A game theoretic model for two types of customers competing for service

Two types of customers arrive at a single server station and demand service. If a customer finds the server busy upon arrival (or retrial) he immediately departs and conducts a retrial after an exponential period of time and persists this way until he gets served. Both types of customers face linear costs for waiting and conducting retrials and wish to find optimal retrial rates which will minimize these costs. This problem is analysed as a two-person nonzero sum game. Both noncooperative strategies are studied.

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.