Existence Of Nash Equilibrium Point Research Articles

Non-Zero Sum Nash Game for Discrete-Time Infinite Markov Jump Stochastic Systems with Applications

This paper is to study finite horizon linear quadratic (LQ) non-zero sum Nash game for discrete-time infinite Markov jump stochastic systems (IMJSSs). Based on the theory of stochastic analysis, a countably infinite set of coupled generalized algebraic Riccati equations are solved and a necessary and sufficient condition for the existence of Nash equilibrium points is obtained. From a new perspective, the finite horizon mixed robust H2/H∞ control is investigated, and summarize the relationship between Nash game and H2/H∞ control problem. Moreover, the feasibility and validity of the proposed method has been proved by applying it to a numerical example.

Nash Equilibrium Points for Generalized Matrix Game Model with Interval Payoffs

Game theory-based models are widely used to solve multiple competitive problems such as oligopolistic competitions, marketing of new products, promotion of existing products competitions, and election presage. The payoffs of these competitive models have been conventionally considered as deterministic. However, these payoffs have ambiguity due to the uncertainty in the data sets. Interval analysis-based approaches are found to be efficient to tackle such uncertainty in data sets. In these approaches, the payoffs of the game model lie in some closed interval, which are estimated by previous information. The present paper considers a multiple player game model in which payoffs are uncertain and varies in a closed intervals. The necessary and sufficient conditions are explained to discuss the existence of Nash equilibrium point of such game models. Moreover, Nash equilibrium point of the model is obtained by solving a crisp bi-linear optimization problem. The developed methodology is further applied for obtaining the possible optimal strategy to win the parliament election presage problem.

Differentiated Cournot duopoly game with fractional-order and its discretization

PurposeFractional calculus provides powerful tool to build more realistic and accurate mathematical models in economic field. This paper aims to explore a proposed fractional-order differentiated Cournot duopoly game and its discretized game.Design/methodology/approachConditions for existence and uniqueness of the proposed game’s solution are derived. The existence of Nash equilibrium point and its local and global stability are obtained. Furthermore, local stability analysis of the discretized game is investigated. The effects of fractional-order on game’s dynamics are examined, along with other parameters of the game, via the 2D bifurcation diagrams in planes of system’s parameters are acquired.FindingsTheoretical and numerical simulation results demonstrate rich variety of interesting dynamical behaviors such as period-doubling and Neimark–Sacker bifurcations, attractors’ crises in addition to chaotic attractors. The results demonstrated that the stability Nash equilibrium point of the game can be lost by period doubling or Neimark–Sacker bifurcations.Practical implicationsOligopoly games are pivotal in the mathematical modeling of some substantial economic areas such as industrial organization, airline, banking, telecommunication companies, international trade and also macroeconomic analysis of business cycles, innovation and growth.Originality/valueAlthough the Cournot game and its variants have attracted great interest among mathematicians and economists since the time of its proposition till present, memory effects in continuous-time and discrete-time Cournot duopoly game have not been addressed yet. To the best of author’s knowledge, this can be considered as the first attempt to investigate this problem of fractional-order differentiated Cournot duopoly game. In addition, studying more realistic models of Cournot oligopoly games plays a pivotal role in the mathematical investigation and better understanding of some substantial economic areas such as industrial organization, airline, banking, telecommunication companies, international trade and also in macroeconomic analysis of business cycles, innovation and growth.

IMPLEMENTING TASK AND RESOURCE ALLOCATION ALGORITHM BASED ON NON-COOPERATIVE GAME THEORY IN CLOUD COMPUTING

Now a day’s cloud computing is becoming most popular computing model in the information industry, and cloud is a large and complex system with a more number of servers and users, so it is necessary to manage all the tasks and users depending on the traffic of the network. To manage tasks and users it has to schedule tasks frequently among the servers and manage its computing resource flexibly to meet the demand of the users. With the growing service demand and higher QoS requirement of the users, the performance of the system is facing a big challenge, and with the expanding scale of cloud computing, its energy waste problem is becoming more and more serious due to the invalid resource organization and failed task scheduling. To improve the energy efficiency of heterogeneous servers in the cloud computing system, this paper puts forward a non-cooperative game based task scheduling and computing resource allocation algorithm NG_TSRA. Firstly, we use non-cooperative game to model the task scheduling and computing resource allocation process of the servers in the cloud computing system, and the server's utility function is modeled as unit power efficiency, then we prove the existence of Nash Equilibrium point of the game, and finally use a Lagrange multiplier-based distributed iteration algorithm to solve the game. The experimental results show that the proposed algorithm can improve the average power efficiency of the cloud computing system.

Ergodic and thermodynamic games

Let [Formula: see text] and [Formula: see text] be compact sets, and [Formula: see text], [Formula: see text] be continuous maps. Let [Formula: see text] where [Formula: see text] is [Formula: see text]-invariant and [Formula: see text] is [Formula: see text]-invariant, be payoff functions for a game (in the usual sense of game theory) between players that have the set of invariant measures for [Formula: see text] (player 1) and [Formula: see text] (player 2) as possible strategies. Our goal here is to establish the notion of Nash equilibrium for the game defined by these payoffs and strategies. The main tools come from ergodic optimization (as we are optimizing over the set of invariant measures) and thermodynamic formalism (when we add to the integrals above the entropy of measures in order to define a second case to be explored). Both cases are ergodic versions of non-cooperative games. We show the existence of Nash equilibrium points with two independent arguments. One of the arguments deals with the case with entropy, and uses only tools of thermodynamical formalism, while the other, that works in the case without entropy but can be adapted to deal with both cases, uses the Kakutani fixed point. We also present examples and briefly discuss uniqueness (or lack of uniqueness). In the end, we present a different example where players are allowed to collaborate. This final example shows connections between cooperative games and ergodic transport.

Existence of Nash equilibrium points for Markovian non-zero-sum stochastic differential games with unbounded coefficients

This paper is related to non-zero-sum stochastic differential games in the Markovian framework. We show existence of a Nash equilibrium point for the game when the drift is no longer bounded and only satisfies a linear growth condition. The main tool is the notion of backward stochastic differential equations which, in our case, are multidimensional with continuous coefficient and stochastic linear growth.

Complexity Analysis of a Cournot-Bertrand Duopoly Game Model with Limited Information

A Cournot-Bertrand mixed duopoly game model with limited information about the market and opponent is considered, where the market has linear demand and two firms have the same fixed marginal cost. The principles of decision-making are bounded rational. One firm chooses output and the other chooses price as decision variable, with the assumption that there is a certain degree of differentiation between the products offered by firms to avoid the whole market being occupied by the one that applies a lower price. The existence of Nash equilibrium point and its local stability of the game are investigated. The complex dynamics, such as bifurcation scenarios and route to chaos, are displayed using parameter basin plots by numerical experiment. The influences of the parameters on the system performance are discussed from the perspective of economics.

Nash equilibria in bandwidth allocation for non-cooperative peer-to-peer networks

In peer-to-peer networks, peers act as clients and servers, i.e., they can download files from others and allow others to download from them, at the same time. Since the bandwidth of a peer acting as server is shared among all its clients, the download rate experienced by a peer depends on the server choices of the other peers. We focus our investigation on the bandwidth allocation among the peers and model this system with non-cooperative game theory. We assume that peers are rational players that maximize their utility, corresponding to minimize their download time. We study the existence of Nash equilibrium points under a very simple server selection policy and show that it is efficiency from the point of view of the bandwidth utilization.

Nonzero sum linear–quadratic stochastic differential games and backward–forward equations

We study the link between linear backward-forward stochastic differential equations and the problem of existence of Nash equilibrium points in nonzero sum linear–quadratic stochastic differential games. For such games we show the existence of an equilibrium point whose expression is given

Nash equilibria in N-person games without convexity

In this note, an extended version of the Zhou and Chen minimax inequality is obtained for reflexive Banach spaces. An application of the result to the existence of Nash equilibrium points without assuming that cost functions are convex is given.

On the existence of nash equilibrium point for a game between infinitely many players with noncompact strategy sets

On the existence of nash equilibrium point for a game between infinitely many players with noncompact strategy sets

On the existence of Nash equilibrium points

AbstractThe existence of Nash equilibrium points of game between n players has been established under the conditions dual to known ones.

A further generalization of the Kakutani fixed theorem, with application to Nash equilibrium points

Abstract : Kakutani's Fixed Point Theorem states that in Euclidean n-space a closed point to (non-void) convex set map of a convex compact set into itself has a fixed point. Kakutani showed that this implied the minimax theorem for finite games. The object of this note is to point out that Kakutani's theorem may be extended to convex linear topological spaces, and implies the minimax theorem for continuous games with continuous payoff as well as the existence of Nash equilibrium points.