A fuzzy fixed-point theorem and its applications to maximal elements and Nash equilibria in noncompact CAT(0) spaces

In this paper, based on the KKM method, we prove a new fuzzy fixed-point theorem in noncompact CAT(0) spaces. As applications of this fixed-point theorem, we obtain some existence theorems of fuzzy maximal element points. Finally, we utilize these fuzzy maximal element theorems to establish some new existence theorems of Nash equilibrium points for generalized fuzzy noncooperative games and fuzzy noncooperative qualitative games in noncompact CAT(0) spaces. The results obtained in this paper generalize and extend many known results in the existing literature.

GAME MODEL OF ONTOLOGICAL PROJECT SUPPORT

Context. In today’s information society with advanced telecommunications through mobile devices and computer networks, it is important to form a variety of virtual organizations and communities. Such virtual associations of people by professional or other interests are designed to quickly solve various tasks: to perform project tasks, create startups to attract investors, network marketing, distance learning, solving complex problems in science, economics and public administration , construction of various Internet services, discussion of political and social processes, etc. Objective of the study is to develop an adaptive Markov recurrent method based on the stochastic approximation of the modified condition of complementary non-rigidity, valid at Nash equilibrium points for solving the problem of game coverage of projects. Method. In this work the multiagent game model for formation of virtual teams of executors of projects on the basis of libraries of subject ontologies is developed. The competencies and abilities of agents required to carry out projects are specified by sets of ontologies. Intelligent agents randomly, simultaneously and independently choose one of the projects at discrete times. Agents who have chosen the same project determine the current composition of the team of its executors. For agents’ teams, a current penalty is calculated for insufficient coverage of competencies by the combined capabilities of agents. This penalty is used to adaptively recalculate mixed player strategies. The probabilities of selecting those teams whose current composition has led to a reduction in the fine for non-coverage of ontologies are increasing. During the repetitive stochastic game, agents will form vectors of mixed strategies that will minimize average penalties for non-coverage of projects. Results. For solve the problem of game coverage of projects, an adaptive Markov recurrent method based on the stochastic approximation of the modified condition of complementary non-rigidity, valid at Nash equilibrium points, was developed. Conclusions. Computer simulation confirmed the possibility of using the stochastic game model to form teams of project executors with the necessary ontological support in conditions of uncertainty. The convergence of the game method is ensured by compliance with the fundamental conditions and limitations of stochastic optimization. The reliability of experimental studies is confirmed by the repeatability of the results obtained for different sequences of random variables.

Dynamical Study of Competition Cournot-like Duopoly Games Incorporating Fractional Order Derivatives and Seasonal Influences

Cournot’s game is one of the most distinguished and influential economic models. However, the classical integer order derivatives utilized in Cournot’s game lack the efficiency to simulate the significant memory characteristics observed in many economic systems. This work aims at introducing a dynamical study of a more realistic proposed competition Cournot-like duopoly game having fractional order derivatives. Sufficient conditions for existence and uniqueness of the new model’s solution are obtained. The existence and local stability analysis of Nash equilibrium points along with other equilibrium points are examined. Some aspects of global stability analysis are treated. More significantly, the effects of seasonal periodic perturbations of parameters values are also explored. The multiscale fuzzy entropy measurements for complexity are employed for this case. Numerical simulations are presented in order to verify the analytical results. It is observed that the time-varying parameters induce very complicated dynamics in perturbed Cournot duopoly game compared with the unperturbed game.

On Maximal Vector Spaces of Finite Noncooperative Games

We consider finite noncooperative [Formula: see text] person games with fixed numbers [Formula: see text], [Formula: see text], of pure strategies of Player [Formula: see text]. We propose the following question: is it possible to extend the vector space of finite noncooperative [Formula: see text]-games in mixed strategies such that all games of a broader vector space of noncooperative [Formula: see text] person games on the product of unit [Formula: see text]-dimensional simplices have Nash equilibrium points? We get a necessary and sufficient condition for the negative answer. This condition consists of a relation between the numbers of pure strategies of the players. For two-person games the condition is that the numbers of pure strategies of the both players are equal.

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.