|
|
 Выпуск 26.1. Математическая теория игр и ее приложения
- Vasiliev V. A. An axiomatization of generalized Owen extension
To introduce a generalized Owen extension we propose a new approach based on the non-additive integration. Besides, we pay strong attention to the axiomatization problem for the extension introduced. One of the main results of the paper demonstrates that the axiomatization in question can be chosen similar to that elaborated by R. J. Aumann and L. S. Shapley for the multiplicative extension of non-atomic cooperative games.
- Vinnichenko S. V. Continuous NIM game
A continuous version of NIM game is considered. Two players take water from the tank one by one. The player wins the game if she makes the last turn. Optimal strategies and the value of the game are calculated.
- Katsev I. V., Yanovskaya E. B. Between the prekernel and the prenucleolus
A collection of TU games solutions intermediate between the prekernel and the prenucleolus is considered. All these solutions are Davis-Maschler consistent, symmetric and covariant. Each solution from the collection is parametrized by a positive integer k such that for all games with the number of players not greater than k the solution for parameter k coincides with the prenucleolus, and for the games with more than k players it is maximal, i.e. satisfies the "k-converse consistency". The properties of solutions are described and their characterization in terms of balancedness is given.
- Mazalov V. V., Sakaguchi M. Equilibrium in n-player competitive game of timing
Each player in the game of timing has to decide his time to shoot under the condition that he is not informed of the shooting times of his rivals. That is, we deal with silent games of timing. Games of timing are used to model auctions, games of war of attrition, competitive predictions of a random variable, etc. Using the symmetry of the model we derive the equation to determine the equilibrium of the game.
- Naumova N. I. Associated consistency based on utility functions of coalitions
A cooperative game problem is treated as a bargaining problem with claim point. For given continuous strictly increasing utility functions of coalitions we suppose that for every partition of the player set the result does not change after equal sacrifice w.r.t. these functions overestimation of characteristic function values for partition members. This supposition and continuity assumption lead to a special value and give an iterative method for its results computation. In particular, for equal logarithmic utility functions of coalitions we get proportional overestimation of characteristic functions for partition members and the value is the weighted entropy solution. The anonymity assumption and the "dummy" property give the Shapley value. The weighted entropy solution follows from the positive homogeneity assumption.
- Petrosyan L. A., Zenkevich N. A. Principles of dynamic stability
There are three important aspects which must be taken into account when the problem of stability of long-range cooperative agreements is investigated: time-consistency of the cooperative agreements, strategic stability and irrational behavior proofness. The mathematical results based on imputation distribution procedures (IDP) are developed to deal with the above mentioned aspects of cooperation. We proved that for a special class of differential games time-consistent cooperative agreement can be strategically supported by Nash equilibrium. We also consider an example where all three conditions are satisfied.
- Petrosyan L. A., Sedakov A. A. Multistage networking games with full information
Multistage networking games with full information are considered. The network structure which connects the players is defined at every time moment. We assume that each verge has a utility (the player's profit from the connection with another player), and players have a right to change the network structure at every stage. The approach to define optimal players' behavior is proposed.
- Tur A. V. Linear-quadratic non-antagonistic discrete-time dynamic games
Linear-quadratic discrete-time dynamic games are considered. The necessary and sufficient conditions of the existence of Nash equilibrium in such class of games are presented. Different cooperative solutions are obtained. D.W.K. Yeung's condition for linear-quadratic discrete-time dynamic games is studied. As an example, the model of production planning under competition is examined.
- Chuyko Yu. V. Routing problem with splitable traffic and incomplete information
We investigate the equilibria in Bayesian routing game in network with selfish users' behavior where each user chooses his route trying to minimize the expected delay of the traffic he sends. This scheme is based on~\cite{inc} and modified for model with parallel links where user's traffic is splittable. Our interest are equilibria: Wardrop Equilibrium, that always exists and can be found using potential function, and its special case Bayesian Wardrop Equilibrium, that can be more easily understood by users, but its existence is an open question.
- Galegov A. I., Garnaev A. Yu. A tax game in a Cournot duopoly
Stackelberg models for hierarchical oligopolistic markets with a homogenous product were studied by researchers extensively. The goal of this paper is to extend the classical solution in closed form of the Stackelberg model for a general hierarchical structures composed by firms arranged into groups of different hierarchical levels.
- Garnaev A. Yu., Toricyn A. O. Fair resource allocation in the presence of a jammer
A game-theoretical model between a base station distributing power among users and a jammer trying to harm the base station is considered. The goal of the base station is to distribute the power among users fairly taking into account its cost. The goal of the jammer is to harm the work of the base station also taking into account the cost of the employed power. The existence and uniqueness of Nash equilibrium are proved and its properties are investigated.
- Gubanov D. A., Novikov D. A., Chkhartishvili A. G. Models of reputation and information control in social networks
The models of social networks are considered which allow formulating and solving reputation formation problems. The reputation is further used in information control.
- Zenkevich N. A., Kolabutin N. V., Yeung D. V. K. Stable joint venture stochastic model
Dynamic joint venture model is investigated. Through knowledge diffusion participating firms can gain core skills and technology that would be very difficult for them to obtain on their own. The stochastic evolution of the technology level of company under joint venture is known as a multivariate stochastic Ito's process. The profit of the joint venture is the expected sum of the participating firms' profits. The member firms would maximize their joint profit and share their cooperative profits according to the Shapley value. Applying the method of regularization for dynamic cooperation problem, we constructed the control in the form of special payments, paid at each time instant on the optimal trajectory. The dynamic stable solution is obtained for the stochastic joint venture dynamic model.
- Ivashko A. A. Full-information best-choice game with two stops
We consider a full-information best-choice game in which each player wants to hire two secretaries. The aim of a player is to maximize the sum of expected applicant' quality values. Two models are considered: m-person best-choice game with the possibility for an applicant to refuse an offer and two-person best-choice game with dominant player. Optimal strategies are obtained. We prove that in the best-choice game with the possibility for an applicant to refuse an offer the players' payoffs don't depend on the total number of players in the game.
- Iskakov M. B., Pavlov P. A. Secure strategy equilibrium in Hotelling's model of spatial competition
The problem of spatial competition was formulated in 1929 by Harold Hotelling. He considered two firms playing a two-stage game. They choose locations in stage 1 and prices in stage 2. If locations are chosen by competitors Nash equilibrium do not always exist. For studying these cases we employ the concept of secure strategy equilibrium (SSE) which allows to solve the game of choosing prices for any locations. We examine a nontrivial particular case when prices grow if market moves from a monopoly to a duopoly.
- Korgin N. A. Equivalence and strategy-proofness of non-anonymous priority resource allocation mechanisms
We provide characterizations of strategy-proof mechanisms of sequential resource allocation, which are equivalent to mechanisms of direct and reverse priorities. Previously known equivalency of anonymous priority mechanisms is extended to non-anonymous case. Equivalency of all non-anonymous direct priorities mechanisms is shown. We provide characterization of class of reverse priorities mechanisms, that have equivalent mechanisms of direct priorities.
- Ougolnitsky G. A. Optimization and game theoretic models in real estate development
A system of optimization and game theoretic models in real estate development is described. The system includes models of sales optimization, competence and cooperation, hierarchical relations, control of sustainable development.
- Rettieva A. N. Cooperative incentive condition in bioresource sharing problem
The discrete-time game model for bio-resource management problem (fish catching) is considered. The center (referee) shares a reservoir between the competitors, and the players (countries) capture the fish. We assume that there is a migratory exchange between the regions of the reservoir. The Nash and cooperative equilibria are obtained for infinite planning horizon. Time-consistent imputation distribution procedure is considered as a method for cooperation maintenance. The new condition which offers an incentive to players to keep cooperation is introduced and called "incentive cooperative condition".
- Shevkoplyas E. V. The Hamilton-Jacobi-Bellman equation for a class of differential games with random duration
The class of differential games with random duration is studied. It turns out that the problem with random duration of the game can be simplified to the standard problem with infinite time horizon. The Hamilton-Jacobi-Bellman equation which help us to find the optimal solution under condition of random duration of the processes is derived. The results are illustrated with a game-theoretical model of non-renewable resource extraction. The problem is analyzed under condition of Weibull distribution for the random terminal time of the game.
|
|
