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 Выпуск 31.1. Математическая теория игр и ее приложения II
- Akimova A. N., Zakharov V. V. A method for estimating the core of root game
It is shown that the base of the grand (shadow) subcore coincides with the core of the root game in any TU-cooperative game. Comparing the definitions of the grand subcore and the grand shadow subcore with the description of the aggregate-monotonic core leads to formal geometrical coincidence of the aggregate-monotonic core with either the grand subcore or the grand shadow subcore. The method for estimating the simplest set of equations and inequalities describing the core of a root game in a TU-game with any number of players (n > 3) is proposed. To develop the method the duality theory and an inductive method by B. Peleg are used.
Original text was published in "Mathematical game theory and applications, 2010. V. 2. No 1. P. 3-26".
- Grigorieva K. V. Solutions for a class of stochastic coalitional games
In this paper one of classes of multistage stochastic games with various coalition structures is considered. The game under research is set on a tree graph. In each vertex z of the tree the coalition structure of players is defined, along with the payoff function of coalitions, and the probability of transition to the following vertices of the tree depending on the situation realized in the game in vertex z. The new mathematical method is offered to building a solution of stochastic coalition games on the basis of calculation of the generalised PMS-vector as a solution of a coalition game. The offered method is illustrated by the example of three-step stochastic game of three persons with variable coalition structure.
Original text was published in "Mathematical game theory and applications, 2010. V. 2. No 1. P. 47-66".
- Zenkevich N. A., Zyatchin A. V. Strong equilibrium construction in a noncooperative differential game
In this paper a special technique based on scalarization of a vector criterion is used to construct a strong equilibrium in a differential game. Sufficient conditions for the existence of strong equilibrium are proved. This approach is tested on an example of an asymmetrical differential game of two players, where the strong equilibrium was found in the explicit form.
Original text was published in "Mathematical game theory and applications, 2010. V. 2. No 2. P. 42-65".
- Zinchenko A. B., Mironenko G. V., Provotorova P. A. A consensus value for games with coalition structure
The class of TU-games for which almost all solution concepts, except the consensus value, yield paradoxical results is selected. It is proved that it is big boss games. Generalisation of the consensus value for games with coalition structure is introduced.
Original text was published in "Mathematical game theory and applications, 2010. V. 2. No 1. P. 93-106".
- Kovshov A. M. Parallel pursuit strategies in a simple motion game on the sphere. Geodesic pursuit
The two-person zero-sum differential simple pursuit game on the sphere is considered. The strategy of geodesic pursuit, having some properties of parallel pursuit strategy on the plane, is defined, the existence and uniqueness in general are proved. All special cases are considered. The fastest property of the geodesic pursuit strategy is proved.
Original text was published in "Mathematical game theory and applications, 2009. V. 1. No 4. P. 41-62".
- Raygorodskaya A. V. A 2x2 epselon-best response stochastic two-step game
A 2x2 epsilon-best response repeated game, in which each player in each subsequent round chooses a pure strategy based on the result of a random test, is analyzed. The random test is generated by the player's arbitrary mixed strategy prescribing the player to choose his/her best response to his/her partner's previously chosen pure strategy with a high probability. The so defined decision making patterns (called epsilon-best response functions) are interpreted as the players' behavioral strategies. These strategies define a stochastic game, in which the expected benefits averaged over all the rounds act as the players' benefits. The game is analyzed in the two-step case. A classification of the Nash equilibrium points is provided, and the equilibrium values are compared with the average benefits gained through the deterministic usage of the players' best response pure strategies.
Original text was published in "Mathematical game theory and applications, 2010. V. 2. No 4. P. 80-101".
- Stepanov D. S. Two types of players in the endogenous coalition formation model
A model of coalition formation by players whose payoff depends on the value of the parameter (e.g., geographical location, bliss point) is considered. In this model a small portion of new players with the different payoff function is injected into the main population. This paper considers different types of coalition stability and corresponding stability criteria. The derived conditions are then compared with the similar criteria in the game with a single type of players.
Original text was published in "Mathematical game theory and applications, 2010. V. 2. No 2. P. 79-98".
- Shevkoplyas E. V. Stable cooperation in differential games with random duration
The problem of time-consistency of cooperative solutions is investigated in the paper. This problem was stated by Petrosyan L.A. in 1977 for differential games with a finite time horizon. In this paper a modification of the game with a finite time horizon is considered, namely, the random time horizon of the game is supposed. The Shapley value is used as an optimality principle under cooperative behavior of players. For this formulation the definition of the imputation distribution procedure (IDP) is given and the analytic formula for IDP is derived. Moreover, the irrational behavior proofness condition by D.W.K. Yeung (2006) is modified for the problem with random duration. The tool is based on using IDP. Theoretical results are illustrated by an example of the differential game of non-renewable resource extraction.
Original text was published in "Mathematical game theory and applications, 2010. V. 2. No 3. P. 79-105".
- Parilina E. M. Cooperative data transmission game in wireless network
The paper considers the problem of data transmission in a simple wireless network. The process of data transmission is modelled with the help of a stochastic game. The paper proposes the system of rewards and costs to the network users to regulate the process of data transmission. The cooperative version of the game is considered. For this purpose the characteristic function is found. The Shapley value is proposed as a cooperative decision of the game. The condition of subgame consistency of the Shapley value and the method of construction of the cooperative payoff distribution procedure are taken. The cooperative payoff distribution procedure allows to redistribute payoffs to the players (network users) at each time slot to overcome the natural inconsistency of the Shapley value. The paper considers the numerical example which demonstrates all obtained theoretical results.
Original text was published in "Mathematical game theory and applications, 2009. V. 1. No 4. P. 93-110".
- Vasin A. A., Gusev A. G., Sharikova A. A. Game-theoretic analysis of one-stage and two-stage homogeneous good auctions
Forward market is a known instrument for reduction of market power of large producers. This paper examines a two-stage oligopoly environment with constant marginal cost. The outcome at both the forward and the spot market is a Cournot outcome dependent on correspondent demand and supply at the market. Producers aim to maximize their profits via choosing subgame perfect equilibrium of the two-stage game as their strategies. In the first part of the current research we extend the model by Bushnell (2005) considering a capacity constraint. Our results show that the optimal way of market organization in such a model strongly depends on the difference between the maximal production volume and the demand volume at price equal to the marginal cost. In the second part of the paper we consider proportional rationing instead of surplus maximizing rationing at the forward market. We show that for such a model there exists only an SPE in correlated mixed strategies. Producers' behavior should depend on some random variable that determines one of two possibilities for the spot market: either the ''bear market'', or the ''bull market''. We compare this SPE with Nash equilibria of one-stage markets.
Original text was published in "Mathematical game theory and applications, 2009. V. 1. No 4. P. 3-30".
- Gladkova M. A., Zenkevich N. A. Game-theoretical model of quality management under competition
In this paper a game-theoretical model of quality management under competition is suggested. The game-theoretical model is represented as a two-stage game where production companies compete on an industrial market and consumer's taste to quality is non-uniformly distributed. The strong Nash equilibrium in the investigated game was obtained in explicit form which allowed us to evaluate prices, companies market shares and revenues in the equilibrium. A case study for Internet-trading systems was used to approve the suggested quality management mechanism.
Original text was published in "Mathematical game theory and applications, 2010. V. 2. No 4. P. 3-24".
- Kalugina A. M. Method of generating functions for procedure of committee' electing
In this paper we consider the minimax and minisum procedures for electing committee. These procedures were proposed by S.F. Brams et al. We introduce the method of generating functions for these procedures. This method can be used for electing with a large number of candidates. The "Mathematica" symbolic calculus software can be used to implement our method.
Original text was published in "Mathematical game theory and applications, 2009. V. 1. No 4. P. 31-40".
- Mazalov V. V., Tokareva J. S. Game-theoretic models of tender design
We consider a n-person non-zero-sum game related to design of a tender. Players present some projects, which are characterized by a vector of parameters. Arbitrator or some juri chooses one of the projects using a stochastic procedure with a certain distribution function, which is known to players. The winner receives a payoff, which depends on the parameters of the project. The game-theoretic model of a tender is presented and equilibrium in two and three-dimensional models is derived.
Original text was published in "Mathematical game theory and applications, 2010. V. 2. No 2. P. 66-78".
- Mentcher A. I. Discrete arbitration procedure with nonuniform distribution
We consider a zero-sum game related to an arbitration scheme. The arbitrator's offers are concentrated in three or four points with nonuniform distribution. The equilibrium in mixed strategies is derived.
Original text was published in "Mathematical game theory and applications, 2009. V. 1. No 4. P. 78-92".
- Zenkevich N. A., Kozlovskaya N. V. Stable Shapley value in cooperative game of territorial environmental production
A game-theoretic model of territorial environmental production is studied. The process is modeled as a cooperative differential game. The stable mechanism of distribution of common cooperative benefit among players is proposed. We prove that the cooperative total stock of accumulated pollution is strictly less than the pollution under Nash equilibrium for the whole duration of the game. The perfect Nash equilibrium is found. We design a stable Shapley value as a cooperative solution, which is time-consistent. The Shapley value is also strategic stable and satisfies the irrational-behavior-proofness condition. The numerical example is given.
Original text was published in "Mathematical game theory and applications, 2010. V. 2. No 1. P. 67-92".
- Korolev Y. M., Golubtsov P. V. Two-level competitive structures in common resource development
This paper studies the effects of owner-developer interaction in resource development. Resource is supposed to be allocated among several proprietors, and several companies are permitted to develop it. Marine fishery gives one of the most interesting examples. We study this game in the context of multinational management of transboundary marine fishery. It is well known that unconstrained harvesting often leads to resource depletion. This effect is often called "the tragedy of commons". The situation becomes more complex when we take into account competition among resource proprietors. Such interaction can be described as a game with players of two different types: proprietors and developers, called first- and second-level players, respectively. The first-level players establish the rules (taxes on development efforts) for the second-level players, who, in their turn, optimize their strategies reasoning from these rules. Every developer receives profit from resource selling and returns part of it to the owner as a tax. The systems described here appear in management problems for energy resources, mineral resources, biological resources, water resources, etc.
Original text was published in "Mathematical game theory and applications, 2010. V. 2. No 4. P. 25-48".
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