A two-person non-zero-sum game model of the world oil market

The world oil market is modelled as a two-person non-zero-sum game in normal form with each player having a continuum of strategies. The two players are the oil importing nations (OPIC) and the oil exporting nations (OPEC). The game is solved in the noncooperative sense using the equilibrium point solution concept due to Nash. The Nash equilibrium point solution yields an analytic expression for the optimal price per barrel of oil for OPEC and the optimal level of imports of oil for OPIC assuming noncooperation between the players. The cooperative solution to the game is also investigated using the von Neumann-Morgenstern negotiation set solution and Nash's bargaining point solution. Again, we give analytic expressions for the optimal price of a barrel of oil and the optimal level of imports of oil assuming that the players cooperate (negotiate, bargain, etc., for a binding agreement) in arriving at a solution.     

A differential game formulation of a controlled network

This paper considers multiclass queueing network systems with fixed routing. With the service control as one player and the arrival and service rates as the other, the problem of network regulation can be formulated as a differential game. Representations of the value function are developed by studying the geometric properties of the associated Hamiltonians and are expressed in terms of related simpler halfspace problems. Also, a method of constructing the optimal feedback controls through the representation and the projected Isaacs equations is provided. The controls so constructed give both a guaranteed level of performance and robust stability over a range of rate perturbations.     

The sigma-core of a cooperative game

This paper is concerned with the existence of (σ-additive) measures in the core of a cooperative game. The main theorem shows, for a capacityu on the Borel sets of a metric space, that to each additive set function, majorized byu and agreeing withu on a system of closed sets, there exists a measure having these same properties. This theorem is applied, in combination with known core theorems, to the case of a cooperative game defined on the Borel sets of a metric space and whose conjugate is a capacity.     

A cooperative location game based on the 1-center location problem

In this paper we introduce and analyze new classes of cooperative games related to facility location models defined on general metric spaces. The players are the customers (demand points) in the location problem and the characteristic value of a coalition is the cost of serving its members. Specifically, the cost in our games is the service radius of the coalition. We call these games the Minimum Radius Location Games (MRLG).We study the existence of core allocations and the existence of polynomial representations of the cores of these games, focusing on network spaces, i.e., finite metric spaces induced by undirected graphs and positive edge lengths, and on the ?_p metric spaces defined over R^d.     

一种资源投入不确定情形下的合作博弈形式及收益分配策略

首先,将经典合作博弈进行扩展,提出了一类模糊联盟合作博弈的通用形式,涵盖常见三种模糊联盟合作博弈,即多线性扩展博弈、比例模糊博弈与Choquet积分模糊博弈.比例模糊博弈、Choquet积分模糊博弈的Shapley值均可以作为一种特定形式下模糊联盟合作博弈的收益分配策略,但是对于多线性扩展博弈的Shapley值一直关注较少,因此利用经典Shapley值构造出多线性扩展博弈的Shapley值,以此作为一种收益分配策略.最后,通过实例分析了常见三类模糊联盟合作博弈的形式及其对应的分配策略,分析收益最大的模糊联盟合作对策形式及最优分配策略,为不确定情形下的合作问题提供了一定的收益分配依据.     

A note on a core catcher of a cooperative game

In Driessen (1986) it is shown that for games satisfying a certain condition the core of the game is included in the convex hull of the set of certain marginal worth vectors of the game, while it is conjectured that the inclusion holds without any condition on the game. In this note it is proved that the inclusion holds for all games.

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Zusammenfassung In Driessen (1986) wurde für Spiele, die eine gewisse Bedingung erfüllen, gezeigt, da\ der Kern des Spieles in der konvexen Hülle von gewissen Vektoren der Marginalwerte liegt. Es wurde vermutet, da\ diese Inklusion ohne weitere Bedingung an das Spiel gilt. In dieser Note wird nun gezeigt, da\ die Inklusion für alle Spiele gilt.

     

A dynamic solution in N-person cooperative game theory

We formulate a cooperative game as an extended form game in which each player in turn proposes payoffs to a coalition over M steps. Payoffs at time t are discounted by a penalty function f(t). If all players in a coalition agree to their payoffs, they receive them. Under a convergence hypothesis verified by computer for three players in many cases, we compute the payoffs resulting from a coalition pattern and give necessary conditions for particular patterns. The resulting solution is related to the Nash bargaining solution and the competitive solution.     

The positive core of a cooperative game

The positive core is a nonempty extension of the core of transferable utility games. If the core is nonempty, then it coincides with the core. It shares many properties with the core. Six well-known axioms that are employed in some axiomatizations of the core, the prenucleolus, or the positive prekernel, and one new intuitive axiom, characterize the positive core for any infinite universe of players. This new axiom requires that the solution of a game, whenever it is nonempty, contains an element that is invariant under any symmetry of the game.     

Maximizing selections from the core of a cooperative game

A core selection is a function which for each cooperative game with non-empty core selects a subset, possibly a single element, of the core. In this paper, we study selections which maximize some function or binary relation on the core. We present an axiomatic characterization of such core selections, as well as a local characterization using duality which can be applied to investigate properties of core selections. We give an application of the local characterization to a question of coalitional monotonicity of core selections.     

On finding the nucleolus of an n-person cooperative game

Kohlberg (1972) has shown how the nucleolus for ann-person game with side-payments may be found by solving a single minimization LP in case the imputation space is a polytope. However the coefficients in the LP have a very wide range even for problems with 3 or 4 players. Therefore the method is computationally viable only for small problems on machines with finite precision. Maschler et al. (1979) find the nucleolus by solving a sequence of minimization LPs with constraint coefficients of either –1, 0 or 1. However the number of LPs to be solved is o(4 ⁿ ). In this paper, we show how to find the nucleolus by solving a sequence of o(2 ⁿ ) LPs whose constraint coefficients are –1, 0 or 1.     

On the computation of the nucleolus of a cooperative game

We consider classes of cooperative games. We show that we can efficiently compute an allocation in the intersection of the prekernel and the least core of the game if we can efficiently compute the minimum excess for any given allocation. In the case where the prekernel of the game contains exactly one core vector, our algorithm computes the nucleolus of the game. This generalizes both a recent result by Kuipers on the computation of the nucleolus for convex games and a classical result by Megiddo on the nucleolus of standard tree games to classes of more general minimum cost spanning tree games. Our algorithm is based on the ellipsoid method and Maschler's scheme for approximating the prekernel. Received February 2000/Final version April 2001     

A differential game model of a market of substitutable products

Modern industrial organization often classifies groups of differentiated products that are fairly good substitutes to belong to the same market. This paper develops a differential game model of a market of substitutable products. To avoid the problem of time-inconsistency, we solve a feedback Nash equilibrium solution for the game. A set of state-dependent equilibrium strategies is derived. Extensions of the model to a stochastic formulation and to an infinite time horizon specification are also provided.     

重复n人随机合作对策的核心

以Su ijs等人(1995)引入的随机合作对策的模型为基础,建立了重复n人随机合作对策的理论,定义了重复n人随机合作对策的支付序列以及支付序列的优超关系,并由此给出了重复n人随机合作对策的核心、超可加性和凸性的定义,并讨论了该核心的一些特征和性质.     

The semireactive bargaining set of a cooperative game

The semireactive bargaining set, a solution for cooperative games, is introduced. This solution is in general a subsolution of the bargaining set and a supersolution of the reactive bargaining set. However, on various classes of transferable utility games the semireactive and the reactive bargaining set coincide. The semireactive prebargaining set on TU games can be axiomatized by one-person rationality, the reduced game property, a weak version of the converse reduced game property with respect to subgrand coalitions, and subgrand stability. Furthermore, it is shown that there is a suitable weakening of subgrand stability, which allows to characterize the prebargaining set. Replacing the reduced game by the imputation saving reduced game and employing individual rationality as an additional axiom yields characterizations of both, the bargaining set and the semireactive bargaining set. Received September 2000/Revised version June 2001     

On the theory of three-person differential games

A differential game of three players with dynamics described by linear differential equations under geometric constraints on the control parameters is considered. Sufficient conditions are obtained for the existence of the first player’s strategy guaranteeing that the trajectory of the game reaches a given target set for any admissible control of the second player and avoids the terminal set of the third player. An algorithm of constructing the first player’s strategy guaranteeing the game’s termination in finite time is suggested. A solution of a model example is given.