Dynamically stable cooperative solutions in randomly furcating differential games

The paradigm of randomly furcating differential games incorporates stochastic elements via randomly branching payoffs in differential games. This paper considers dynamically stable cooperative solutions in randomly furcating differential games. Analytically tractable payoff distribution procedures contingent upon specific random events are derived. This new approach widens the application of cooperative differential game theory to problems where future environments are not known with certainty.     

Additive stable solutions on perfect cones of cooperative games

Closed kernel systems of the coalition matrix turn out to correspond to cones of games on which the core correspondence is additive and on which the related barycentric solution is additive, stable and continuous. Different perfect cones corresponding to closed kernel systems are described. Received: December 2001/Revised: July 2002 RID="*" ID="*"  This note contains the new results, which were presented by the first author in an invited lecture at the XIV Italian Meeting on Game Theory and Applications in Ischia, July 2001. The lecture was dedicated to Irinel Dragan on the occasion of his seventieth birthday.     

On cooperative games with large monotonic core

Cooperative games with large core were introduced by Sharkey (Int. J. Game Theory 11:175–182, 1982), and the concept of Population Monotonic Allocation Scheme was defined by Sprumont (Games Econ. Behav. 2:378–394, 1990). Inspired by these two concepts, Moulin (Int. J. Game Theory 19:219–232, 1990) introduced the notion of large monotonic core giving a characterization for three-player games. In this paper we prove that all games with large monotonic core are convex. We give an effective criterion to determine whether a game has a large monotonic core and, as a consequence, we obtain a characterization for the four-player case.     

Population monotonic solutions on convex games

The Dutta-Ray solution and the Shapley value are two well-known examples of population-monotonic solutions on the domain of convex games. We provide a new formula for the Dutta-Ray solution from which population-monotonicity immediately follows. Then we define a new family of population-monotonic solutions, which we refer to as “sequential Dutta-Ray solutions.” We also show that it is possible to construct several symmetric and population-monotonic solutions by using the solutions in this family. Received September 1998/Revised version: December 1999     

Minimum norm solutions for cooperative games

A solution f for cooperative games is a minimum norm solution, if the space of games has a norm such that f(v) minimizes the distance (induced by the norm) between the game v and the set of additive games. We show that each linear solution having the inessential game property is a minimum norm solution. Conversely, if the space of games has a norm, then the minimum norm solution w.r.t. this norm is linear and has the inessential game property. Both claims remain valid also if solutions are required to be efficient. A minimum norm solution, the least square solution, is given an axiomatic characterization.      

Monotonic solutions of cooperative games

The principle of monotonicity for cooperative games states that if a game changes so that some player's contribution to all coalitions increases or stays the same then the player's allocation should not decrease. There is a unique symmetric and efficient solution concept that is monotonic in this most general sense — the Shapley value. Monotonicity thus provides a simple characterization of the value without resorting to the usual “additivity” and “dummy” assumptions, and lends support to the use of the value in applications where the underlying “game” is changing, e.g. in cost allocation problems.     

Distributive solutions for absolutely stable games

Every absolutely stable game has von Neumann-Morgenstern stable set solutions. (Simple games and [n, n?1]-games are included in the class of absolutely stable games.) The character of these solutions suggests that the distributive aspect of purely discriminatory solutions is of as much conceptual importance as the discriminatory aspect.     

Monotonic stable solutions for minimum coloring games

For the class of minimum coloring games (introduced by Deng et al. Math Oper Res, 24:751–766, 1999) we investigate the existence of population monotonic allocation schemes (introduced by Sprumont Games Econ Behav 2:378–394, 1990). We show that a minimum coloring game on a graph $G$ has a population monotonic allocation scheme if and only if $G$ is $(P_4,2K_2)$ -free (or, equivalently, if its complement graph $\bar{G}$ is quasi-threshold). Moreover, we provide a procedure that for these graphs always selects an integer population monotonic allocation scheme.     

On dominance core and stable sets for cooperative ellipsoidal games

Abstract

The allocation problem of rewards or costs is a central question for individuals and organizations contemplating cooperation under uncertainty. The involvement of uncertainty in cooperative games is motivated by the real world where noise in observation and experimental design, incomplete information and further vagueness in preference structures and decision-making play an important role. The theory of cooperative ellipsoidal games provides a new game theoretical angle and suitable tools for answering this question. In this paper, some solution concepts using ellipsoids, namely the ellipsoidal imputation set, the ellipsoidal dominance core and the ellipsoidal stable sets for cooperative ellipsoidal games, are introduced and studied. The main results contained in the paper are the relations between the ellipsoidal core, the ellipsoidal dominance core and the ellipsoidal stable sets of such a game.     

Totally monotonic games and flow games

Equivalences between totally balanced games and flow games, and between monotonic games and pseudoflow games are well-known. This paper shows that for every totally monotonic game there exists an equivalent flow game and that for every monotonic game, there exists an equivalent flow-based secondary market game.     

On three Shapley-like solutions for cooperative games with random payoffs

Three solution concepts for cooperative games with random payoffs are introduced. These are the marginal value, the dividend value and the selector value. Inspiration for their definitions comes from several equivalent formulations of the Shapley value for cooperative TU games. An example shows that the equivalence is not preserved since these solutions can all be different for cooperative games with random payoffs. Properties are studied and a characterization on a subclass of games is provided.2000 Mathematics Subject Classification Number: 91A12.The authors thank two anonymous referees and an associate editor for their helpful comments.This author acknowledges financial support from the Netherlands Organization for Scientific Research (NWO) through project 613-304-059.Received: October 2000     

A marginalistic value for monotonic set games

In this paper we characterize a value, called a marginalistic value, for monotonic set games, which can be considered to be the analog of the Shapley value for TU-games. For this characterization we use a modification of the strong monotonicity axiom of Young, but the proof is rather different from his.     

Subgame-consistent cooperative solutions in randomly furcating stochastic differential games

The paradigm of randomly-furcating stochastic differential games incorporates additional stochastic elements via randomly branching payoffs in stochastic differential games. This paper considers dynamically stable cooperative solutions in randomly furcating stochastic differential games. Analytically tractable payoff distribution procedures contingent upon specific random realizations of the state and payoff structure are derived. This new approach widens the application of cooperative differential game theory to problems where the evolution of the state and future environments are not known with certainty. Important cases abound in regional economic cooperation, corporate joint ventures and environmental control. An illustration in cooperative resource extraction is presented.     

Properties of solutions of cooperative games with transferable utilities

We understand a solution of a cooperative TU-game as the α-prenucleoli set, α ∈ R, which is a generalization of the notion of the [0, 1]-prenucleolus. We show that the set of all α-nucleoli takes into account the constructive power with the weight α and the blocking power with the weight (1 ? α) for all possible values of the parameter α. The further generalization of the solution by introducing two independent parameters makes no sense. We prove that the set of all α-prenucleoli satisfies properties of duality and independence with respect to the excess arrangement. For the considered solution we extend the covariance propertywith respect to strategically equivalent transformations.     

Predicting stable configurations of coalitions in cooperative games and exchange economies

Uniform competitive solutions are stable configurations of proposals predicting coalition formation and effective payoffs. Such “solutions” exist for almost all properly defined cooperative games and, therefore, can be proposed as substitute of the core. The new existence results obtained in the present paper concern also the case when the coalitional function of a game has empty values. All concepts and results are implemented in the competitive analysis of the exchange economies. Received: July 1997/Final version: February 2000     

最短路博弈群体单调分配方案构造

群体单调分配方案(Population Monotonic Allocation Scheme, 后简称PMAS)是合作博弈的一类分配机制。在合作博弈中, PMAS为每一个子博弈提供一个满足群体单调性的核中的分配方案, 从而保证大联盟的动态稳定性。本文主要贡献为利用线性规划与对偶理论构造与求解一类基于最短路问题的合作博弈(最短路博弈)的PMAS。我们首先借助对偶理论, 利用组合方法为最短路博弈构造了一个基于平均分摊思想的PMAS。然后借鉴计算核仁的Maschler方案, 将PMAS的存在性问题转化为一个指数规模的线性规划的求解问题, 并通过巧妙的求解得到了与之前组合方法相同的最短路博弈的PMAS。