Ilya V. Katsev

The prenucleolus for games with restricted cooperation

A game with restricted cooperation is a triple (N,v,Ω), where N is a finite set of players, Ω⊂2N is a nonempty collection of feasible coalitions such that N∈Ω, and v:Ω→R is a characteristic function. The definition implies that if Ω=2N, then the game (N,v,Ω)=(N,v) is the classical transferable utility (TU) cooperative game.

The Shapley Value for Games with Restricted Cooperation

The traditional assumption in cooperative game theory is that every coalition is feasible and can form to attain its payoff. However, in many real life situations not every group of players has the opportunity to cooperate and to collect their own payoff. We say that we deal with cooperative games with restricted cooperation when not all coalitions can form. In this paper we will deal with generalizations of the Shapley value for games with restricted cooperation. Three solutions for games with restricted cooperation will be considered. One of them (the Myerson value) is well known. Two others are based on the same principle: to construct some restricted game and to use the Shapley value for this game. We consider the class of all solutions which can be constructed in an analogous way. We will show that this class coincides with the class of all solutions with balanced contribution property. Also each solution from this class can be described by either a potential function or a consistency property.

A polynomial time algorithm for computing the nucleolus for a class of disjunctive games with a permission structure

Recently, applications of cooperative game theory to economic allocation problems have gained popularity. In many such allocation problems there is some hierarchical ordering of the players. In this paper we consider a class of games with a permission structure describing situations in which players in a cooperative TU-game are hierarchically ordered in the sense that there are players that need permission from other players before they are allowed to cooperate. The corresponding restricted game takes account of the limited cooperation possibilities by assigning to every coalition the worth of its largest feasible subset. In this paper we provide a polynomial time algorithm for computing the nucleolus of the restricted games corresponding to a class of games with a permission structure which economic applications include auction games, dual airport games, dual polluted river games and information market games.

Games on Union Closed Systems

A situation in which a finite set of players can obtain certain payoffs by cooperation can be described by a cooperative game with transferable utility, or simply a TU-game. A solution for TU-games assigns a set of payoff distributions to every TU-game. In the literature various models of games with restricted cooperation can be found. So, instead of allowing all subsets of the player set N to form, it is assumed that the set of feasible coalitions is a subset of the power set of N. In this paper we consider such sets of feasible coalitions that are closed under union, i.e. for any two feasible coalitions also their union is feasible. Properties of solutions (the core, the nucleolus, the prekernel and the Shapley value) are given for games on union closed systems.

An algorithm for computing the nucleolus of disjunctive non-negative additive games with an acyclic permission structure

A cooperative game with a permission structure describes a situation in which players in a cooperative TU-game are hierarchically ordered in the sense that there are players that need permission from other players before they are allowed to cooperate. In this paper we consider non-negative additive games with an acyclic permission structure. For such a game we provide a polynomial time algorithm for computing the nucleolus of the induced restricted game. The algorithm is applied to a market situation where sellers can sell objects to buyers through a directed network of intermediaries.

Between the Prekernel and the Prenucleolus

A collection of TU games solutions intermediate between the prekernel and the prenucleolus is considered. Each solution from the collection is parametrized by a positive integer k > 1 and is called the k-prekernel for properties extending those verifying by the prekernel such that the 2-prekernel coincides with the prekernel. If the number of players in a game is less than k, then the k-prekernel of this game coincides with the prenucleolus. All k-prekernels are efficient, covariant, consistent in the sense of Davis-Maschler, and satisfy equal treatment property. K-analogs of balancedness of collections of coalitions and of consistency properties are defined, and with thehelp of such properties an axiomatic characterization of the collection of the k-prekernels is given for the class of TU games with an infinite universe set of players.

An algorithm for computing the nucleolus of disjunctive additive games with an acyclic permission structure

A situation in which a finite set of players can obtain certain payoffs by cooperation can be described by a cooperative game with transferable utility, or simply a TU-game. A (single-valued) solution for TU-games assigns a payoff distribution to every TU-game. A well-known solution is the nucleolus. A cooperative game with a permission structure describes a situation in which players in a cooperative TU-game are hierarchically ordered in the sense that there are players that need permission from other players before they are allowed to cooperate. The corresponding restricted game takes account of the limited cooperation possibilities by assigning to every coalition the worth of its largest feasible subset. In this paper we consider the class of non-negative additive games with an acyclic permission structure. This class generalizes the so-called peer group games being non-negative additive games on a permission tree. We provide a polynomial time algorithm for computing the nucleolus of every restricted game corresponding to some disjunctive non-negative additive game with an acyclic permission structure. We discuss an application to market situations where sellers can sell objects to buyers through a directed network of intermediaries.