An optimal bound to access the core in TU-games (original) (raw)

The Core, the Objection-Free Core and the Bargaining Set of Transferable Utility Games

It is well-kown that the (Aumann-Maschler) bargaining set of a transferable utility game (or simply a game) with less than five players coincides with the core of the game, provided that the core is nonempty. We show that this coincidence still holds for a superset of the core, the objection-free core which is the set of all imputations with no bargaining set type objection. Furthermore, for any game and for any coalition structure, the objection-free core contains the core, is a subset of the bargaining set and is a polyhedron when it is nonempty.

The core and balancedness of TU games with infinitely many players ∗

2017

Transferable utility cooperative games with infinitely many players are considered. We generalize the notions of core and balancedness, and present a generalized Bondareva-Shapley Theorem for games without and with restricted cooperation. Our generalized BondarevaShapley Theorem extends previous results by Bondareva (1963), Shapley (1967), Schmeidler (1967), Faigle (1989), and Kannai (1969, 1992) among others.

Fundamental cycles of pre-imputations in non-balanced TU-games

International Journal of Game Theory, 2003

In this paper we prove a characterization for the subclass of non-balanced T U-games. The result is stated in terms of certain class of cycles of pre-imputations. A cycle is a finite sequence of pre-imputations, where each pair of neighbouring elements are interrelated to each other through a transfer of some amount of utility from members of a certain coalition to the members of the complementary coalition, with the understanding that individual gains or losses within any coalition are proportional to the number of members of the coalition. These cycles are strongly connected with a transfer scheme designed to reach a point in the core of a T U-game provided this set is non-empty. The main result of this paper provides an alternative characterization of balanced TU-games to Shapley-Bondareva’s theorem.

U-Cycles in N-Person Tu-Games with Equal-Sized Objectionable Families of Coalitions

2010

It has recently been proven that the non-existence of certain types of cycles of pre-imputation, fundamental cycles, is equivalent to the balancedness of a TU -game (see [3]). In some cases, the class of fundamental cycles can be narrowed and a characterization theorem may still be obtained. In this paper, we deal with n-person TU -games for which the only coalitions with nonzero value, aside from the grand coalition, are all coalitions of the same size k ≤ n, which also form a balanced family of coalitions. This class of games includes those studied in previous papers where the non-zero value coalitions are the family of coalitions with n − 1 players. The main result obtained in this framework is that it is always possible to find a U -cycle, a certain type of fundamental cycle, provided the game under consideration is non-balanced and n and k are relatively prime. A computational procedure to get the cycle is provided as well. In many situations, these cycles turn out to be maxima...

On a games theory of random coalitions and on a coalition imputation

2002

The main theorem of the games theory of random coalitions is reformulated in the random set language which generalizes the classical maximin theorem but unlike it defines a coalition imputation also. The theorem about maximin random coalitions has been introduced as a ...