U-cycles in n-person TU-games with equal-sized objectionable families of coalitions

Abstract

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 maximal U-cycles, an even more restricted class of fundamental cycles.