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

Abstract

TU-games, Transfer scheme, Cycles, Non-balanced games, Characterization,

A general characterization for non-balanced games in terms of U-cycles

“…We now introduce some kinds of cycles of preimputations and state, without proof, several results proved in (Cesco and Aguirre, 2002;Cesco, 2003).…”

“…We will refer to BðcÞ as the family of coalitions supporting c, and to the entries of the vector ðl k Þ m k¼1 as the transfer amounts. BðcÞ is a balanced family of coalitions for every fundamental cycle c (Cesco, 2003, Theorem 1). In the latter paper it was also shown that the existence of a fundamental cycle implies the non-balancedness of the game and that every non-balanced TU-game has a fundamental cycle (Cesco, 2003, Theorem 3 and Theorem 9).…”

“…BðcÞ is a balanced family of coalitions for every fundamental cycle c (Cesco, 2003, Theorem 1). In the latter paper it was also shown that the existence of a fundamental cycle implies the non-balancedness of the game and that every non-balanced TU-game has a fundamental cycle (Cesco, 2003, Theorem 3 and Theorem 9). These two results together provide a characterization of the non-balanced TU-games (i.e., games with non-empty core) in terms of fundamental cycles.…”

“…It is already known (see for instance Cesco, 2003) that the existence of any kind of the above mentioned cycles implies the non-balancedness of the game. Here we show that every non-balanced game has a U-cycle.…”

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In a paper by Cesco [Cesco, J.C., 2003. Fundamental cycles of pre-imputations in non-balanced TU-games.

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