关键词: 合作博弈, Shapley值, 均分值, 联盟结构, Owen值

Abstract: For cooperative games with transferable utilities or short TU-games, many allocation rules or values are defined to allocate the worth of the grand coalition to all players in coalition. For instance, the Shapley value, the egalitarian value, the solidarity value and Banzhaf value and so on are the famous single-values. MYERSON (1980) used the balanced contribution axiom to give a characterization of the Shapley value, which means that for each pair of players, each loses (or gains) the same amount if the other leaves the coalition. Moreover, the Shapley value is a unique efficient value satisfying balanced contributions but there is no literature to study the characterization of the solidarity and egalitarian values by employing the balanced contribution axiom because the axiom is so strong.
KAMIJO and KONGO (2010) proposed balanced cycle contribution motivated by the idea of equilibrium in economics and this property states that for any order of all the players, the sum of each player's claims on his predecessor equals that of each player's claims on his successor. Finally, they gave a characterization of the Shapley value by invoking the properties of efficiency, balanced cycle contribution and null player out property. Moreover, KAMIJO and KONGO (2010) found that not only does the Shapley value satisfy the balanced cycle contribution axiom but also some other values for TU-games do so, such as the solidarity value, the egalitarian value and Banzhaf value. Hence, the balanced cycle contribution axiom is a less restrictive requirement than the balanced contribution property. In order to characterize these above values, KAMIJO and KONGO (2012) introduced the invariance axiom, which states that the removal of a particular player from the game does not affect the payoffs of other players, and that the removal is different in value. They also gave the axiomatic characterization of these above values for TU-games. Concretely, the egalitarian value is a unique value satisfying efficiency, the invariance from proportional player deletion and balanced cycle contribution. The solidarity value is a unique value satisfying efficiency, the invariance from quasi-proportional player deletion and balanced cycle contribution. The Banzhaf value is a unique value on TU-games that satisfies 2-efficiency, efficiency with respect to 1-person games, balanced cycle contribution and the invariance from null player deletion.
In this paper we extend the balanced cycle contribution to TU-games with coalition structure and coin the phrase—the balanced cycle contribution between the unions. Furthermore, we characterize the Owen value by invoking efficiency, balanced cycle contribution between the unions, null priori union out property and balanced contribution within the unions. Moreover, we find that the difference between the Owen value and the equal division value for TU-games with coalition structures is that the deletion of a specific union from games does not affect the other unions' payoffs, and this deletion is different in both values. Finally, we characterize the equal division value for TU-games with coalition structures by using the axioms of efficiency, balanced cycle contribution between the unions, invariance of the priori union's payoff and balanced contribution within the unions.

Key words: cooperative game, Shapley value, egalitarian value, coalition structure, Owen value