Abstract: A cooperative game with transferable utility (a TU-game) is a pair(N,v), N being the finite set of players and v : 2^N→R with v(Ø)=0, the characteristic function of the game, that is a real valued map that assigns to each coalition S$\subseteq$N the worth v(S) that its members can obtain by cooperating. The worth v(S) represents the economic possibilities of the coalition S if it is formed. A central issue is to find a method to distribute the benefits of cooperation among these players. A (single-valued) solution for TU-games is a function that assigns to every TU-game a vector with the same dimension as the size of the player set, where each component of the vector represents the payoff assigned to the corresponding player. The Shapley value (SHAPLEY, 1953) probably is the most eminent single-valued solution concept for this type of games. In 1977, Owen suggested an extension of the Shapley value to TU-games with a coalition structure. The coalition structure is represented by a partition of the set of players. He defined and characterized the coalitional value for TU-games with coalition structures. This coalitional value is also called the Owen value. The Owen value can be seen as the two-step application of the Shapley value, which first regards the unions as players, uses the Shapley value for the first allocation between unions, and then uses the Shapley value for the second allocation within each union.
The Owen value (OWEN, 1977) is axiomatically characterized by employing efficiency, additivity, null player axiom, symmetry across unions and symmetry within unions. In 2007, KHMELNITSKAYA and YANOVSKAYA established another axiomatization via four axioms of efficiency, marginality, symmetry across unions, and symmetry within unions. Symmetry among unions requires that the unions with the same marginal contribution in quotient game should get the same total union payment. While symmetry within unions requires that if two players in the same union have the same marginal contribution, they should get the same payment. However, both symmetry across unions and symmetry within unions have the payment comparison between the players which have the same marginal contribution. But sometimes it is hard to achieve it in reality. For this reason, the two kinds of symmetries are weakened respectively in this paper. We propose weak symmetry across unions and weak symmetry within unions axioms for the axiomatizations of the Owen value. For the unions with the same marginal contribution in quotient game, weak symmetry across unions axiom requires the unions to obtain the total payment with the same sign. Similarly, for the players with the same marginal contribution in the same union, weak symmetry within the union only requires them to get the same sign of payment.
First of all, by using weak symmetry across unions and weak symmetry within unions axioms, we give an axiomatic characterization of the Owen value by combining efficiency, additivity and null player axiom. The axiomatization can be obtained by showing that the value with the above five properties must satisfy symmetry across unions and symmetry within unions. Furthermore, by replaced additivity and null player axiom with marginality, we provide an alternative characterization of the Owen value. Marginality states that if a player has the same marginal contribution in any two games, then he will receive the same payment in both games. In other words, marginality demands that a player’s payoff only depends on her own productivity. This axiom plays a significant role in the axiomatizations of values for TU-games and has been successfully applied to characterize a variety of values. In order to give the axiomatic characterization, we propose a lemma which plays a key role in the proof of the axiomatization. Finally, we show that the axioms involved in the two axiomatic characterizations of the Owen value presented in this paper are logically independent.

Key words: cooperative game, Owen value, weak symmetry across unions, weak symmetry within unions

摘要： Owen值是Shapley值在具有联盟结构的合作对策中的推广,联盟间对称性和联盟内对称性是Owen值唯一性刻画中的重要公理。联盟间对称性要求具有相同边际贡献的优先联盟获得相同的联盟总支付,联盟内对称性要求同一优先联盟内具有相同边际贡献的参与者获得相同的支付。然而,考虑到联盟间对称性和联盟内对称性在现实情景中的应用限制较强,本文将两种对称性分别进行了弱化,提出了联盟间弱对称性和联盟内弱对称性,弱化后的对称性只要求对称联盟或对称参与者具有相同符号的支付。本文首先利用弱化后的两种对称性结合有效性、可加性和零元性给出了Owen值的公理化刻画。其次,用边际性替代了可加性和零元性,利用有效性、边际性、联盟间弱对称性和联盟内弱对称性给出了Owen值的另一公理化刻画。

关键词: 合作对策, Owen值, 联盟间弱对称性, 联盟内弱对称性