Abstract: TIJS (1981) proposed the famous τ-value on the quasi-balanced games based on the marginal contribution of players to the grand coalition. Significantly, this value is an effective compromise value between the maximum and minimum potential benefits of players in TU games. We take the situations, in real life (especially economic activities), in which players are unable to cooperate with others due to conflicts or their cooperative relationships are limited by coalition structures into account. CASAS-MÉNDEZ(2003)extended τ-value to cooperative games with coalition structures. In their research on this value, LI Dengfeng and HU Xunfeng (2017) urther extended the τ-value to cooperative games with level structures. WU Meirong et al. (2014),YANG Dianqing and LI Dengfeng (2016) respectively studied the τ-values on bicooperative quasi-balanced games and fuzzy cooperative games.
In the study of cooperative game’s solution, it is not only necessary to propose the concept of corresponding solution and provide mathematical expression as much as possible, but more importantly to characterize the fairness and rationality of solution. Although the scholars have defined the τ-value under different game models and provided the axiomatic conclusions, the axiomatic properties used in those conclusions are all variations of the criteria used by TIJS (1987). The ideas of subsequent conclusions is completely similar to the former. Therefore, the new axiomatic method for exploring this value has important theoretical significance. The axiomatization research of classical cooperative game solutions such as Shapley value has produced rich results, and although the vast majority of axiomatization studies have proposed new axiomatic properties, they often need to combine with efficiency to achieve the purpose of characterizing uniqueness of the Shapley value. BÉAL et al. (2015) defined the covariance and invariance of solutions and used the direct-sum decomposition of game space to provide an axiomatized conclusion for Shapley value. The interesting aspects of this conclusion are that they abandon the classical property of effectiveness and provide a new algebraic base for the TU games space.
In this paper, inspired by Béal’s ideas, we first prove that all uniform games constitute a set of algebraic bases for the quasi-balanced games space, and give a direct-sum decomposition of the linear space of the quasi-balanced games. Secondly, based on the individual covariance (BÉAL, 2015), the new axiom property, synergistic covariance, of cooperative game solution are introduced, and we obtain a new axiomatic characterization of τ-value by using the direct-sum decomposition theory of Euclidean space and the classical axiom of maximum concession to proportionality (TIJS, 1987). Finally, another axiomatic conclusion of τ-value is given by using inessential game and individual covariance instead of synergistic covariance.

Key words: quasi-balanced games, τ-value, inessential games, decomposition, synergistic covariance

摘要： 文章首先证明所有全体一致对策构成了准均衡对策空间的一组代数基,并给出准均衡对策空间的一个直和分解结果。其次,基于个体共变性(BÉAL,2015)提出了合作对策博弈解新的公理性质,即协同共变性,并结合经典公理最大让步成比例性(TIJS,1987),利用欧式空间的直和分解理论给出τ值的一种新公理化刻画。最后,利用非本质对策性与个体共变性代替协同共变性给出τ值的另一公理化结论。

关键词: 准均衡对策, τ值, 非本质对策, 直和分解, 协同共变性