Abstract: The Banzhaf value and Shapley value are two famous allocation rules in cooperative games with transferable utility, both of which determine the payment of the change participant by the marginal expectation of each player to all coalitions. The difference is that the Banzhaf value originates from the voting game, which assumes that players have the same probability of joining any coalition of any size, while the Shapley value only assumes that players have the same probability of joining any coalition of the same size. In addition, both Banzhaf value and Shapley value assume that any player can form a feasible alliance. However, the cooperation between players is affected by factors such as geography and intimacy. Players with closer cooperation in the same region are more likely to form a priori union. In 1974, Aumann and Dréze first studied the cooperative games with coalition structures. They define each coalition structure as a partition of the grand coalition, and each subset in the partition is called a priori union. In 1977, Owen assumed that any priori union could cooperate with all or part of the players in other priori unions. On this basis, he proposed the famous Owen value. The Owen value is to first assign the Shapley value to each priori union, and then use the Shapley value to perform secondary allocation within each priori union. Then, based on the Banzhaf value, Owen gives another allocation rule in cooperative games with coalition structures, which is called Banzhaf-Owen value. In 2009, Kamijo considered a situation different from Owen’s assumption that players in each priori union can only cooperate with other priori unions as a whole. Under this assumption, he proposes a new allocation rule in cooperative games with coalition structures, called Ka value. In order to distinguish, he refers to the allocation rules based on the Owen’s assumption as the coalition value, and the allocation rules under this assumption are collective value.
In fact, we often need to consider the collective value under the Kamijo’s assumption. For example, a company holds a general meeting of shareholders to make decisions. Due to the different number of shares held by each shareholder, the corresponding voting power is also different. Some shareholders with the same or similar ideas will form the priori union in order to achieve a certain purpose. The priori unions are all involved in decision-making as a whole, with more shares to improve the bargaining power in decision-making or negotiation to obtain more benefits, and then distribute vested interests within the priori union. At this time, we can consider using the collective value to estimate the power index of each shareholder. In order to better estimate the power index of different participants in such problems, as the Banzhaf value originates from the voting game, it is particularly important to define the Banzhaf value with coalition structure under the Kamijo’s assumption.
In this paper, based on that there may exist a coalition structure formed by priori unions among all participants, a new Banzhaf value with a coalition structure, called the C-Banzhaf value, is introduced into cooperative games with coalition structures. First, we show that the C-Banzhaf value is uniquely determined by pairwise merging of partition and standardness. Secondly, by the decision-making process of a company’s general meeting of shareholders as an example, the C-Banzhaf value is applied to analyze the power index of each shareholder and compared with other values. The results show that the C-Banzhaf value is a good power evaluation method when shareholders are more inclined to form the priori union to seek more benefits with their strong bargaining power.

Key words: cooperative games, coalition structures, Banzhaf value, collective value

摘要： Banzhaf值是(效用可转移)合作对策中一个著名的分配规则,它起源于投票对策,而后被Owen推广到一般合作对策中。经典的Banzhaf值以每个参与者对所有联盟边际贡献的期望值决定该参与者的支付,并假设任何参与者间均可达成合作形成可行联盟。然而,现实参与者间的合作受地域、亲疏关系等因素的影响,相同地域或历史合作关系更加紧密的参与者间更容易达成合作,形成优先联盟。在合作对策中,考虑参与者之间可能存在由优先联盟形成的联盟结构,本文定义了一种新的具有联盟结构的Banzhaf值,称为C-Banzhaf值。首先,证明了C-Banzhaf值可以由分割内合并性和标准性所唯一刻画。其次,以某公司股东大会进行决策为例,利用C-Banzhaf值分析了各股东的权力指数,并与其他值做了对比分析。

关键词: 合作对策, 联盟结构, Banzhaf值, 集体值