Abstract: A cooperative game with transferable utility(TU-game) assumes that all players can communicate with one another and focus on how to rationalize the distribution of group benefits. TU-game assumes that all possible coalitions are feasible in the classical TU-game. In practice, however, many collaborations do not really happen due to geographical, communication technology or organizational influences.We therefore need to place limits on cooperation, and there are two main types of constraint structures in cooperative games. Myerson (1977) introduced restrictions to the communication among players through a graph, and claimed that players can cooperate only when there is a link between them. Aumann (1974)constructed a cooperative game with coalition structures, and declared that only players belong to a same priori union can they communicate. Usually, we call the allocation in a cooperative game the value. The Shapley value (Shapley, 1953) is the most famous value in TU-game, claiming that benefits should be distributed according to marginal utility. However, the equal division value divides the benefits equally, and believes that equal distribution makes cooperation possible.
   China built a large number of multi-storey buildings from the 1970s to 1980s, and due to technical and financial pressures, most of them were not fitted with elevators. In order to improve the quality of life of the residents and solve thetravel difficulties of the elderly, governments accelerate the process of installing elevators in existing multi-storey residential buildings. There are many factors that affect the installation work, and the most controversial one is the cost sharing scheme. Because the residents on different floors have different demands for elevators, the residents on lower floors often do not agree to install elevators or share a lot of costs, which has seriously delayed the progress of installing elevators. Therefore, this paper proposes a weight system to describe this demand difference, or named allocation difference. Considering the cooperative game with coalition structures, as residents on the same floor often have the same interests, we take the residents on the same floor as a priori union. In this way, the problem with installing elevators in existing multi-storey residential buildings can be reduced to a weighted cooperative game model with coalition structures.
   Based on this model, this paper proposes an allocation method called “weighted division value”. We use the four properties of additivity, weighted symmetry within unions, weighted symmetry among unions and nullifying player property to characterize this value. Then we apply this value to the cost sharing scheme of installing elevators in existing multi-storey residential buildings in Shanghai. When the appropriate weights are taken, the allocation rule implies the guiding standard for cost sharing of owners in existing multi-storey residential buildings in most cities of China, which provides an accurate theoretical basis for the perfection, promotion and implementation of this standard. The results show that residents on lower floors will share less under a staggered entry and those on higher floors will share less under a flat entry, which explains the reason for disputes among residents over different methods of installation. And if we only use floorage as the basis for determining weights, there will also be a large cost difference between residents on the same floor, which is likely to undermine the original union structure. Finally, the “weighted distribution value” proposed in this paper can also be used in the allocation of enterprise performance bonus, stock dividend and other construction costs of facilities (wells, monitoring, fitness equipment).

Key words: cooperative game, allocation rule, coalition structure, weighted distribution value, existing multi-storey residential building, install elevator

摘要： 本文结合我国当前城市既有多层住宅加装电梯费用分摊问题的实际,引入了具有联盟结构的赋权合作博弈模型。在该类博弈中,参与者将依据现实情况进行结盟,形成所谓的“优先联盟”。在此基础上,提出了被称为权分值的分摊规则,并证明了它可以由可加性、联盟内比例对称性、联盟间比例对称性和空化参与者性四个公理唯一确定。作为这个合作博弈模型的应用,本文将我国城市既有多层住宅加装电梯费用分摊问题可归结为具有联盟结构的赋权合作博弈模型,其中业主为参与者,而每一层的参与者们可看作一个优先联盟。当取适当的权值时,该分摊规则涵盖了我国大部分城市出台的既有多层住宅加装电梯业主费用分摊的指导标准,这为这些标准的完善、推广和实施提供了合理的理论依据。

关键词: 合作博弈, 分配规则, 联盟结构, 权分值, 既有多层住宅, 加装电梯