Abstract: More than 200 rivers around the world are shared by different countries and much more rivers flow through various regions. On the one hand, inhabitants or firms along the river can utilize the water natural resources. On the other hand, they may also discharge household or industrial waste into the rivers, which has a substantial influence on ecological environment. Therefore, how to share the cost of controlling a polluted river becomes a more and more important issue.
NI and WANG (2007) introduced two allocation methods, namely the Local Responsibility Sharing (LRS for short) method and the Upstream Equal Sharing (UES for short) method, which are based on the principles of Local Responsibility (LR) and Downstream Responsibility (DR) respectively. The LR principle requires that a polluter should take the full responsibility for the cost of cleaning river pollutants in the segment where it locates. The DR principle states that an upstream polluter bears some responsibilities for its downstream segments. Most existing allocation rules are based on the DR principle, but there are also allocation demands that contradict the principle in practice. For example, the southeastern regions of China (such as the Yangtze Delta and the Pearl River Delta) have benefited from the policy of reform and opening up and have achieved significant development. With the industrial upgrading of the Chinese economy, a large number of high-energy-consuming and high-polluting industrial have been relocated to the central and western region (upstream and midstream areas). Considering the historical sacrifices made by the central and western regions (mainly in the economic aspect), it is necessary to discuss compensation from downstream to upstream regions. Although the DES method proposed by DONG et al. (2012) partially reflects this requirement, its essence is still based on the DR principle, which addresses the dual problem of cost-sharing in water pollution.
To embody the compensation demands of downstream regions from upstream regions, this paper proposes two new allocation methods: SLRS (Solidarity Local Responsibility Sharing) and SUES (Solidarity Upstream Equal Sharing) solidarity methods. Both adopt downstream compensation principles, but the methods differ. The SLRS rule achieves compensation by reducing the costs of upstream regions, while the SUES rule directly allocates upstream costs to downstream regions. Furthermore, the equivalence between the two allocation rules and the Solidarity value is revealed. It is worth noting that the LRS and UES method are the Shapley value of two corresponding games induced by the LR and DR principles respectively. The null player property of the Shapley value in cost games results in the fact that null players do not have to bear any costs. NOWAK and RADZIK (1994), based on the potential social or psychological factors in the game (often difficult to quantify), proposed the Solidarity value. Unlike the Shapley value, the Solidarity value provides a certain allocation to zero players. NOWAK and RADZIK (1996) pointed out that if there are friendly relationships among game participants that cannot be characterized by characteristic functions (possessing solidarity-related social attributes), the Solidarity value is more likely to be accepted by the participants. The two methods proposed in this paper are equivalent to the Solidarity value, and therefore SLRS and SUES are referred to as solidarity allocation methods.
To characterize the fairness of the two new allocation methods, we propose three properties. Nonzero dominance states that if a player is the only one with nonzero cost, then the withdraw of this player has a greater impact on other players. Most upstream nonzero dominance keeps the dominance of the unique nonzero cost player only when the players is located on the most upstream position. Zero balanced contribution implies that the withdrawal of zero cost players brings the same influence to other players if there exists a unique nonzero cost segment. To prove the uniqueness that the SLRS and SUES method satisfies certain properties respectively, we first consider a basis of original problem space and then decompose the problem into n sub-problems. By using mathematical inductions on the size of coalitions, we show the uniqueness on sub-problems. Together with additivity, we characterize the proposed allocation methods with different properties.
Further research can be conducted on the mechanism design method of allocation methods, by constructing non-cooperative game model and utilizing equilibrium solutions to achieve stable allocation methods for water pollution control cost allocation problems.

Key words: cost allocation, cooperative game, Solidarity value, allocation method, axiomatic characterization

摘要： 水污染治理成本分摊问题受到越来越多的关注,寻求公平合理的分配规则是该问题的关键,合作博弈为解决此类问题提供了有效的理论工具与方法指导。现有分配规则大都基于DR(Downstream Responsibility)原则,即上游区域承担其下游区域的部分治理成本,但实际中亦存在与之对应的分配需求,即下游污染主体也要承担其上游区域污染物的部分治理成本。为了分析和描述此类特征,本文提出了水污染治理成本分摊问题两种新的分配规则:SLRS(Solidarity Local Responsibility Sharing)和SUES(Solidarity Upstream Equal Sharing)团结分配规则,并揭示了两种规则与相关合作博弈Solidarity值的等价性。为了刻画两种团结分配规则的公平合理性,提出非零元支配性、最上游非零元支配性、零元均衡贡献性对SLRS和SUES团结分配规则进行了公理化刻画。本文提出的两种分配规则,对具有线性合作结构的成本分摊问题具有一定的启示意义和理论价值。

关键词: 成本分摊, 合作博弈, Solidarity值, 分配规则, 公理化