Players cooperating in a line is a special while essential phenomenon in real life collaborating activities such as assembly line production, pipeline supply chain management and other streamlining operational settings. In this paper, we study the scenario of cooperative sewage discharge with multiple participants positioning in a line along a river such that the optimization decision and cooperation strategy are mutually affected by both upstream and downstream players. We make three main contributions accordingly: Firstly, we formalize the sewage discharge problem (SDP) for different groups of players, and use greedy strategy and dynamic programming to design the optimal algorithms to solve the SDP in polynomial time. Secondly, we show that the cooperative game defined on sewage discharge problem, referred to as SDG, has a non-empty core due to its special line-positioning structure. Therefore, a grand stable cooperation is guaranteed. Furthermore, inspired by the fact that the SDG is core non-empty while non-convex, we successfully identify a relaxed concept of convexity -- directional-convexity, which can also serve as a sufficient condition for a cooperative game having a non-empty core.

翻译：沿河线性分布的参与者合作排污是现实协作活动（如流水线生产、管道供应链管理及其他流线化运营场景）中一种特殊且重要的现象。本文研究了多个参与者沿河线性分布的协作排污场景，其中上下游参与者的优化决策与合作策略相互影响。我们据此做出三项主要贡献：首先，针对不同参与者群体形式化定义了污水排放问题（SDP），并采用贪心策略与动态规划设计了多项式时间内求解SDP的最优算法；其次，证明基于污水排放问题定义的合作博弈（SDG）因其特殊的线性分布结构而具有非空核，从而保证了大联盟稳定合作的存在；最后，受SDG核非空但非凸这一事实启发，我们成功识别出一种松弛化的凸性概念——方向凸性，该条件同样可作为合作博弈具有非空核的充分条件。