Finding core for coalition structure utilizing dual solution

When forming the grand coalition is not possible or optimal, agents need to create a coalition structure. The idea of the core can be extended to such a case. In this paper, we propose an innovative exact algorithm called CoreD to check core-non-emptiness for coalition structures. A more straightforward exact algorithm based on existing techniques, which we call CoreP, first obtains the value of optimal coalition structure by solving an integer programming problem. Then, it checks whether that value can be divided without making a blocking (dissatisfied) coalition. In contrast, CoreD first finds a minimal value of the optimal coalition structure so that there exists no blocking coalition. Next, it checks whether the optimal value equals the minimal value We empirically show that when the core is empty, CoreD is by far superior to CoreP. Also, to find a second-best payoff vector when the core is empty, we propose a new solution concept called the weak ε-core⁺, which can utilize the approximate value of the optimal coalition structure. Based on the idea of CoreD, we further develop an algorithm for checking the non-emptiness of the weak ε-core⁺.