Optimal coalition splitting with heterogenous strategies

We consider a group of players initially members of a coalition managing cooperatively a public bad, in this case, the stock of pollution. Countries are technologically heterogeneous but the pollution damage is uniform. We essentially attempt to characterize the conditions under which a country may eventually split and when it splits within an infinite horizon multi-stage differential game. In contrast to the existing literature, we do not assume that after splitting, the splitting player and the remaining coalition will adopt Markovian strategies. Instead, we assume that the latter will remain committed to the collective control of pollution and play open-loop, while the splitting player plays Markovian. Within a full linear-quadratic model, we characterize the optimal strategies. We later compare with the outcomes of the case where the splitting player and the "remaining" coalition play both Markovian. We highlight several interesting results in terms of the implications for longterm pollution levels and the duration of coalitions with heterogenous strategies.