Why and when coalitions split? An alternative analytical approach with an application to environmental agreements

We use a parsimonious two-stage differential game setting where the duration of the first stage, the coalition stage, depends on the will of a particular player to leave the coalition through an explicit timing variable. By specializing in a standard linear-quadratic environmental model augmented with a minimal constitutional setting for the coalition (payoff share parameter), we are able to analytically extract several nontrivial findings. Three key aspects drive the results: the technological gap as an indicator of heterogeneity across players, the constitution of the coalition and the intensity of the public bad (here, the pollution damage). We provide with a full analytical solution to the two-stage differential game. In particular, we characterize the intermediate parametric cases leading to optimal finite time splitting. A key characteristic of these finite-time-lived coalitions is the requirement of the payoff share accruing to the splitting country to be large enough. Incidentally, our two-stage differential game setting reaches the conclusion that splitting countries are precisely those which use to benefit the most from the coalition. Constraining the payoff share to be low by Constitution may lead to optimal everlasting coalitions only provided initial pollution is high enough, which may cover the emergency cases we are witnessing nowadays.