How Do Coalitions Break Down? An Alternative View

We propose an alternative dynamic theory of coalition breakdown. Motivated by recent coalition-splitting events of unilateral country withdrawals, we assume that: (i) the payoff-sharing rule within coalitions is not necessarily set according to any optimality and/or stability criterion and (ii) players initially behave as if the coalition will last forever. If the sharing rule is non-negotiable or if renegotiation is very costly, compliance with these rules may become unbearable for a given member because the rule, being too rigid, would make exit preferable as time passes. We examine this endogenous exit problem in the case of time-invariant sharing rules. Assuming a Nash non-cooperative game after a (potential) split where players play Markovian strategies, we characterize the solutions of the endogenous exit problem in a linear-quadratic frame with endogenous splitting time. We find that splitting countries are precisely those that used to benefit the most from the coalition. Suitable sharing rules should be used to prevent coalition splitting. When initial pollution is high, all shares should be low enough and none of the players should receive a payoff share larger than 1/2. If initial pollution is low, we provide an explicit interval for sharing-rule values to prevent the collapse of the coalition. Finally, we demonstrate that the latter properties are qualitatively consistent with the optimal behaviors and equilibrium outcomes resulting from players anticipating the end of the coalition and acting accordingly.