Generalized consistent ranking and the formation of self-enforcing coalitions

Agents endowed with powers compete for a divisible prize by forming coalitions. When a coalition wins, all non-members are eliminated. The winning coalition then divides the prize among its members according to a given sharing rule. We investigate the case where the sharing rule satisfies a property we call consistent ranking. Consistent ranking ensures that agents’ rankings of competing coalitions coincide. Sharing rules such as equal and proportional sharing satisfy this property. We also examine a larger class of sharing rules that satisfy a property we call generalized consistent ranking where agents can rank coalitions even though the sharing rule does not satisfy consistent ranking. For instance, a convex combination of equal and proportional sharing, which we call combination sharing, violates consistent ranking but satisfies generalized consistent ranking under certain conditions. For these different sharing rules, we characterize rules on choosing coalitions (called transition correspondence) that satisfy two main axioms: self-enforcement, which requires that no further deviation happens after a coalition has formed, and rationality, which requires that agents pick the coalition that gives them their highest payoff. We find that a transition correspondence that satisfies self-enforcement and rationality always exists for sharing rules that satisfy generalized consistent ranking (and hence, consistent ranking). In the case of combination sharing, one way to satisfy generalized consistent ranking is to restrict the configuration of powers in society to satisfy size-power monotonicity, where larger coalitions have higher powers.