Coalition formation: a game-theoretic analysis

There are many examples of individuals forming coalitions to obtain or protect a valuable resource. We present an analytical model of coalition formation in which individuals seek alliances if they judge themselves too weak to secure the resource alone. We allow coalition seeking to carry an investment cost (theta) and let contest outcomes depend probabilistically on the relative fighting strengths of contesting parties, with effective coalition strength directly proportional to combined partner strength. We identify the evolutionarily stable strength thresholds, below which individuals within triads should seek a coalition. We show that if theta exceeds a critical value, then unilateral fighting over resources is an evolutionarily stable strategy (ESS). Universal (3-way) coalitions are also an ESS outcome if theta is less than a second critical value. Both of these extreme solutions are less likely to arise, the greater the variance in fighting strengths and the greater the benefit from dominating opponents. Our analysis also identifies intermediate solutions in which only the weaker individuals seek coalitions: only then can a true coalition (2 vs. 1) form. We characterize these ESSs and show that true coalitions are more likely to arise when the effective strength of a coalition is less than the sum of its individual strengths (antergy). Alliances in primates are characterized by antergy, high reliability of strength as a predictor of contest outcome, and high variability in strengths. These are precisely the conditions in which in our model most favors true coalition formation.