Barrett's paradox of cooperation in the case of quasi-linear utilities

Abstract

This paper fits in the theory of international agreements by studying the success of stable coalitions of agents seeking the preservation of a public good. Extending Baliga and Maskin, we consider a model of $N$ homogeneous agents with quasi-linear utilities of the form $u_j(r_j;r)=r^{\alpha }-r_j$, where $r$ is the aggregate contribution and the exponent $\alpha$ is the elasticity of the gross utility. When the value of the elasticity $\alpha$ increases in its natural range $\left(\mathrm{0,1}\right)$, we prove the following five main results in the formation of stable coalitions: (i) the gap of cooperation, characterized as the ratio of the welfare of the grand coalition to the welfare of the competitive singleton coalition grows to infinity, which we interpret as a measure of the urge or need to save the public good; (ii) the size of stable coalitions increases from 1 up to $N$; (iii) the ratio of the welfare of stable coalitions to the welfare of the competitive singleton coalition grows to infinity; (iv) the ratio of the welfare of stable coalitions to the welfare of the grand coalition “decreases” (a lot), up to when the number of members of the stable coalition is approximately $N/e$ and after that it “increases” (a lot); and (v) the growth of stable coalitions occurs with a much greater loss of the coalition members when compared with free-riders. Result (v) has two major drawbacks: (a) A priori, it is difficult to “convince” agents to be members of the stable coalition and (b) together with results (i) and (iv), it explains and leads to the “pessimistic” Barrett's paradox of cooperation, even in a case not much considered in the literature: The ratio of the welfare of the stable coalitions against the welfare of the grand coalition is small, even in the extreme case where there are few (or a single) free-riders and the gap of cooperation is large. “Optimistically,” result (iii) shows that stable coalitions do much better than the competitive singleton coalition. Furthermore, result (ii) proves that the paradox of cooperation is resolved for larger values of $\alpha$ so that the grand coalition is stabilized.