A key step in the derivation of Chapter 4's equilibrium existence theorem was to establish that the assumptions (in the model specified in Section 4.2) were sufficient for each candidate's payoff function to be concave in her possible strategies. Then, in deriving the theorem that identified the implicit social objective function that is maximized by these equilibria, a key step was to establish that the assumptions (in the model specified in Section 4.2) were sufficient for the implicit social objective function to be concave. What is more, concavity for a (social or individual) objective function is also of inherent interest in and of itself; as Avriel et al. (1981, p. 24) put it, “In utility theory, the definition of concavity is precisely equivalent to the notion that an individual would never prefer an actuarially fair gamble.”

The assumption in Section 4.2 that played the greatest role in the derivation of the (intermediate) “concavity results” was, not surprisingly, the assumption that, for each group θ ∈ Θ, the corresponding scaling function f(s | θ) is concave on S. Because of the important role that this assumption played in the derivation of those key concavity results, we now concern ourselves with the possibility of identifying alternative conditions on the groups' scaling functions, which (when all of the other assumptions in Section 4.2 are maintained) also assure either that the society's implicit social objective function is necessarily concave or that the candidates' objective functions are necessarily concave. The approach used to obtain these alternative conditions is described in Section 5.1.