A duality theorem on a pair of simultaneous functional equations

Given P and Q convex compact sets in R^kandR^s, respectively, and u a continuous real valued function on P × Q, we consider the following pair of dual problems: Problem I-Minimize f{hook} so that f{hook}: P × Q → R and f{hook} ≥ Cav_pVex_q × max(u, f{hook}). Problem II-Maximize g so that g: P × Q → R and g ≤ Vex_q × Cav_pmin(u, g). Here Cav_p is the operation of concavification of a function with respect to the variable p ε{lunate} P (for each fixed q ε{lunate} Q). Similarly, Vex_q is the operation of convexification with respect to q ε{lunate} Q. Maximum and minimum are taken here in the partial ordering of pointwise comparison: f{hook} ≤ g means f{hook}(p, q) ≤ g(p, q) ∀(p, q) ε{lunate} P × Q. It is proved here that both problems have the same solution which is also the unique simultaneous solution of the following pair of functional equations: (i) f{hook} = Vex_qmax(u, f{hook}). (ii) f{hook} = Cav_pmin(u, f{hook}). The problem arises in game theory, but the proof here is purely analytical and makes no use of game-theoretical concepts.