A class of extremum problems related to agency models with imperfect monitoring

The cost minimization problem in an agency model with imperfect monitoring is considered. Under the first order approach, this can be stated as a convex minimization problem with linear inequality and equality constraints in a generally infinite dimensional function space. We apply the Fenchel Duality Theorem, and obtain as a dual problem a concave maximization problem of finite dimension. In particular, a Lagrange multiplier description of the optimal solution to the cost minimization problem is derived, justifying and extending thus the approach of Kim. By the duality, the dependence of the minimum cost value on the information system used becomes particularly visible. The minimum cost value behaves monotonically w.r.t. the convex ordering of certain distributions induced by the competing information systems. Under the standard inequality constraint, one is led to the distributions of the score functions of the information systems and their convex order relation. It is shown that also for multivariate actions, Blackwell sufficiency implies the convex order relation of the score function distributions. A further result refers to a multi-agents model recently considered by Budde, when the maximum of n independent and identically distributed (i.i.d.) univariate output variables is focused. If two univariate information systems have monotone likelihood ratios, then the convex ordering between the two score function distributions implies the weaker convex increasing ordering between the distributions of the same score functions under the maximum distributions.