Constrained Majorization: Applications in Mechanism Design

Abstract

Classical frameworks in mechanism design often specify an objective function and maximize it by choosing an allocation rule. We extend these frameworks by allowing maximizing an objective function (such as allocative efficiency in allocation of a scarce public resource) subject to additional constraints (such as guarantees for the Gini index or welfare). The additional complexity arising in the optimal mechanism due to each additional constraint corresponds to at most one jump discontinuity in intervals where agent types are pooled. In the case of allocating a scarce resource, this complexity corresponds to one additional price-quantity option on the menu of optimal mechanism. The framework includes other applications such as auction and contract design.

By varying the right-hand side of a constraint, we can also use this framework for comparative statics analysis of the mechanisms on a Pareto frontier. For example, in the case of allocating a scarce resource, we find a monotonic relation between equality and welfare on the efficiency-welfare Pareto frontier: Mechanisms that achieve higher welfare also achieve lower inequality, in the sense of Lorenz order. A direct corollary is that the Gini index preserves the Lorenz order on the Pareto frontier.

Keywords: Mechanism design, Majorization