Narrow Results

We show that, within the class of three-element unary algebras, there is a tight connection between a finitely based quasi-equational theory, finite rank, enough algebraic operations (from natural duality theory) and a special injectivity condition.

We give an infinite family of finite dualizable unary algebras that are not fully κ-dualizable, for all cardinals κ.

Fix a finite set M with at least three elements. We find uncountably many different clones on M, each of which is the clone of term functions of a strongly dualisable algebra. This provides a solution to the Finite Type Problem of natural duality theory: there are finite algebras that are dualisable but not via a structure of finite type.

Adverse selection is a version of the principal-agent problem that includes monopolist nonlinear pricing, where a monopolist with known costs seeks a profit-maximizing price menu facing a population of potential consumers whose preferences are known only in the aggregate. For multidimensional spaces of agents and products, Rochet and Choné (Econometrica, 1998) reformulated this problem as a concave maximization over the set of convex functions, by assuming agent preferences combine bilinearity in the product and agent parameters with a quasilinear sensitivity to prices. We characterize solutions to this problem by identifying a dual minimization problem. This duality allows us to reduce the solution of the square example of Rochet–Choné to a novel free boundary problem, giving the first analytical description of an overlooked market segment.