A geometric approach to mechanism design

An important result in convex analysis is the duality between a closed convex set and its support function. We exploit this duality to develop a novel geometric approach to mechanism design. For a general class of social choice problems we characterize the feasible set, which is closed and convex, and its support function. We next provide a geometric interpretation of incentive compatibility and refine the support function to include incentive constraints using arguments from majorization theory. The optimal mechanism can subsequently be derived from the support function using Hotelling's lemma. We first assume that values are linear in types and types are independent, private, and one-dimensional. For this environment we provide a simple geometric proof that Bayesian and dominant strategy implementation are equivalent by showing that the feasible sets that remain after imposing either type of incentive constraints coincide. Furthermore, we derive the optimal mechanism for any social choice problem and any linear objective, including revenue and surplus maximization. As an illustration, we determine the optimal multi-unit auction for a class of value functions that exhibit decreasing marginal valuations. Other types of constraints, such as capacity constraints and budget balancedness, can be interpreted geometrically as well, which facilitates a unified approach to a range of social choice problems, including auctions, bargaining, and public goods provision. We discuss how our geometric approach extends to environments with value interdependencies, non-linear valuations, and correlated or multi-dimensional types. Specifically, we illustrate that with interdependent valuations the equivalence between Bayesian and dominant strategy implementation breaks down, and our approach naturally produces the second-best outcomes for both types of incentive constraints.