A Geometric Approach to Mechanism Design

University of Zurich Department of Economics Working Paper No. 56

34 Pages Posted: 21 Dec 2011 Last revised: 12 Jun 2013

See all articles by Jacob K. Goeree

Jacob K. Goeree

University of Zurich

Alexey Kushnir

Carnegie Mellon University - David A. Tepper School of Business

Date Written: June 5, 2013

Abstract

We develop a novel geometric approach to mechanism design using an important result in convex analysis: the duality between a closed convex set and its support function. By deriving the support function for the set of feasible interim values we extend the well known Maskin--Riley--Matthews--Border conditions for reduced-form auctions to social choice environments. We next refine the support function to include incentive constraints using a geometric characterization of incentive compatibility. Borrowing results from majorization theory that date back to the work of Hardy, Littlewood, and P\'olya (1929) we elucidate the "ironing" procedure introduced by Myerson (1981) and Mussa and Rosen (1978). The inclusion of Bayesian and dominant strategy incentive constraints result in the same support function, which establishes equivalence between these implementation concepts. Using Hotelling's lemma we next derive the optimal mechanism for any social choice problem and any linear objective, including revenue and surplus maximization. We extend the approach to include general concave objectives by providing a fixed-point condition characterizing the optimal mechanism. We generalize reduced-form implementation to environments with multi-dimensional, correlated types, non-linear utilities, and interdependent values. When value interdependencies are linear we are able to include incentive constraints into the support function and provide a condition when the second-best allocation is ex post incentive compatible.

Keywords: Mechanism design, convex set, support function, duality, majorization, ironing, Hotelling's lemma, reduced-from implementation, BIC-DIC equivalence, concave objectives, interdependent values, second-best mechanisms

JEL Classification: D44

Suggested Citation

Goeree, Jacob K. and Kushnir, Alexey I., A Geometric Approach to Mechanism Design (June 5, 2013). University of Zurich Department of Economics Working Paper No. 56, Available at SSRN: https://ssrn.com/abstract=1974922 or http://dx.doi.org/10.2139/ssrn.1974922

Jacob K. Goeree (Contact Author)

University of Zurich ( email )

Rämistrasse 71
Zürich, CH-8006
Switzerland

HOME PAGE: http://www.econ.uzh.ch/faculty/jgoeree.html

Alexey I. Kushnir

Carnegie Mellon University - David A. Tepper School of Business ( email )

5000 Forbes Avenue
Pittsburgh, PA 15213-3890
United States

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