Quadratic Vote Buying∗

A group of individuals with access to transfers seeks to make a binary collective decision. All known mechanisms they might use are either are often inefficient (e.g. voting), subject to severe collusion problems (e.g. the Vickrey-Clarke-Groves mechanism) or require the planner being informed about the distribution of valuations (e.g. the Expected Externality mechanism). I propose a simple, budget-balanced mechanism inspired by the work of Hylland and Zeckhauser (1979). Individuals purchase votes with the cost of a marginal vote being linear in the number of votes purchased; thus the total cost of votes is quadratic in the number purchased. The revenues earned from that individual are then refunded to other individuals. When there are a large number of individuals, this Quadratic Vote Buying mechanism is efficient in any Bayesian equilibrium under symmetric independent private values and is usually nearly efficient even with aggregate uncertainty. Collusion by a small group or individuals’ taking on (a small number of) multiple identities does not significantly reduce efficiency. Nota bene: Many of the results in this paper are conjectural, and full formal proofs of many results are missing. They will be added, hopefully, by mid-summer. ∗I am grateful to David Ahn, Eduardo Azevedo, Eric Budish, Drew Fudenberg, Jacob Goeree, Scott Duke Kominers, Steven Levitt, Paul Milgrom, Stephen Morris, Michael Ostrovsky, Azeem Shaikh and especially Roger Myerson, as well as seminar participants at the University of Chicago Booth School of Business, for helpful comments and for the financial support of the Institut D’Économie Industrielle and the Social Sciences Division at the University of Chicago. Tim Rudnicki, Daichi Ueda and especially Joshua Bosshardt supplied excellent research assistance. All errors, and I expect there are many in this draft, are my own. †Department of Economics, University of Chicago: 1126 E. 59th Street, Chicago, IL 60637 and Toulouse School of Economics, Manufacture de Tabacs, 21 allées de Brienne, 31000 Toulouse, France; [email protected], http://www.glenweyl.com.