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Quadratic Voting in Finite Populations

34 Pages Posted: 28 Feb 2015 Last revised: 22 May 2017

Bharat Chandar

Oracle Corporation

E. Glen Weyl

Microsoft Research; Yale University

Date Written: May 21, 2017


We study the performance of the Quadratic Voting (QV) mechanism proposed by Lalley and Weyl (2016) in finite populations of various sizes using three decreasingly analytic but increasingly precise methods with emphasis on examples calibrated to the 2008 gay marriage referendum in California. First, we use heuristic calculations to derive conservative analytic bounds on the constants associated with Lalley and Weyl’s formal results on large population convergence. Second, we pair numerical game theory methods with statistical limit results to approximate equilibria for moderate population sizes. Finally, we use purely numerical methods to analyze small populations. The more precise the methods we use, the better the performance of QV appears to be in a wide range of cases. In our most precise results, we have not found an example where QV sacrifices more than 10% of potential welfare for any population size. However, convergence to full efficiency in large populations may be much slower with fat tails than with bounded support.

Keywords: Quadratic Voting, small populations, analytic approximations, computational game theory

JEL Classification: D47, D61, D71, C72, D82, H41, P16

Suggested Citation

Chandar, Bharat and Weyl, E. Glen, Quadratic Voting in Finite Populations (May 21, 2017). Available at SSRN: or

Bharat Chandar

Oracle Corporation ( email )

500 Oracle Parkway
Redwood Shores, CA 94065
United States

Eric Glen Weyl (Contact Author)

Microsoft Research ( email )

One Memorial Drive
Cambridge, MA 02142
United States
(857) 998-4513 (Phone)


Yale University ( email )

28 Hillhouse Ave
New Haven, CT 06520-8268
United States

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