One man, one bid

Abstract

We compare two mechanisms to implement a simple binary choice, e.g. adopt one of two proposals. We show that when neither alternative is ex ante preferred, simple majority voting cannot implement the first best outcome. The fraction of the surplus lost rises with the number of voters and the total surplus loss diverges in the limit when the number of voters grows large. We introduce a simple bidding mechanism where votes can be bought at a quadratic cost and voters receive rebates equal to the average of others' payments. This mechanism is budget-balanced, individually rational, and fully efficient in the limit. Moreover, the mechanism redistributes from those that gain from the outcome to those that lose and everyone is better off under bidding compared to voting. We test the two mechanisms in the lab using an environment with "moderate" and "extremist" voters. In the first part of the experiment, subjects gain experience with both bidding and voting. Then they collectively decide which mechanism applies in the second part. This endogenous choice of institutions provides clear evidence in favor of bidding: with groups of size eleven, 90% of the groups opt for the bidding mechanism. The observed efficiency losses under voting are close to theoretical predictions and significantly larger than under bidding. Because of redistribution, the efficiency gain from bidding benefits mostly the moderate voters. Observed behavior under the bidding mechanism deviates from theoretical predictions to some extent, which can be explained by a quantal response equilibrium model if we assume that voters overestimate the chance of being pivotal. These deviations hardly reduce any of the efficiency-improving effect of the bidding mechanism, which supports its adoption in real-world applications.

The relationship between the normalized gradient addition mechanism and quadratic voting

“…We are heartened to see the research interest that QV has justifiably garnered (as evidenced, for example, by this special issue). In our view, an important next step is more empirical tests of these mechanisms, like the experiments reported by Cárdenas et al (2014), Goeree and Zhang (2016), and Quarfoot et al (2016).…”

“…Because of concerns that QV favors the rich (Laurence and Sher 2016), that in practice the mechanism will not generate efficient outcomes if people mistakenly spend the wrong amount of money on voting (Kaplow and Kominers 2016), and that implementing it with actual monetary vote buying may often be impractical, there has been interest in implementations of QV in which people buy votes with tokens (instead of money) that are equally allocated by the mechanism designer. Moreover, the experiments with QV to date that we are aware of (Cárdenas et al 2014; Goeree and Zhang 2016; Quarfoot et al 2016) have been conducted with such equally allocated (though at times monetarily valuable) tokens. As far as we are aware, however, there has been no prior formal analysis of QV with tokens.…”

Radical Markets by Eric Posner and E. Glen Weyl: A Review Essay

“…As QV is a brainchild of one of the authors of the book and as the book does not explain the idea particularly well (mostly with a table showing what the squares of numbers look like), I turned to scientific literature to see what was actually being proposed. Here is what I gleaned from Goeree and Zhang (2017) and Lalley and Weyl (2018). Suppose you can buy voting credits at a utility cost equal to the square of the credits purchased, that there are no liquidity constraints, and that there are k items to vote on with item j worth v j i to voter i and that ϵ j is the increased probability of winning that comes with a single credit.…”

Does Vote Trading Improve Welfare ?

“…for some constant κ , 40 so that the assumption of constant marginal pivotality is approximately satisfied. The first order condition yields 39 We sketch here the argument of Goeree & Zhang (2017) for an arbitrary normalization of the cost function (the parameter κ), and thus introduce the proportionality parameter α N . 40 We have…”