Quadratic Voting

Steven P. Lalley

Department of Statistics, University of Chicago

Glen Weyl

Microsoft Research New England; University of Chicago


While the one-person-one-vote rule often leads to the tyranny of the majority, alternatives proposed by economists have been complex and fragile. By contrast, we argue that a simple mechanism, Quadratic Voting (QV), is robustly very efficient. Voters making a binary decision purchase votes from a clearinghouse paying the square of the number of votes purchased. If individuals take the chance of a marginal vote being pivotal as given, like a market price, QV is the unique pricing rule that is always efficient. In an independent private values environment, any type-symmetric Bayes-Nash equilibrium converges towards this efficient limiting outcome as the population grows large, with inefficiency decaying as 1/N. We use approximate calculations, which match our theorems in this case, to illustrate the robustness of Quadratic Voting, in contrast to existing mechanisms. We discuss applications in both (near-term) commercial and (long-term) social contexts.

Gauging The Efficiency of Quadratic Voting – Introduction

Prohibitions on gay marriage seem destined to be remembered as classic examples of the “tyranny of the majority” that has plagued democracy since the ancient world. While in many countries a(n increasingly narrow) majority of voters oppose the practice, the value it brings to those directly affected seems likely to be an order of magnitude larger than the costs accruing to those opposed, as we discuss in greater detail in Section 2. However, one-person-one-vote (1p1v) offers no opportunity to express intensity of preference, allowing such inefficient policies to persist. While most developed countries have institutions, such as independent judiciaries and log-rolling, that help protect minorities, these are often slow, insufficient and plagued with their own inefficiencies. To address these limitations, in this paper we analyze a simple and robust mechanism, Quadratic Voting (QV), that we believe may offer a practical paradigm for collective decision-making.

The basic problem is that 1p1v rations rather than prices votes, resulting in externalities across individuals. This contrast with the market mechanisms for allocating private goods where individuals pay the opportunity cost of their purchases, leading to social efficiency by the classic arguments of Smith (1776). We therefore, in Section 3, consider a simple class of costly voting rules under which individuals can purchase any continuous number of votes they wish using a quasi-linear numeraire. To study such rules, we propose a “price-taking” model where individuals take as given the vote-price of influence, the number of votes it takes to have a unit of influence on the outcome. This price for short plays the same role in coordinating behavior that prices do in a standard market for private goods. Optimization given price-taking, together with a fixed total supply of influence, inspired by general statistical limits on influence as derived by Al-Najjar and Smorodinsky (2000), and an essentially arbitrary rule for returning funds thus raised, define a price-taking equilibrium.

We show that for any convex vote costs there is a unique equilibrium. Limiting cases yield familiar predictions: in (nearly) linear vote buying equilibrium approaches the dictatorship of the single individual with the most intense preference typically derived from linear vote buying models and as the cost becomes extremely convex 1p1v results. We use this concept in Section 4 to extend to discrete decisions Hylland and Zeckhauser (1980)’s argument that efficiency occurs in equilibrium for all value configurations if and only the pricing rule is quadratic. This uniqueness contrasts with the complete information, game theoretic environment studied by Groves and Ledyard (1977a) where many rules are efficient (Maskin, 1999). Individuals in this model have an assumed-linear value of acquiring “influence”, a concept without clear micro-foundation in a non-stochastic price theoretic environment. In Section 5 we therefore study a canonical, quasi-linear independent private values model with a small aggregate noise that smooths payoffs in the limit as the population size grows large.

We prove that in any type-symmetric, monotone Bayes-Nash equilibrium, at least one of which exists, any social waste associated with equilibrium is eliminated as the population grows large. Except in the non-generic case when the mean of the value distribution = 0, equilibrium takes a surprising form, where a vanishingly small tail of “extremists”, from the side of the distribution opposite to its mean, purchase enough votes to win the election with high probability.

Quadratic Voting

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