This working paper was published in August 2022. You might also be interested in this more recent post by one of the authors on a similar topic.

## Abstract

Quadratic funding is a public good provision mechanism that satisfies desirable theoretical properties, such as efficiency under complete information, and has been gaining popularity in practical applications. We evaluate this mechanism in a setting of incomplete information regarding individual preferences, and show that this result only holds under knife-edge conditions. We also estimate the inefficiency of the mechanism in a variety of settings and show, in particular, that inefficiency increases in population size and in the variance of expected contribution to the public good. We show how these findings can be used to estimate the mechanism’s inefficiency in a wide range of situations under incomplete information.

## Introduction

The non-excludability and non-rivalry of public goods poses a challenge for public good provision that has long received considerable attention in both the theoretical and the applied economic literature (Samuelson 1954; Lindahl 1958). Several mechanisms for providing efficient levels of a public good have been proposed (Clarke 1971; Groves and Ledyard 1977; Hylland and Zeckhauser 1979; Walker 1981), and while these mechanisms are of considerable theoretical importance, there has been to date limited practical application of these solutions, resulting at least in part from undesirable properties they were shown to possess (Walker 1981; Healy 2006; Rothkopf 2007). On the other hand, there are various solutions commonly used in practice, such as majority voting, 1:1 donation matching, and private provision of public goods, all of which lead to inefficient outcomes in the general case (Bergstrom 1981; Bergstrom et al. 1986).

The quadratic funding (QF) mechanism, proposed by Buterin et al. (2019), appears to be promising in both a theoretical and a practical sense. This mechanism provides a public good level that is equal to the square of the sum of the square roots of individual contributions. That is, if every individual contributes some quantity for funding a public good, then the resulting funding for the public good through QF is Besides efficiency under complete information, one characteristic that distinguishes this mechanism from previously proposed ones is that it does not require any assumptions about the set of public goods to be funded, making it particularly well-suited to cases in which it is important that individuals be able to propose new public goods. It also stands out for its simplicity, and satisfies other desirable properties such as individual rationality and homogeneity of degree one. These characteristics make QF particularly promising for usage in a broad range of situations. In fact, QF has been employed to allocate significant sums of money for funding open-source software projects and matching donations to charity.^{[1]}

This paper aims to analyze the efficiency of the quadratic funding mechanism in a more general informational context, and we do so in two main ways. First, we adapt the framework introduced by Buterin et al. (2019) to allow for the possibility of incomplete information regarding individual preferences. Besides showing the existence of equilibria, we present necessary and sufficient conditions for efficiency, and show that QF is only efficient under knife-edge conditions, which stands in contrast with the efficiency of the mechanism under complete information. In particular, we show that QF is inefficient whenever an individual is uncertain about whether the efficient provision is positive, and that QF is efficient for individuals with isoelastic utility functions for the public good if and only if the elasticity coefficient of these functions is equal to 1/2. We show that the latter condition can be interpreted as saying that QF is efficient when the optimal individual contributions is a dominant strategy, i.e., do not depend on the contribution by others, thus presenting an easily verifiable test for efficiency in applications of this mechanism.

Second, and motivated by the large class of models in which the private provision of the public good is inefficient, we use numerical estimations to quantify the inefficiency of QF under incomplete information. We define two measures of inefficiency, and then analyze how these measures respond to changes to parameters of our setup. We show that inefficiency is increasing in the number of players and in the variance of the expected value of the fund, and we characterize conditions under which this response is more or less intense. The results presented in our analysis can be used to assess how QF would perform even when it does not lead to efficient public good provision.

Besides the importance of these findings to quadratic funding, our results also bring implications to quadratic voting (Lalley and Weyl 2019), and more broadly to the growing literature on quadratic pricing (Tideman and Plassmann 2017). Quadratic voting is a voting mechanism that has gained attention both from academia (Kaplow and Kominers 2017; Park and Rivest 2017; Quarfoot et al. 2017; Weyl 2017) and from policymakers, having been applied by the Democratic Party of the United States for political decision-making.^{[2]} Quadratic funding can be understood as an adaptation of quadratic voting to a context of continuous public good provision, which makes our findings particularly surprising, given that Lalley and Weyl (2019) showed that the outcome chosen through quadratic voting under incomplete information converges to efficiency as the population grows. Therefore, our results suggest that incomplete information might pose an important challenge to other quadratic pricing mechanisms, despite the efficiency result for quadratic voting, and, more generally, the properties pertaining to one mechanism may fail to be held by the other.

The paper is divided into seven sections. Section 2 presents the setting used in the paper, and shows the existence of equilibria for quadratic funding. Section 3 presents efficiency results under complete information, and section 4 analyzes efficiency under incomplete information. Section 5 gives an economic intuition and an efficiency result for the special case where individuals have isoelastic utility functions for the public good, and section 6 employs this class of utility function to develop quantitative estimates of inefficiency under incomplete information. Section 7 summarizes some conclusions of the paper. The Appendix provides the proofs of the stated propositions.