Update (11/3): I develop these ideas more fully in my paper “Weyl Versus Rasmey: A Bayesian Voting Primer,” available here.

We stipulated in our previous post how beliefs and preferences can come in degrees. I also mentioned how this simple insight has radical implications for any domain in which voting is used to make collective decisions. Why? Because the various voting methods that are used to make decisions in legal and political domains pay no heed to the decision makers’ degrees of belief or to the intensity of their preferences. Simply put, whether the decision makers consist of a  population of registered voters, a group of legislators, a panel of appellate judges, or even a 12-man jury, the traditional “one-man, one-vote” rule does not accurately measure the strength or intensity of the decision makers’ beliefs and preferences.

To remedy this defect of traditional voting methods, my colleague Glen Weyl has recently proposed a new voting method called “quadratic voting,” a simple but ingenious mechanism that does allow voters to express the strength or intensity of their preferences. This new voting procedure has not only received extensive academic attention (see, for example, all the formal papers published in the July 2017 issue of the journal Public Choice); it has also been tested in some real-world and artificial settings (by way of introduction, see this entry for “history of quadratic voting” in Wikipedia). In this post, I will explain how quadratic voting works.

There are several concise and good introductions to quadratic voting–including this primer by Vitalik Buterin as well as this formal paper by Stephen Lalley and Glen Weyl–so this post will be brief. Boiled down to its two most essential features, quadratic voting is a simple voting procedure in which (1) voters are allowed to buy as many votes as they want but in which (2) the price of each additional extra vote purchased is the square of the number of votes purchased. Now, before proceeding any further, an important clarification is in order: the payment of votes can be made using a real currency or cryptocurrency, such as Bitcoin, Dollars, Pesos, etc., or these payments can be made using tokens or some other form of artificial currency. Lalley and Weyl, for example, have proposed that an equal amount of tokens or “voice credits” be distributed to all the voters before they cast their votes.

Let’s stick with Lalley & Weyl’s innovative idea of “vote credits,” which is less amenable to ethical objections than QV systems with actual money in play (see, e.g., Laurence & Sher, 2017), and let’s use a concrete real-world example to see how their ingenious quadratic system would work in practice. By way of illustration, let’s consider the upcoming Democratic presidential primary to be held in South Carolina on 29 February 2020. To simplify, since billionaire candidate and late-entrant Mike Bloomberg is not yet on the ballot, assume the voters in the Palmetto State must choose among the following five viable candidates:

Under the traditional method of voting (one-man, one-vote), each voter must vote for one — and only one — candidate. Under Lalley and Weyl’s quadratic system, by contrast, each voter is allocated, say, 25 “vote credits” before the election, and each is then allowed to allocate these credits among the candidates in whatever way they wish. But there is a catch. The more votes you allocate to a given candidate, the more each additional vote will cost you. For example, let’s say you are a die-hard Bernie Sanders supporter, so you want to allocate all of your votes to Senator Sanders. The price of one vote will cost you only one vote credit, but the price of two votes will cost you four credits (two squared), while the price of three votes will cost you nine credits (three squared), and so on, as illustrated in the following vote-payment table below:

As a result, if you wish to allocate your entire vote budget to Bernie (or to some other candidate, for that matter), it will cost you 25 vote credits, but at the same time, your preferred candidate will receive only five votes because of the quadratic cost structure of this voting method. Okay, now that we have presented the least objectionable form of quadratic voting, I will identify and discuss the main flaws with this method — and then present a simpler alternative to quadratic voting that I call “Bayesian Voting” — in my next two posts.

Reference: Ben Laurence and Itai Sher, “Ethical considerations on quadratic voting,” Public Choice, Vol. 172 (July, 2017), pp. 195-222.