Glen Weyl and I just completed revisions on our paper, “Quadratic Voting in Finite Populations.” Quadratic voting (QV) is a voting rule proposed by Glen in which individuals purchase votes by paying the square of the votes they buy using some currency. QV has been written about extensively, most notably in Glen’s book Radical Markets with Eric Posner. I give an overview of QV here, but there are many other places online that explain it well if you would like to read more1. Among our contributions in this paper is that we show QV leads to desirable social choices even in small groups of people, giving formal justification for experimenting with QV in real-world, smaller-scale settings.
A fundamental problem societies face is making collective decisions, such as who to choose as representatives, how much tax to levy, or what rights should be guaranteed to minorities. As just one example, societies decide whether to allow same-sex marriage. A particularly common method for making collective decisions is majority voting, or one-person-one-vote: whichever option is favored by the most people becomes law. In the United States, prior to the Supreme Court’s ruling in Obergefell v. Hodges in 2015, several states put the decision of whether to allow same-sex marriage to direct popular vote. In 2008, California voters rejected same-sex marriage by a 52% to 48% margin; in Maryland, on the other hand, voters approved same-sex marriage in 2012 by a 52% to 48% margin.
The limitations of one-person-one-vote have been understood for thousands of years. It can lead to “mob rule” or “tyranny of the majority,” scenarios in which the strong interests of the minority are trampled by the majority. Many modern states have various institutions to prevent tyranny of the majority, among them written constitutions, bicameral legislatures, and judiciaries. In the US, the Supreme Court ultimately ensured minority LGBT individuals would have the right to marry across the country.
However, the institutions that the state uses to protect minorities have many flaws. The Supreme Court, for instance, concentrates a large amount of power among a few unelected officials. Further, there is no guarantee that any of the above-mentioned institutions will actually act in the interest of the minority when needed, or that the benefits they generate outweigh the inefficiencies they cause.
Designing better mechanisms to make desirable collective decisions is an old, critical open problem in the social sciences. I cannot hope to list all of the mechanisms proposed over the years or discuss their merits2. Quadratic voting is a relatively recent contribution to the field, proposed by Glen originally in 2012.
Under QV, each individual in the society can purchase votes in an election, either for or against a resolution passing; in the near- and medium-term, most applications will likely use artificial currency distributed equally to each member and which can be spent only across each election the society faces. Individuals can purchase as many votes as they would like in an election (as long as they still have money left), but each additional vote they buy in an election increases in cost. The total cost paid is the square of the number of votes purchased, so 1 vote costs $1, 2 votes cost $4, 2.5 votes cost $6.25, etc. In the case of artificial currency, buying more votes in one election leaves less money left over to spend on other elections. The resolution passes if the total votes purchased for the resolution exceeds the total votes purchased against the resolution. Since individuals can buy more than one vote, they can choose to vote more in elections they really care about and vote less in elections that are not as important to them. On the other hand, the increasing cost of voting prevents individuals with extreme preferences from dominating the election. Buying a lot of votes becomes costly quickly.
Each individual either prefers that the resolution passes or that it fails. If we make (quite strong) assumptions about people’s preferences, then how strongly they feel about their preferred outcome can be measured by the amount of currency they would be willing to pay to get it. If someone in favor of the resolution is indifferent between having it fail and having it pass but giving up $10, then their value for the outcome as measured by willingness to pay is $10. Lalley and Weyl (2019) give conditions under which, when there are many people taking part in the election, QV gives the outcome that maximizes total willingness to pay. In fact, the squared cost of votes in QV is the unique cost function that accomplishes this feat in the election framework Lalley and Weyl consider. If everyone in favor of the resolution passing would be willing to pay $1000 in total, and everyone opposed would be willing to pay $900 in total, then the resolution would pass under QV under the conditions outlined by Lalley and Weyl3.
To see why maximizing willingness to pay could be a desirable outcome, consider a simple, extreme example. Suppose voters are voting between candidate A and candidate B. 49% of voters really dislike candidate A. They would be willing to pay $10,000 each to prevent A from winning. The other 51% of voters are mostly indifferent, but would each be willing to pay $1 to have candidate A win. Under one-person-one-vote, since the majority of people favor candidate A, candidate A would win. Under QV, the total willingness pay is clearly higher for the side that favors candidate B, who would end up winning. Thus, QV allows people to express how much they care, whereas one-person-one-vote behaves as if everyone cares the same amount. Under QV, the minority shows that they really want B to win by spending more of their scarce voting resources to prove it.
How often is there tyranny of the majority? It is difficult to quantify exactly (not least because modern elections and polling fail to measure intensity of preference well), but the case of same-sex marriage offers one case study. It seems quite likely that the people with the strongest preferences in the 2008 same-sex marriage referendum in California were LGBT individuals. LGBT voters constituted about 4% of the population of California in 2010. Since the ban on same-sex marriage passed, a small majority of non-LGBT voters must have been in favor of the ban. While we do not have hard numbers on people’s intensity of preference, it is easy to imagine this as a case where the weak preferences of the majority against same-sex marriage dominated the stronger preferences of the minority in favor of same-sex marriage. This is a case where QV could have allowed the minority to have greater voice in the election and swing it in their direction.
However, as I mentioned previously, Lalley and Weyl (2019) only show that under their conditions QV maximizes willingness to pay when there are very many people voting in the election. The reason they require a lot of people is that having many voters resolves uncertainty that could muddle analysis of possible outcomes. One way to think about this is to imagine randomly drawing 5 people from California to take part in the election. When there are only 5 voters, the direction of total willingness to pay for and against the resolution can vary depending on which 5 voters are drawn, which means each individual voter has a meaningful chance of swinging the outcome by themselves. The outcome can vary a lot depending on how much each person votes. Accounting for the possible outcomes in such an election is hard to model mathematically. Instead, if every voter in California takes part, there is little chance any one individual will be influential enough to swing the election themselves. As a result, there is much greater certainty over which outcome will receive the most votes and how this relates to total willingness to pay4.
In real-world applications in the near future, it is hard to imagine QV being tested in elections with millions of people. This creates a dilemma: how can we test if QV actually works in the real-world if the theory only makes predictions about its efficacy in massive elections we should be hesitant to experiment in? Answering this question is a major contribution of our paper. Through simulation, we show that even in small-scale elections QV usually leads to the outcome that maximizes total willingness to pay in a broad number of scenarios. In fact, it very often outperforms one-person-one-vote.
There are intuitive reasons one-person-one-vote should actually do well in small-scale elections. As a toy example, suppose there are only two people, and they need to decide whether to go to Chipotle or Qdoba for lunch. For the sake of argument, suppose most people in society prefer Chipotle, but the minority of people who prefer Qdoba are very passionate that it’s better and have greater willingness to pay to go to Qdoba on average. If the two individuals both are in favor of Chipotle or are both in favor of Qdoba, then one-person-one-vote will definitely choose the outcome that maximizes total willingness to pay. Either of these scenarios should happen with fairly high probability. Instead, if the two people disagree about going to Chipotle or Qdoba, then suppose the tie is broken via a coin flip. The correct outcome will still be chosen half the time. Thus, in small groups, one-person-one-vote can often lead to the right outcome just by chance, even if in the overall society the minority has higher total willingness to pay. As the society gets larger, these chance beneficial outcomes become more and more unlikely if the minority has greater total willingness to pay, leading to tyranny of the majority. In a group of 20 people, it is likely people who prefer Chipotle outnumber people who prefer Qdoba, but people who prefer Qdoba have greater total willingness to pay.
A similar, but stronger, logic occurs for QV. Similar to one-person-one vote, the total willingness to pay for each side will frequently coincide by chance. However, unlike one-person-one-vote, QV also allows people to spend more or less of their budget depending on how much they care. People can express how intense their preferences are. In the case with two voters, instead of relying on a coin flip to break ties, we rely on the number of votes purchased by each side, which should almost certainly break the tie in favor of the side with stronger intensity of preference. We show that in elections with approximately 2 to 15 voters, QV almost always chooses the side that has greater willingness to pay.
We believe our findings give formal basis for experimentation with QV in smaller-scale elections. In all cases we simulate, QV chooses the wrong outcome only a small fraction of times, and typically only when the willingness to pay on both sides is nearly equal (10% is the largest inefficiency for QV we found in any setting). Numerous real-world experiments with QV have in fact already occurred, and so far give promising results5. Our work gives intuition for why QV can do well in these cases, and provides software for researchers to test it out on their own and explore conditions where QV can do especially well or poorly. We hope our results encourage more people to experiment with QV in the future.
- See here, here, and here, for example.
- A very small sampling: Borda count, VCG, storable voting.
- Lalley and Weyl assume an independent private values setting where each voter knows the value distribution. See Weyl (2017) for discussion of how QV holds up in more general settings that allow for collusion among voters, fraud, uncertainty about the value distribution, and behavioral mistakes.
- The case with an intermediate number (something like 50-1000) of voters is more complex, as we discuss in the paper. The outcome that maximizes total willingness to pay usually wins, which decreases people’s incentive to buy a lot of votes. If the outcome that maximizes total willingness to pay is almost certainly going to pass anyway, there is no point in spending a lot of money in the election. However, if people decrease their vote levels too much, this creates an incentive for someone with extreme preferences to “steal” the election inefficiently, profiting off everyone’s complacency. We show such extremists exist and can swing the election even when the other side has greater total willingness to pay, but the likelihood this happens falls as the number of total people in the election increases, so that asymptotically the outcome that maximizes the total willingness to pay is always chosen. Results concerning the inefficiency from extremists in fact constitute the bulk of our paper, as we use various analytical and computational approaches to show this inefficiency is small in a broad class of settings.
- See here, here, and here for a few examples.