Over twenty years ago, I proposed that Congress enact a public financing program for candidates for federal office, in which each voter would receive $100 in campaign finance vouchers to allocate to candidates, political parties, and interest groups.1 Building on an unpublished paper by Aanund Hylland and Richard Zeckhauser,2 I proposed that the rule for allocating such vouchers be quadratic: candidates, parties, and interest groups would receive only the square root of the amount voters allocate.3 This system would encourage voters to spread rather than “plump” their donations: a single $100 voucher contribution would be worth only $10 to the recipient, while eleven $9 donations would be worth $33 collectively to the recipients. I argued that this pluralist and egalitarian reform, in which public financing replaces private funding of campaigns, would enhance the formal equal political power of each voter, promote economic efficiency by limiting the amount of rent seeking by the wealthy, and increase social welfare by using a voting system that takes intensity of preference into account.

As radical as such a plan was in 1996 (among other things, it would require overturning the Supreme Court’s 1976 opinion in Buckley v Valeo4 and later opinions reaffirming it holding that campaign spending may not be limited consistent with the First Amendment),5 Eric Posner and Glen Weyl have produced a much more radical plan: the use of quadratic voting, or QV, for conducting voting for ballot measures and (perhaps) representative government. The original version of Posner and Weyl’s QV would allow literal vote buying, in which the wealthy could decide how many votes to allocate in an election such as one to decide whether to build a public park.6 In the current version of their proposal, contained in their delightfully provocative book, Radical Markets: Uprooting Capitalism and Democracy for a Just Society, Posner and Weyl advocate a modified version of QV in which all voters receive an equal number of “credits” which they can then vote quadratically across ballot measures over time or among a number of candidates for office.7 They argue that such a system will allow “a passionate minority” to “outvote an indifferent majority” and that “the outcome of the vote should maximize the well-being of the entire group, not the well-being of one subset at the expense of another.”8 Their chapter on QV begins with a vignette of a rural Japanese voter saving up his money to cast a large number of votes in favor of a measure repealing gun control in rural areas.9

In this short response, I express some skepticism about the creation of the radical egalitarian voting market Posner and Weyl suggest, and argue instead for QV-based public financing of elections. First, I argue that the kind of tradeoffs that modified QV elections require are both normatively undesirable and likely beyond the rational capacity of voters, especially in a politically polarized political environment fueled by social media and disinformation. Second, I argue that the caveats and constraints Posner and Weyl build into their model makes modified QV risky and impractical as an actual voting reform, as compared to other voting methods that reflect intensity, such as cumulative voting. Third, I briefly argue that QV public financing of elections is both practical and can achieve many of the benefits Posner and Weyl suggest for their voting market without some of the drawbacks I have identified.

I.  Bad Tradeoffs

Posner and Weyl’s proposal for modified QV voting appears to have had its genesis in a 2013 working paper of Weyl’s that has now been published as a co-authored paper with Steven Lalley.10 Building on Hylland and Zeckhauser’s work proposing quadratic voting for choosing continuous public goods (for example, how much money to invest in public parks), Lalley and Weyl extended the idea to binary public goods (for example, a single vote whether or not to build a public park).11 They argue that a voting system would increase overall social welfare if it allowed the buying of votes but awarded buyers a number of votes equal the square root of their purchase. In a 2017 paper, Weyl offers an example in which people in the LGBT community and their supporters would have paid (dollars) under QV to successfully defeat California’s Proposition 8, a ballot measure banning same-sex marriage.12 Weyl argued that by allowing voters to use dollars to register their intensity of preference, which they cannot do in a one person, one vote system, QV leads to more socially desirable outcomes.

Posner and Weyl’s 2015 Vanderbilt Law Reviewarticle defended QV using dollars, though the authors recognized that “some readers will not take seriously a political voting system that allows people to buy votes.”13 In a 2017 Public Choicesymposium on quadratic voting, Posner and Nicholas Stephanopoulos conceded the unconstitutionality of a QV system using dollars.14 Remaining “agnostic as to whether a QV-based system is, in fact, politically and practically feasible,”15 they argued for the constitutionality of “modified QV” in which votes could not be purchased, but voters each receive equal vote credits to spread across referenda or elections of representatives.16

In their 2018 book, Posner and Weyl thank Stephanopoulos for devising with Posner “the practical vision of egalitarian election law based on” QV,17 and the book proposes only modified QV with voter credits.18 For this reason, the remainder of my analysis considers only modified QV with voter credits, which I will refer to simply as QV; a QV system using real dollars raises distinct problems.

Consider the paradigmatic case of QV that Posner and Weyl offer for ballot measures. As with the rest of their book chapters, Chapter 2 on “Radical Democracy” begins with a fictitious vignette. The authors describe a conservative rural Japanese voter whose father had been killed by a bear during a time when Japan barred the use of guns even in rural areas.19 The voter had accumulated 800 voice credits over the past forty years and used 400 of those credits to cast 20 votes (the square root of 400) in favor of a measure allowing hunting rifles in rural areas. Even though a majority of peoplevoting on the measure opposed it, the fictitious measure passed because it obtained a majority ofvotesfrom people using their voting credits.20

Posner and Weyl do not spell out the details of the credit system, such as how to determine in each voting cycle how many credits each voter gets and whether that is related to the number of referenda each cycle. In their Japanese example, it appears that voters get twenty credits per year for potentially using on an unknown number of referenda. Nor do they explain the benefit of the QV portion of their plan for storable votes, as opposed to letting voters vote the full value of votes accruing to them over time.

A key feature of a QV scheme dependent on voters hoarding votes is that each vote decision by a voter involves making a tradeoff against an uncertain future. Whereas Weyl’s original proposal involved a tradeoff between dollars and votes on a single issue at a single election (such as a ban on same-sex marriage), voters now would have to make tradeoffs about artificial credits under conditions of great uncertainty.

For example, suppose one feels pretty strongly about gun rights and very strongly about the issue of abortion. This year there is a gun rights measure on the ballot but next year there might or might not be a measure on the issue of abortion. What is the rational way to decide how many, if any, votes to cast on the gun rights measure? For any voters who are not single-issue voters (even the Japanese voter in the vignette was “angry” about other actions taken by the liberal government21), this seems like a tough position for all voters, who will have to hold back on voting as many votes as they otherwise would like given the lack of adequate information.

Tradeoffs like these are normatively problematic, and the problem is not wholly solved by giving an expiring basket of credits for each voter to use in each election. Suppose the gun rights measure and the abortion measure appear on the same ballot in the same year, and voters have ten credits to use in the single election. Given current polarization, it seems likely that people will fall into the category of either (1) for gun control andabortion rights or (2) against gun control and abortion rights. In an ordinary election, we would expect each coalition (for and against) to work together to register voters, get them to turn out, and vote for both measures because in the current system voters get one vote to use for or against each measure.

The QV system, meant to help register intensity of preference, would cause likely allies to turn on each other, because lots of credits spent for votes on the gun control measure means fewer credits available to buy votes on the abortion measure. This would distort political incentives, and therefore vote totals may not reflect public opinion. This distortion would be especially likely now given how social media may be used to inflame passions and heighten voter intensity.

The system is even worse when you take into account the possibility of strategic gaming during polarized periods. For example, imagine the insurance industry wants to pass a state initiative eliminating the ability of injured parties to receive damages for pain and suffering. Polling indicates that liberal voters (who also support gun control and abortion rights) oppose the measure, but others will vote for it in exchange for a promise of lower insurance premiums. The industry might put abortion and gun control measures on the ballot so that liberal voters will exhaust their credits. The insurance industry then has a better chance of getting the insurance measure through, especially if credits carry over for multiple years and the industry can put its preferred measure on the ballot in a subsequent year. That the passage of the insurance measure would be affected by how people vote on unrelated measures (potentially in different elections) suggests that QV violates Kenneth Arrow’s “independence of irrelevant alternatives” requirement for a fair election system.22

Posner and Weyl’s examples seem to stack the deck by offering (what they consider to be) normatively desirable outcomes. The rural Japanese voter wants the gun to protect himself from bears, after a bear killed his father. “Most Japanese people live in cities and disapprove of guns. However, most of them have other priorities than gun control, given how low crime is in Japan and how the proposed reform exempts urban areas.”23 Supporters of same-sex marriage get to block the vote using the traditional version of QV. In both examples, Posner and Weyl seem to believe these are situations of an intense minority on one side and a mild majority on the other side (something which does not seem like it would accurately describe voter attitudes on either same-sex marriage or gun rights).

It is easy to imagine less normatively desirable examples, such as if a demagogic candidate convinces rural voters to plump all their credits on a measure which completely deregulates gun use, and mass shootings become as prevalent in Japan as they are in the United States. The rural voters, now having spent all their credits, cannot cast their votes against liberal abortion policies which liberal voters are able pass during a later referendum period when conservatives have blown their vote credits eliminating gun control. It is possible that aggregate welfare goes way down in my counter-anecdote.

The point here is that other than by way of anecdote, the authors cannot say that their QV system of voting credits, which will require inevitable and difficult tradeoffs, necessarily will increase, rather than decrease, overall social welfare. Weyl’s earlier formal demonstration that QV increases overall social welfare applied to QV with dollars and not QV with voter credits across multiple elections.24 He has offered no such formal argument proving the latter.

And while the authors consistently present QV as simply a system for the allocation of public goods,25 many of the kinds of things that people would vote upon in referenda, such as a ban on same-sex marriage, are not fairly characterized as public goods in which all share. Same-sex marriage is not like a public park with non-rivalrous beneficial consumption. A better model is that of a zero-sum game in which there are winners and losers depending upon the vote’s outcome.

Even with voters voting for a basket of referenda, it is  simply impossible to demonstrate that a QV system would    necessarily improve voters’ well-being overall. This is especially true when views on apparently disparate subjects such as     climate change and abortion are correlated with each other given today’s contemporary politics, and therefore voters’ strategic tradeoffs under conditions of uncertainty could lead to suboptimal results.26

The authors’ future work on QV should build upon Alessandra Casella’s work on “storable votes,”27 which is “the possibility of shifting one’s own votes from one contest to another, of storingvotes not spent on decisions that are low priorities for use on decisions that matter more.”28 Casella offers a formal argument that storable votes (without a quadratic component) can increase efficiency and protect minority voting rights when a body or board must vote on a number of binary proposals with a fixed agenda, such as the Federal Reserve Board deciding each quarter whether or not to raise interest rates, or US Senators voting on a set slate of nominees to federal appeals courts. Casella’s requirement of a fixed agenda29 solves some of the informational and tradeoff problems I have raised about QV. However, she notes that, even in a simple version of her proposal, strategizing to protect minority rights is complicated, and its workability depends upon some important informational constraints.30

II.  Caveats and Constraints

As a thought experiment, QV with credits raises fascinating issues. But I take it that the authors are trying to promote something practical and not just theoretical, as demonstrated by their experimentation with apps to prove QV’s feasibility.31 For this reason, it is important to look at the assumptions the authors have built into their model so that we do not simply “assume a can opener”32 about QV’s real-world viability.

Some of the assumptions about what it would take for QV to function properly are reasonable, such as the requirement that a QV system guard against fraud and abuse in the voter credits market.33 But the authors require much more from voters for this system to work. QV requires voters to be “rational and selfish.”34 It requires “social norms against pressure, vote-buying, and collusion; and a sense of civic duty to participate in proportion to one’s knowledge.”35 Voters cannot be “induced by social motives” to vote for a measure that “would be appropriate to fail” if everyone were knowledgeable and voted sincerely.36

Aside from the norm against vote buying, most of these assumptions and expectations defy our understanding of contemporary politics. Voters are swayed by social motives to vote in all kinds of ways. We encourage “collusion” in voting: people form coalitions to advocate for a basket of candidates, ballot measures, or a combination. Political parties are an example of such “collusion,” which provides a shortcut for busy voters to make decisions consistent with their interests. We rely on the opinions of others and band together to overcome collective action and information problems. We do not condition votes based on knowledge, and current norms encourage everyone to vote as a way of dividing power among political equals, rather than telling voters to sit elections out unless they are knowledgeable enough.

Further, as noted in the last Part, even assuming voter rationality, a QV system with credits requires voters to make both incommensurable tradeoffs and decisions under conditions of extreme uncertainty about how many future votes there will be and on what topics.

Compare QV with other systems meant to allow voters to register intensity of preference, such as cumulative voting. In a cumulative voting system, voters have votes equal to the number of candidates for a given office (such as a city council) and can plump all or some of their votes on a single candidate. This system too allows an intense minority of voters to get some representation in a system. Real world use of cumulative voting shows that it can achieve the promised results.37 Unlike QV, cumulative voting does not require voters to make unacceptable tradeoffs over time or across unrelated issues; it does not require deep voter knowledge; and it does not depend upon a lack of “collusion” among political allies—indeed it depends upon collusion. So, while QV might be better than cumulative voting in theoryin dealing with the intensity of preference problem, there is little reason to believe it would be superior in factwhen used on the ground.

III.  QV Public Financing as a Superior Alternative

Given the impermissible tradeoffs and unreasonable and unrealistic assumptions built into QV, what is the best way for allowing a minority of voters with intense interests to express their views and try to promote the greatest social welfare? A radically democratized publicly financed campaign finance system would be far better to achieve these goals than QV. Given the word constraints on this symposium paper, I will just briefly sketch my argument, which I have laid out more fully elsewhere.38

In my original 1996 proposal, every voter would be given $100 in campaign finance vouchers in each election cycle to distribute among candidates, parties and interest groups.39 Recipients would receive the square root of each donation, encouraging voters to spread their votes around.40 In my original proposal, voucher money would be the only type of money allowed to be used in federal elections, a plan that would require either a constitutional amendment or the Supreme Court rethinking its jurisprudence in this area.41

In my 2016 book, Plutocrats United, I proposed a modified voucher system, keeping the $100 limit but allowing some private spending (up to $25,000 per person per race, and $500,000 across all races during a two-year election cycle) to strengthen the constitutional case and assure that there is enough money to keep the system competitive.42 I dropped the quadratic aspect of the proposal in my book because most people rolled their eyes when they heard that they needed to compute square roots in order to participate in politics.

Although campaign finance voucher proposals have been around for a long time,43 Seattle only recently became the first US jurisdiction to implement them (without quadratic voting and without spending limits). While the early evidence is mixed,44 some studies look promising.45

Using vouchers (especially with a quadratic feature) will help create an influence market that better approximates the views and intensity of voters in society, regardless of wealth. A voucher-funded market for campaign money should limit rent seeking by the wealthy, who could no longer use large campaign contributions and spending to purchase access. Voters could use cues such as party to decide how to allocate voucher dollars. Voucher intermediaries would be the new power brokers, and their power would be reflected based upon how many voucher dollars they can drive to candidates, parties or groups. Those with intense preferences can band together and pool their voucher dollars to make their voices heard, much like we have seen the rise of small donations in modern campaigns facilitated by social media.

Overlaying the campaign finance voucher system would be our normal voting rules. By allowing each vote choice to be independent and allowing voters to collude or abstain from voting, voters can still have a chance to express their intensity of interest. They would still have a say in a system less likely to be distorted by strategic gamesmanship and the risk of decisions made under conditions of poor information and polarization.

As a matter of theory, a QV campaign finance voucher program perhaps does not register intensity of preference as well as QV voting. QV vouchers nonetheless are preferable in the real world because they do not raise the same risks of distortion from difficult tradeoff decisions, lack of voter knowledge, and strategic voting behavior.

  • 1. See generally Richard L. Hasen, Clipping Coupons for Democracy: An Egalitarian/Public Choice Defense of Campaign Finance Vouchers, 84 Cal L Rev 1 (1996).
  • 2. Aanund Hylland and Richard Zeckhauser, Efficient Public Goods Decisions under an Established Tax System13–17 (1984) (unpublished manuscript); Hasen, 84 Cal L Rev at 36 (cited in note 1) (discussing work of Hylland and Zeckhauser).
  • 3. Hasen, 84 Cal L Rev at 35–37 (cited in note 1) (explaining mechanics and benefits of use of square root formulate for allocation of campaign finance vouchers).
  • 4. 424 US 1 (1976).
  • 5. Id at 57–58. See also, for example, Citizens United v Federal Election Commission, 558 US 310, 361–62 (2010) (holding unconstitutional a prohibition on campaign spending by all corporations); Federal Election Commission v Massachusetts Citizens for Life, Inc, 4789 US 238, 263–64 (1986) (holding unconstitutional a prohibition on campaign spending by certain nonprofit corporations).
  • 6. See Eric A. Posner and E. Glen Weyl, Voting Squared: Quadratic Voting in Democratic Politics, 68 Vand L Rev 441, 446 (2015).
  • 7. Eric A. Posner and E. Glen Weyl, Radical Markets: Uprooting Capitalism and Democracy for a Just Society99–110 (Princeton 2018).
  • 8. Id at 106.
  • 9. Id at 80–82. See also id at 105–08.
  • 10. Weyl posted a paper dated April 1, 2013 on SSRN with the title “Quadratic Vote Buying.” The same URL now links to the published paper, Steven Lalley and E. Glen Weyl, Quadratic Voting: How Mechanism Design Can Radicalize Democracy, 1 Am Econ Assoc Papers and Proceedings 1, archived at
  • 11. Id at 1.
  • 12. E. Glen Weyl,The Robustness of Quadratic Voting, 172 Pub Choice 75, 80–81 (2017).
  • 13. Posner and Weyl, 68 Vand L Rev at 446 (cited in note 6).
  • 14. Eric A. Posner and Nicholas A. Stephanopoulos, Quadratic Election Law, 172 Pub Choice 265, 270 (2017):

    It is reasonably clear that the pure form of QV is unconstitutional because it differentiates among voters based on their ability to pay and violates the principle of equal voter treatment. But the modified form of QV, in which each voter is given the samenumber of credits, is likely lawful, though it would require some extension of existing precedent.

  • 15. Id at 266.
  • 16. Id at 270–73.
  • 17. Posner and Weyl, Radical Markets at x (cited in note 7).
  • 18. Id at 105–10. Near the end of the book, the authors defend monetized QV as an even better form of QV. Id at 263–64. But they most fully develop their ideas under modified (non-monetized) QV.
  • 19. Id at 80–82.
  • 20. Id.
  • 21. Id at 80.
  • 22. Kenneth J. Arrow, Social Choice and Individual Values11–13 (Yale 2d ed 1963).
  • 23. Posner and Weyl, Radical Markets at 106 (cited in note 7).
  • 24. Lalley and Weyl, 1 Am Econ Assoc Papers and Proceedings at 2–3 (cited in note 10).
  • 25. See, for example, Posner and Weyl, Radical Markets at 256 (cited in note 7).
  • 26. The authors also offer extensions of QV to other types of elections, including a system for the election of representatives in which a voter could spread voting credits across federal, state, and local offices, spending “as much or as little of her budget on as many votes as she could afford, spread among all of the candidates at all levels—for, against, or zero.” Id at 120–121. This system too creates major tradeoff problems: it would pit state, local, and federal candidates in the same party against each other to compete for votes; it would lead many voters not to vote in elections where they would like to have a say, simply because they care more about a different election; and it would be open to appeals of demagogues during conditions of polarization.
  • 27. Most notably, Alessandra Casella, Storable Votes: Protecting the Minority Voice(Oxford 2012). Posner and Weyl do not discuss the work around Casella’s research in their book.
  • 28. Posner and Weyl, Radical Markets at xviii–xix (cited in note 7).
  • 29. See id at 3–4 (noting requirement of a fixed agenda).
  • 30. Id at 24, 94–95.
  • 31. Posner and Weyl, Radical Markets at 111–116 (cited in note 7).
  • 32. Robert P. Greenspoon and Catherine M. Cottle, Don’t Assume A Can Opener: Confronting Patent Economic Theories with Licensing and Enforcement Reality, 12 Colum Sci and Tech L Rev 194, 194 n 1. See also Wikipedia, Assume a Can Opener, archived at
  • 33. Posner and Weyl, Radical Markets at 109 (cited in note 7).
  • 34. Id.
  • 35. Id at 109–10.
  • 36. Id at 109.
  • 37. Richard H. Pildes and Kristen A. Donoghue, Cumulative Voting in the United States, 1 U Chi Legal F 241, 272–76 (1995).
  • 38. See Hasen, 84 Cal L Rev at 21–27 (cited in note 1).
  • 39. Id at 21.
  • 40. Id.
  • 41. Id at 39–44.
  • 42. Richard L. Hasen, Plutocrats United: Campaign Money, the Supreme Court, and the Distortion of American Elections94–103 (Yale 2016).
  • 43. For the history of campaign finance voucher proposals in the United States, see id at 212 n 14.
  • 44. Bob Young, Seattle’s Democracy Vouchers Haven’t Kept Big Money Out of      Primary Election(Seattle Times, July 30, 2017), archived at
  • 45. Gene Balk, Do Seattle Democracy Vouchers Work? New Analysis Says Yes      (Seattle Times, Oct 13, 2017), archived at