That is the new paper by Lalley and Weyl.  Here is the abstract:

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 QV, in contrast to existing mechanisms. We discuss applications in both (near-term) commercial and (long-term) social contexts.

Eric Posner has a good summary.  I would put it this way.  Simple vote trading won’t work, because buying a single vote is too cheap and thus a liquid buyer could accumulate too much political power.  No single vote seller internalizes the threshold effect which arises when a vote buyer approaches the purchase of an operative majority.  Paying the square of the number of votes purchased internalizes this externality by an externally imposed pricing rule, as is demonstrated by the authors.  This is a new idea, which is rare in economic theory, so it should be saluted as such, especially since it is accompanied by outstanding execution.

The authors give gay marriage as an example where a minority group with more intense preferences — to allow it — could buy up the votes to make it happen, paying quadratic prices along the way.

My reservation about this and other voting schemes (such as demand revelation mechanisms) is that our notions of formal efficiency are too narrow to make good judgments about political processes through social choice theory.  The actual goal is not to take current preferences and translate them into the the right outcomes in some Coasean or Arrovian sense.  Rather the goal is to encourage better and more reasonable preferences and also to shape a durable consensus for future belief in the polity.

(It is interesting to read the authors’ criticisms of Vickrey-Clarke-Grove mechanisms on p.30, which are real but I do not think represent the most significant problems of those mechanisms, namely that they perform poorly on generating enough social consensus for broadly democratic outcomes to proceed and to become accepted by most citizens.  One neat but also repugnant feature of democratic elections is how they can serve as forums for deciding, through the readily grasped medium of one vs. another personae, which social values will be elevated and which lowered.  “Who won?” and “why did he win?” have to be fairly simple for this to be accomplished.)

I would gladly have gay marriage legal throughout the United States.  But overall, like David Hume, I am more fearful of the intense preferences of minorities than not.  I do not wish to encourage such preferences, all things considered.  If minority groups know they have the possibility of buying up votes as a path to power, paying the quadratic price along the way, we are sending intense preference groups a message that they have a new way forward.  In the longer run I fear that will fray democracy by strengthening the hand of such groups, and boosting their recruiting and fundraising.  Was there any chance the authors would use the anti-abortion movement as their opening example?

If we look at the highly successful democracies of the Nordic countries, I see subtle social mechanisms which discourage extremism and encourage conformity.  The United States has more extremism, and more intense minority preferences, and arguably that makes us more innovative more generally and may even make us more innovative politically in a good way.  (Consider say environmentalism or the earlier and more correct versions of supply-side economics, both innovations with small starts.)  But extremism makes us more innovative in bad ways too, and I would not wish to inject more American nutty extremism into Nordic politics.  Perhaps the resulting innovativeness is worthwhile only in a small number of fairly large countries which can introduce new ideas using increasing returns to scale?

By elevating persuasion over trading in politics (at some margins, at least), we encourage centrist and majoritarian groups.  We encourage groups which think they can persuade others to accept their points of view.  This may not work well in every society but it does seem to work well in many.  It may require some sense of persuadibility, rather than all voting being based on ethnic politics, as it would have been in say a democratic Singapore in the early years of that country.

In any case the relevant question is what kinds of preference formation, and which kinds of groups, we should allow voting mechanisms to encourage.  Think of it as “politics as education.”  When it comes to that question, I don’t yet know if quadratic voting is a good idea, but I don’t see any particular reason why it should be.

Addendum: On Twitter Glenn Weyl cites this paper, with Posner, which discusses some of these issues more.