This sounds a lot like an idea I was toying with for the governance structure of an LLC which would manage a piece of land serving as a community gathering space. I was using a log function rather than a quadratic, but I didn’t go to this level of rigor, either. Neat to see someone actually work out a proof.
Quadratic voting is a procedure that a group of people can use to jointly choose a collective good for themselves. Each person can buy votes for or against a proposal by paying into a fund the square of the number of votes that he or she buys. The money is then returned to voters on a per capita basis. Weyl and Lalley prove that the collective decision rapidly approximates efficiency as the number of voters increases.