I think (but don’t know) that it doesn’t really matter how much each vote costs, as long as it’s increasing, because then it causes you to calculate your price for marginal votes on different options.
For the specific goals that quadratic voting was designed for (figure out how much to fund public goods; assume each voter is a self-interested rational actor; assume no collusion) it is important that cost grows quadratically (and not, say, cubically). (Of course, the LW review is different enough that this wouldn’t apply.)
Nitpick: quadratic voting and quadratic funding are technically different schemes. In the former you vote for specific bills that can either pass or not pass. In the latter you fund projects and your donations are matched according to a particular formula.
However, there is a close correspondence between them. One way to see it is as follows. Quadratic funding can be seen as a vote using the quadratic voting scheme on the following bill:
Voter 0 will distribute $X to a certain project.
and the quadratic funding subsidy formula is the maximum X for which Voter 0 will not pay to stop the bill.
In more detail: to prevent the bill, Voter 0 must buy more votes than all the other voters who voted against them. That is, V0>∑ni=1Vi. If we use the cost-function C(V)=V2 , each of the other voters paid √V1 for their votes. This means that in order to prevent the bill from passing Voter 1 must pay C(V0)>(∑ni=1Vi)2=(∑ni=1√Ci)2. But this is exactly the subsidy formula from quadratic funding.
For the specific goals that quadratic voting was designed for (figure out how much to fund public goods; assume each voter is a self-interested rational actor; assume no collusion) it is important that cost grows quadratically (and not, say, cubically). (Of course, the LW review is different enough that this wouldn’t apply.)
Nitpick: quadratic voting and quadratic funding are technically different schemes. In the former you vote for specific bills that can either pass or not pass. In the latter you fund projects and your donations are matched according to a particular formula.
However, there is a close correspondence between them. One way to see it is as follows. Quadratic funding can be seen as a vote using the quadratic voting scheme on the following bill:
and the quadratic funding subsidy formula is the maximum X for which Voter 0 will not pay to stop the bill.
In more detail: to prevent the bill, Voter 0 must buy more votes than all the other voters who voted against them. That is, V0>∑ni=1Vi. If we use the cost-function C(V)=V2 , each of the other voters paid √V1 for their votes. This means that in order to prevent the bill from passing Voter 1 must pay C(V0)>(∑ni=1Vi)2=(∑ni=1√Ci)2. But this is exactly the subsidy formula from quadratic funding.