## The nonlinearity bogeyman (slain)

`Has thou slain the Jabberwock? Come to my arms, my beamish boy! O frabjous day! Callooh! Callay!' He chortled in his joy. - Charles Lutwidge Dodgson (1832-1898), famous voting-methods researcher.

Q. If I rate a candidate as, say, 20%, this might mean that I think the candidate worthy of the office, but there happens to be someone else who is much more worthy, or it might mean that I consider the candidate a blithering idiot only able to appear in public with extensive handling. A 20% rating might well be a vote against a candidate.

We do not know what voters meant when they rated the Libertarian candidate, for example, at a certain rating – unless it was the highest rating, in which case we can definitely treat it as a full vote for the candidate. But we don't know how to treat the rest of the scale.

Voting for a candidate in Approval is a clear indication of support for that candidate, as it is in Plurality. But I do not see that giving a candidate a range vote of 30% means anything. So it is not legitimate to compare range votes with approval votes. It is not even meanigful to consider the ratio of the numbers of range votes got by two given candidates (although it is legimate to do that with approval or plurality votes), because really with range voting some nonlinear transformation might be required. Is a 2% average Range vote twice the vote of a 1% Approval vote? No.

A. (In this response, for maximum simplicity, we shall be considering range and approval voting without blanks, i.e. the "totalling" version of each.) The questioner is confused. A range vote is not a "meaningless rating" but in fact a "meaningful vote." What is the distinction?

Simple. If I tell you "rate George Bush on a scale of 0-to-100" and you say "51," then yes, the precise meaning of that is pretty inscrutable. But if I tell you your rating will be used in a range voting election, then you are telling me that 51 in the full knowledge that, if twice as many voters give Bush 51 than give his top opponent Kerry 100, then Bush will be elected. That is what 51 "means," ok? If Bush gets elected or not, that is a pretty darn real meaning, you better believe it.

That gives it a meaning. That is the difference between votes and mere ratings – votes are inherently imbued with real meaning because they lead to real election results in a precisely defined way. (And that real meaning is linear not nonlinear, because in range voting with totalling, the vote combination method – totalling – is linear.) OK? So now consider the range-and-approval-voting exit poll we conducted in the 2004 US presidential election.

Libertarian candidate Michael Badnarik got 1/158th as many Plurality votes as Bush; under Approval Voting, he got 1/65th; while under Range Voting, he got 1/4.4 times as many votes as Bush.

It is fully legitimate to compare these results and conclude that Range Voting is way better for Badnarik versus Bush than the other two systems. You might quibble "naah, range voting is not like `real' votes, it is meaningless ratings, so we cannot claim this really means Badnarik should want range voting." Wrong. The correct interpretation of these results by Badnarik would be: "these `vote counts' in each system do have real meaning. They are measuring `the vital juice that I need to get elected.' And I see under range, Bush has only 4.4 times more of that vital juice than I do. I want range voting."

And if you think, say, that plurality votes somehow have "more" real meaning than that, you are again wrong. It would be entirely possible for Bush and Kerry to each get, say, about 50% of the plurality votes, even though 99% of the population felt that neither of them was the best candidate if the voters all were "strategic" i.e. felt "I cannot afford to `waste my vote' on my true favorite third-party candidate, because I think he has no chance of winning." (We are not saying the strategic distortion in the 2004 US presidential election was actually that severe, but there is no doubt that it did happen.) OK? Plurality votes do not count the number of people who think X is the best candidate. They mean nothing of the sort. They simply measure "how much vital juice that candidate got, where getting the most juice means you get elected."

And that is exactly also true for range voting. And in fact range votes have more to do with a candidate's quality than do plurality and approval votes, because there is less strategic distortion and "forced rounding to 0 or 99" distortion. So in that sense, we see, it is range votes that have more "real meaning."

Thanks to Abd ulRahman Lomax for help with this file.