[EM] Linear Spread Median Range Voting

[Abd ul-Rahman Lomax]

At 01:45 AM 12/21/2005, rob brown wrote:

>Your vote for B lowered his score when you would have wanted to 
>raise his score (had you known that it was A, and not C, that was 
>his main competitor).  The reason I had them differ by a small 
>amount was to show the case where the election could be lost by one 
>vote.  But of course that's not the point.  You don't know whether 
>your vote will help or hurt your candidate.  Why do it that way when 
>condorcet and especially DSV methods allow you to be absolutely 
>honest, and always* have it work in your favor?

There is an example being considered on the Range list which shows 
the defect in the Condorcet Criterion, at least as it would usually 
be applied. Basically, ranked ballots assume that the preference gap 
between candidates is equal. A tiny expressed preference is treated 
the same as if it were a preference between a maximally loved 
candidate and a maximally hated one.

The example was originally stated, by Warren, as:
>Theorem:
>Both LTBM and Range can fail to elect a Condorcet winner if one exists
>(and indeed both can elect a Condorcet loser).
>Indeed if in a majority of votes A>B, then B can win in both LTBM and Range.
>26% give A=0.51 and B=0.49
>25% give A=0.26 and B=0.24
>49% give A=0 and B=1
>LTBM result: A-median=0.26   B-median=0.49, B wins
>Range result: B wins with about 0.48 higher average than A
>Condorcet result: A wins 51-to-49.

I should explain that LTBM is Warren's expression for my Spread 
Median, which he did not actually use to show the results; and 
because the data is highly discontinuous (missing ratings), the 
median is badly behaved. I have now proposed a spread median which is 
better behaved in the presence of gaps and which is just about as 
well behaved (and a little easier to calculate) with no gaps. But 
that's not the topic here.

In this election, *almost* a majority of voters expressed a strong 
preference for B over A. I noticed that the Range votes of a slight 
majority of voters were not fully expressive (only middling ratings 
were used). And, indeed, this is why B wins.

The B voters strongly preferred B over A, and the A voters cared 
hardly at all. I'd argue that, if the ratings were accurate, B is the 
best winner even though, strictly, A is the Condorcet winner by a 
slight margin. In fact, I consider it *clear* that B is the best 
winner. The election of B would minimize dissatisfaction with the 
election outcome, the electorate would be essentially united. On the 
other hand, if we presume that the ratings are sincere, the B voters 
will be very unhappy with the outcome. Warren has used extreme 
examples from history to show how this could work.

A has promised to kill all the Catholics.
B has promised religious toleration.

The A voters don't like Catholics, but are a little worried that 
killing them all is not a great idea. They aren't worried enough to 
strongly support B. But enough to not have a strong preference. 
*None* of them have a strong preference. A *very* slight majority 
would off the Catholics.

The Catholics and others horrified by the prospect of A winner vote the limits.

I'm assuming that these votes are all sincere.

And, as Warren has pointed out in similar discussions, if you really 
cared to see A elected, why didn't you rate A at 100?

I'll tell you why. That election is not the full election result. 
There were two other candidates whom A voters actually preferred. 
Called them C and D. C and D were not-so-bad candidates for the 
Catholics and friends, they only want, perhaps to repress them but 
would still protect basic rights; perhaps this was the status quo. 
Again, a majority of voters (51 to 49) preferred C and D to B also. 
So B is the Condorcet *loser* in this election.

Yet, I submit, if the ratings are sincere, B is the best winner.

Condorcet is unable to express strength of preference. In Range, 
every voter may decide for himself whether or not to express full 
strength of preference. It's simple to vote Range as Approval, and 
your vote, if you do, will count for maximum strength in each 
pairwise election, *except*, of course, for pairwise elections where 
you have either totally rejected or totally approved both candidates.

Range *allows* you to express intermediate preferences. It does not 
require you to do so. Why would you want to weaken your vote?

Well, if there are a large number of candidates, strategic 
considerations become quite complex; and strategic voting, with 
Range, brings with it a risk. As you have seen. By rating your 
favorite below the true utility to you of the election of this 
candidate, you acted to cause the candidate to lose.

Range, I believe, will maximize the expected utility of the election 
*to you*, in the general case and with a large number of candidates. 
When there are only a very small number of candidates, with one or 
two frontrunners easily predicted to win (one of them), you might be 
able to safely vote strategically, i.e., to exaggerate your ratings 
in order to move the election more effectively toward your favorite 
of the frontrunners. Approval voting allows you to do this without 
devaluing your favorite. Range also allows you what Approval allows, 
but you can also express a small preference in Range with little 
harm. If you vote 100 for your favorite, and 99 for the second 
favorite, it is essentially impossible that the 99 would cause your 
second favorite to lose. *But* it might cause your second favorite to 
win over your first. And that is the risk.

If you cannot reasonably predict the election outcome, the correct 
Range vote is the utility of the election as perceived by you, 
normalized to the full range, i.e., you vote 0 for at least one and 
100 for at least one.

This is no more difficult than ranking many candidates. Indeed, it 
might be simpler.

>I'm all for broad support.  What I am against is methods that 
>require voters to be strategic and to have knowledge of the current 
>standings of the candidates for their votes to have the most impact.
>
>Especially when this problem has been solved by other methods.

First of all, as has been shown, there is a problem with the 
Condorcet Criterion and with all purely ranked methods. Ranking is a 
binary function, quite unexpressive. Black and white, so to speak. 
Condorcet seems to be more than that, because it does express more 
than Yes or No for each candidate. But it is still all or nothing in 
each pairwise election.

Voters can maximize the impact of their vote by voting Range in 
Approval style. That takes no great knowledge or strategy. But, yes, 
it reduces the choice; there is thus a secondary maximizing approach 
which consists of rating the candidates in order of worthiness, with 
the least worthy, seriously bad, getting 0, and the most worthy, 
you'd be pleased if any of these were elected at 100. And where you 
are indifferent, rate in between. If you really want your vote to 
count maximally, even though you don't really even know why Fred is 
better than Ralph, but your uncle was named Ralph and you disliked 
him so you prefer Fred, you then order the candidates from bottom to 
top and top to bottom. If there are any ranks left, you leave the 
middle ranks blank.

In high granularity range, this would actually be *almost* maximally 
effective, where you really care, with a relatively small number of 
candidates (like less than ten). And if you have strong preferences 
that discriminate strongly between 50 different candidates, you have 
one complex mind. Frankly, impossibly complex.

The error here is in thinking that you and society benefit by 
equating your mildest preference or aversion with what you really 
care about. They are not equal for you and they are not equal for 
society. Condorcet methods, as we have them, make them equal.

In the end, I think, voters will simply rate the candidates as they 
please, without much regard for strategic considerations, per se. 
They will only think of how good the candidates are, just as you 
want. If they have strong preferences, they will express them with 
strong votes (at or near 0 or 100). And if they don't, they are 
completely free to vote strongly or weakly.

*Yes*, the system puts, it would appear, a pressure on voters to 
abandon the narrow idea that "my benefit is maximized by me always 
strongly acting toward getting my preference, no matter how slight." 
The fact is, though, that such an idea is disastrous personally as 
well as socially....

But election methods in general are defective, and not just because 
of Arrow's theorem. It's not the way you would run a business, if you 
owned it, it's not the way the sovereignty of shareholders is 
expressed in corporations (which use elections to create boards, but 
which could, if the shareholders so wished, strongly limit the powers 
of the board -- and corporations almost always use, if I am correct, 
cumulative voting, which, combined with the use of proxies, and in an 
environment where the shareholders can actually communicate with each 
other -- unfortunately too rare --, is equivalent to Asset Voting).