[EM] Linear Spread Median Range Voting

[Abd ul-Rahman Lomax]

The electorama wiki has a page on Median Ratings as a method.

This is basically Range Voting, but the median rating for each 
candidate is determined and the candidate with the highest rating is 
the winner.

There are two basic problems with Median Ratings noted on the wiki page:

(1) Due to the quantized nature of ratings from ballots (and even if 
the ballot allows refined ratings, voters will often choose rounded 
ratings except to express, perhaps, fine distinctions as might be 
wished to vote "almost approval" style, i.e., 100% for one candidate 
and 99.9% for another, ties may be common.

(Note, however, that this may actually be unlikely except where many 
voters vote approval style, *and* there are two candidates with 100% 
approval from a majority of voters. And there is an obvious way to 
resolve this tie.... which will work unless there is a tie among the 
remainder of voters, which is *very* unlikely in an election with a 
substantial number of voters. When there are a small number of 
voters, holding a runoff between the tied candidates would probably 
be the most useful tie resolution method.

(2) Median Ratings appears to fail certain election criteria.

However, the failures shown are dependent upon the tie-resolution 
method, and it is quite possible to design a Median Rating method 
where ties are impossible. I will give an example, which I am 
tentatively calling Linear Spread Median Range. (The spread function 
is linear in the form thus far suggested by me)..

To calculate the Spread Median from the standard median, the standard 
median is first determined. Call this M. There will be N ratings at 
M. In a large election, a coincidence of ratings such that N is 
identical for more than one tied candidate at the median is unlikely 
(and, if it happens, the case is the same as for a tie with FPTP; but 
there is additional information present which can be used to break 
such a tie; the essential method I'd propose is to recalculate the 
election excluding the tied median votes; and this could be iterated 
until and unless the election was a total tie.)

The N ratings at M are adjusted to become N ratings spread in the 
range R of M-0.5 < R < M + 0.5. I'm assuming that integers are the 
only allowed ratings). (The same calculation is done for M = 0 and M 
= max rating.) The exact spread function is not important, it seems; 
what is important is that each rating is unique. The adjusted votes 
are arranged in sequence of rating. Warren Smith stated, I think, a 
general outline of this, except that I haven't been able to 
understand it in the time I've had so far to review it.

Given the new ratings, the median is recalculated. The winner is the 
candidate with the highest spread median rating.

I used a linear spread function that spreads the votes evenly across 
the interval. It is quite possible that a better spread function 
exists. Indeed, I think it likely. However, though I have not 
investigated the question adequately, at this point I suspect that 
the linear function is adequate, and it has the virtue of simplicity.

Ties would be highly unlikely.

Further, every vote counts in determining the median. A vote at the 
extremes will shift the median voter number up or down by 1/2, making 
a small adjustment in the calculated spread median. Unlike the 
standard median, a single vote cannot shift the median rating by a 
whole integer, unless there is only one vote at the median, which 
would be unusual; and in this event, if this swung the election, it 
was essentially a very close election. Practically a tie.

It seems to me, at first glance, that this is a Range method that 
satisfies the Condorcet Criterion as applied to Range Voting (i.e, 
ranking is assumed from the relationship of ratings on a ballot.) 
However, Mr. Smith has asserted that this method can fail the 
Condorcet Criterion; I have not examined this in detail; hopefully it 
will be the subject of another post.

As one possible advantage of this method, it does not reward 
strategic vote exaggeration. A vote is effective in the reducing a 
candidate's election expectation as long as it is less than the 
median, and the opposite with increasing the expectation; it must 
only be greater than the median. Note that "vote exaggeration" of a 
form is expected: that is, voters would wisely, unless they wished to 
dilute their vote, vote at least one candidate at the minimum and one 
at the maximum, with the rest at or in between. Voting more than one 
at one extreme is effectively abstaining from the pairwise election 
between the two, and there is no strategic advantage to this. That 
is, if there are two frontrunners, A and B, and you have a preference 
between them, though you prefer C, not only is there no cost to 
giving C the maximum rating, but, in most elections, you would be 
safe with giving your favorite of the frontrunners the next-highest 
rating. In a closely contested election, as is posited, it is 
unlikely that the front-runners would tie at the next-highest rating 
*unless* it was very low granularity Range; rather, it is much more 
likely that, if they are going to tie, it would be at a lower rating than that.