[EM] Toy election model: 2D IQ (ideology/quality) model

[Jameson Quinn]

Here's a toy model where the math is easy and you can get some interesting
results.

-Voters are distributed evenly from [-1, 1] along the ideology dimension.
-Candidates are represented by an ordered pair (i,q) where i is an ideology
from -1 to 1 and q is a quality from 0 to 2.
-The utility of a voter with ideology v for candidate (i,q) is: U(v) =
|v-i|+q

Some initial results:


   - The overall utility (integral) for a candidate is 2q - (i^2).
   - Assuming that the candidate set for an election does not contain any
   strict dominances (any candidate who is preferred by all voters over
   another) means that if you drew a line connecting all the candidates in
   order of ideology, it would never be sloped more than 45 degrees.
   - If candidate O is at (0,1) (average-quality centrist) and candidate
   Q is at a random location which is neither dominated by nor dominating O,
   then O always wins. (That is, for Q to win, they would have to be
   dominant.) The chances that, despite losing, Q's overall utility is greater
   than O, are 1/6. Roughly speaking, a centrist has a 1/6 chance of
   undeservedly winning a given pairwise contest.
   - If a candidate L is at (-0.5,1) (average-quality "nominee of the
   leftist party", that is, the center of the left half of the distribution),
   and candidate Q is random non-dominated-or-dominant, then the chances Q
   wins are 11/20. Assuming that Q wins, the chances that it's undeserved are
   2/11. This position at -0.5 ideology (or 0.5) is special in that L never
   wins undeservedly.
   - Looking at the above two facts, it is entirely rational to be slightly
   concerned about the problem of a "weak CW" in this model. Though it is
   likely overall that the CW is the utility winner (UW), there is a far
   higher chance of a centrist CW who is not a true CW than a non-centrist CW
   who is not a CW.
   - In particular, it's reasonable to want a method which, in a
   Burlington-like scenario, could give either the centrist CW or the pairwise
   winner of the other two (leftist in the case of Burlington), but not the
   plurality winner/Condorcet loser (rightist in Burlington). That's not
   merely a matter of hypocritical/ad-hoc adherence to the Condorcet pairwise
   principle; it's justified.

I could go further. For instance, it would be possible to find the
probability that honest normalized Range, honest normalized median, or
honest non-normalized median would each elect the utility winner/loser
under given constraints. Generally speaking, I like this model; it's a toy,
but a useful one.

Jameson
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20111106/bb1a8891/attachment-0002.htm>