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
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