One way to look at this is that with one issue, you basically have two types of candidates: liberal (positive CP) and conservative (negative CP). Candidates can be "strong" or "weak" liberals (or conservatives) based on the magnitude of their CP's. Similarly, voters are either liberal (normalized W equal to +1) or conservative (normalized W equal to -1). In the simplest no-strategy case, all liberal voters vote for liberal canditates and conservative voters vote for conservative candidates. If there are 5 liberal candidates, 5 conservative candidates, 51 liberal voters, and 49 conservative voters, then under Approval all 5 liberal candidates will get 51 votes and all 5 conservative voters will get 49 votes. Thus the majority candidates will all be tied, and whatever method is used to break the tie will not likely result in the most liberal candidate (the CR winner) being selected.In this scenario, if the "Candidate Positions" are evenly spaced, the sincere CR winner would be the LEAST liberal
Ken Johnson
candidate. If this candidate has any first-preference supporters, then this candidate will also be the sincere Condorcet
winner (because this candidate's supporters can combine with all the conservative voters to pairwise beat all the more
liberal candidates and with all the other liberal voters to pairwise beat all the conservative candidates.)
If this candidate has no first-prefernce supporters (so, unlike the candidates, the voters are not evenly-spaced) then of
course it is possible that any of the liberal candidates could be the the CW (and maybe also the outright majority favourite)
but given the presence of a bare minority of conservative voters, it isn't possible that the MOST liberal candidate could be
the sincere CR winner (which I take to mean the candidate with the highest "Social Utility").
Chris Benham
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