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The following method (created by Forest Simmons, and called the Simmons method by Michael Ossipoff) is based on score or range style ballots. I believe it satisfies the favorite betrayal criterion, Plurality, the chicken dilemma criterion, Monotonicity, Participation, Clone independence, and the IPDA. It reduces to ordinary Approval when only the extreme ratings are used for all candidates.
I call it MinMaxPairwiseApproval or MinMaxPA for short.
It is based on a concept of “pairwise approval.”
A zero to 100% cardinal ratings ballot contributes the following amount to the “pairwise approval of candidate X relative to candidate Y”:
The amount is either … 100% if X is rated strictly above Y, or Zero if X is rated strictly below Y, or Their common rating if they are rated equally.
According to this definition, the ballot’s contribution to the pairwise approval of X relative to itself is simply the ballot’s rating of X, since it is rated equally with itself.
The method elects the candidate whose minimum pairwise approval (relative to all candidates including self) is maximal.
The motivation for this idea is the question, “If candidates X and Y were the only two candidates with any significant chance of winning the election, what is the probability that the ratings ballot voter would want X approved (in a Designated Strategy Voting system, say)?”
If the voter rated X over Y, this probability would be 100 percent. If the voter rated Y over X, this probability would be zero. If the voter rated both X and Y at 100 percent, this probability would be 100 percent. If the voter rated them both at zero, she would want neither of the approved. If she rated them both at 50%, then our best guess is that there is a fifty-fifty chance that she would approve X. Etc.
Whatever nice properties the method has depends solely on its definition, not the motivation for the definition, so please explore it with an open mind.