Minmax (SOP) and Median (non-finite cost exponent)

First I propose a new Minmax system, applicable to all voting methods, that restores Condorcet to Minmax (pairwise opposition) (MMPO): elect the candidate whose greatest opposition is least, X’s opposition from Y being determined by the lexicographically minimal strength of preference (SOP) for Y (Y=X having 0 strength and Y<X having negative strength) among the ceil[50%] of votes with greatest SOP for Y. For a sufficiently large odd number of votes, this amounts to opposition from Y being determined first by the median voter’s SOP for Y and then (for oppositions tied in median SOP) by the number of votes with above-median SOP for Y.

In traditional ranked voting, Minmax (SOP) lies between MMPO and Minmax (winning votes): like MMPO, it includes non-winning oppositions; unlike MMPO, it ignores (technically, considers strictly weaker) X’s oppositions from candidates a majority prefers X to. In traditional approval voting, Minmax (SOP) reduces to Approval.

Next I explore the implications, for median systems in general, of my interpretation of score (including approval) voting and ranked voting as generalized cumulative voting with an infinite and infinitesimal (respectively) cost exponent.

Under my interpretation, of two voters giving X the same score, the voter giving X a larger fraction of the sum of his (the voter’s) scores is giving X the (infinitesimally) higher score. This makes Median Ratings more decisive (e.g., with approval ballots, it can elect a winner even when there isn’t a unique majority-approved candidate) and alters Majority Judgement (e.g. requiring it to stray less from the median and making it capable of electing a candidate other than the approval winner even when approval ballots are used). However, I disagree with Majority Judgement and prefer electing the candidate with the highest lexicographic minimum score among the ceil[50%] of voter’s scoring him highest (which also may elect a candidate other than the approval winner even when approval ballots are used).

Under my interpretation of ranked voting, a voter’s score for a candidate is effectively equal to the infinitieth exponent of the number of candidates he’s ranked above. The implication for Minmax (SOP) is that SOP is determined primarily by the number of candidates the preferred candidate is ranked above, secondarily by the number of candidates the other candidate is ranked above.