We have seen that in the three-candidate, three-faction, cyclical case , A beats B beats C beats A, considerations of symmetry (thanks to Saari) lead naturally to dA= fpA - fpC as a measure of A's claim to victory.

If we add fpA + fpB + fpC = n to dA, we get

n+dA = 2fpA + fpB,

which we recognize as the Borda count for A in the scenario ...

fpA: ABC

fpB: BCA

fpC: CAB

Our question was how to generalize fpA - fpC to a larger context. Can we just adopt Borda as the answer?

No, because in a less specific context we cannot depend on Borda to satisfy clone-independence. So this is where we respectfully must part company with Saari: our main justification for advancing beyond single-mark Plurality voting (in single winner elections) is to solve the "spoiler" problem, which is an example of dependence, particularly clone-winner failure.

Are there clone-free ways to adapt Borda to a larger context? Yes, there are various ways of decloning Borda, some more complicated than others. One is to trade in rankings for ratings ... leading to Range Voting.

If the ratings are on a scale of zero to two, then the method is obviously equivalent to Borda in the case of three candidates ... provided equal rankings are allowed and counted appropriately in Borda. This version of Borda is equivalent to "the number of equal-top minus the number of equal-bottom" rule. In the pairwise matrix formulation it becomes the "row sum minus the column sum" rule, as long as there are only three candidates.

The "Top minus Bottom" rule is clone-free in general, but the "Row minus Column" rule is clone free only in the three candidate case. However, you can salvage it in the form of the "Min Row member minus Max Column member" maximization rule.

To reduce the number of negative quantities involved, it is convenient to recast the latter rule as the "Max Column member minus Min Row member"

minimization rule.

To be continued ...

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