[EM] Ordinal-cardinal combination methods

Kristofer Munsterhjelm km_elmet at t-online.de
Sun Nov 14 16:00:37 PST 2021

It's quite difficult to make a method that's not either consistently 
ordinal or cardinal. Lottery information somehow is between these, and 
hard to make use of without reverting to either ordinal or cardinal.

My generalized STAR method is vulnerable to cloning. It determines the 
potential winner set as the smallest set of candidates so that in every 
three-candidate cardinal election (cumulative voting, Range, etc.), two 
candidates drawn from that set always beat the third; and then uses a 
Condorcet method to rank the candidates in that set.

However, if there is an ordinal CW and this CW is cloned twice, then the 
elections (A1, A2, X) where A is the CW and X is someone else, is always 
won by one of the clones, assuming the rated convention of a clone. So 
the potential winner set collapses to A1 and A2, and then one of them 
wins by conventional Condorcet.

Cloning A forces what's essentially two-candidate comparisons into the 
three-way comparison logic. The whole point of the three-way comparison 
being to have the least number of candidates required to do a cardinal 
comparison (so that fewer candidates obscure each other and lead to a 
Burr dilemma).

The problem can be solved by not normalizing the inferred Range ballots 
- but then the whole method just turns into Range due to the latter's 
IIA compliance. So the method can either align along the ordinal grain 
and become Condorcet, or the cardinal and become whatever the base 
method is (as long as this method passes IIA).

The cardinal Condorcet lottery idea suggests that comparisons of more 
than three candidates also have their place, since lotteries can have 
any number of candidates with nonzero chance of winning. So another 
"natural" way to imagine a cardinal-ordinal method would be one that 
somehow uses k-candidate comparisons for all k > 1: from pairwise 
(Condorcet) to full data. But how? The obvious way is IRNR (instant 
runoff with cardinal normalization instead of Plurality counting). But 
that method inherits IRV's various undesirable features like 
nonmonotonicity. (Though IRNR wih lp-normalized cardinal ballots might 
not be quite as center-squeezy as plain IRV.)

It's harder than one would think.


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