Approval Sorted Margins is a good way to get an aggregated social ranking of the candidates ... but here's another one with it's own charms inspired by the question what to do when more than one or fewer than one candidate gets majority approval ... First the manual version ... For now let's assume an odd number of voters to keep things simple ... At stage n each voter submits an approval ballot constrained by a consistency rule explained below.* Each alternative gets a point of plus or minus 1/2^n depending on whether or not it was approved on more ballots than not or vice-versa (that is approved on fewer ballots than disapproved ... for the vice-versa part). At any stage n when the candidates' scores attain complete numerical distinction, their numerical order gives the finish order. *If at some previous stage k<n a voter V's approval/disapproval for candidate X is contradicted by the majority decision, then the voter is locked in to its approval/disapproval decision for that candidate at all subsequent stages ... consistency requires this doubling down... think about it! A faithful simulation "instant" version of this method is easy to devise on the basis of Dyadic Approval ballots or "ranked rankings" in general. ...more on this next time ... FWS

How to use dyadic approval ballots for this MA/D method ... Suppose voter V's ranked ranking ballot is A>B>>C>D>>>E>F>>G>H Then in the first round V will approve A, B, C, &D and will disapprove E,F,G,&H. In the second round V's nominal approvals are A,B, E,& F , while nominally disapproving the others. In the third round, V's projected approvals are A, C, E, & G. We say "nominal" because the consistency rule trumps these tentative approval/disapproval projections. Suppose for example, that a majority of voters approve E in the first round, contradicting V's disapproval of E in that round. As a result E ends up with half a point after the first round instead of with the negative half point that voter V judged to be more fitting. If the majority disapproves V in all subsequent rounds, then D's final point total will be 1/2 - 1/4 - 1/8 ... > 0, more than V deemed appropriate ...still the best (i,e, smallest) V could hope for after E's initial (undeserved according to V) positive point. So, if V is consistent with her original judgment of E, she will disapprove E in all subsequent rounds, as the consistency rule requires, even though her nominal (projected) choices for E in the respective rounds were ... Disapprove, Approve, Approve, respectively, ... which would have worked out to give E a point total of -1/2 + 1/4 + 1/8 = -1/8 if V had had her druthers. I hope this example helps to clarify the consistency rule! Next .... how to handle ties ... FWS El mié., 20 de oct. de 2021 10:03 p. m., Forest Simmons < forest.simmons21@gmail.com> escribió: > Approval Sorted Margins is a good way to get an aggregated social ranking > of the candidates ... but here's another one with it's own charms inspired > by the question what to do when more than one or fewer than one candidate > gets majority approval ... > > First the manual version ... > > For now let's assume an odd number of voters to keep things simple ... > > At stage n each voter submits an approval ballot constrained by a > consistency rule explained below.* Each alternative gets a point of plus or > minus 1/2^n depending on whether or not it was approved on more ballots > than not or vice-versa (that is approved on fewer ballots than disapproved > ... for the vice-versa part). > > At any stage n when the candidates' scores attain complete numerical > distinction, their numerical order gives the finish order. > > *If at some previous stage k<n a voter V's approval/disapproval for > candidate X is contradicted by the majority decision, then the voter is > locked in to its approval/disapproval decision for that candidate at all > subsequent stages ... consistency requires this doubling down... think > about it! > > A faithful simulation "instant" version of this method is easy to devise > on the basis of Dyadic Approval ballots or "ranked rankings" in general. > > ...more on this next time ... > > FWS > > > > >