# [EM] Majority Approval/Disapproval based aggregate social order

Forest Simmons forest.simmons21 at gmail.com
Thu Oct 21 13:19:46 PDT 2021

```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

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 at 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
>
> 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
>
>
>
>
>
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