[EM] How to estimate ratings from a set of ranked ballots (was A Framework for adapting single winner methods to the task of Proportional Representation in multi winner districts)

[Forest Simmons]

Kristopher,

Thanks for your constructive comments.  That first version left a lot of
room for improvement, so here goes a second attempt:

As you mentioned methods like PAV based on approval ballots and versions of
Proportional Range Voting based on cardinal ratings style ballots are not
naturally adapted to ordinal ranking style ballots.

In my first attempt at a general framework I basically said if you want to
convert rankings to approval style ballots, just use implicit approval.
Obviously that leaves much to be desired. So the next approximation would
be to give "equal top" rank full approval, while only half approval is
given to rankings strictly between top and bottom, which is what I used in
my first attempt (although omitted from the partial quote below 🙂).

So I want to use this message to take care of this problem, i.e. how to
approximate ratings from rankings:

First let's review why Borda is inadequate.  Borda assumes that ranked
candidates are equally spaced in utility. But this assumption is
incompatible with clone independence:

40 A>B>C>D>E
60 E>A>B>C>D

Assuming equal spacing (as in non-parametric statistics) we have

40 A(4)>B(3)>C(2)>D(1)
60 E(4)>A(3)>B(2)>C(1)

So A is the winner with a score of 4*40 + 3*60, beating out the Condorcet
winner E whose total score is only 4*60, tied with the Pareto dominated
candidate B!

The Pareto dominated candidates B, C, and D artificially prop up candidates
A and B to the point of taking the wind out of the ballot CW.

How do we fix this?

First we tally first place preferences or "favorite" scores for all of the
candidates.  In the above example  A gets forty, E gets 60, and the other
candidates get zero each.

Then we use these tallys to construct the random favorite probability
distribution: P(A)=40%, P(B)=0=P(C)=P(D), and P(E)=60%.

On any given ballot our estimated rating for candidate X will be R=L
/(H+L), where L is the  probability that  (on this ballot) a random
favorite will be ranked Lower (or unranked)) than X, and H is the
probability that a random favorite will be ranked strictly Higher than X on
this ballot.

Notice that the highest ranked candidate will have H = 0. so that its
rating will be L/(0 + L) which is 1, or 100 percent.  Similarly any bottom
candidate on a ballot will have a value of L equal to zero, so its
estimated rating will be 0/(H + 0), which is zero.

If some candidate X on a ballot has the same  values for L and H, which
means that a random favotite is just as likely to be ranked below X as
above X, then the estimated rating is given by L/(H+L) = L/(L+L), which
equals 1/2 or fifty percent.

So on any ballot from the first faction the estimated ratings of the
reaspective candidates are given by R(A) = 60/(0 +60), which equals 1 or
100 percent. While R(E) = 0/(60+0), which equals zero. And R(B) =R(C)=R(D)
which are all equal to 60/(40+60) or 60 percent.

Similarly on any ballot from the second faction in our example the
estimated ratings are given by R(E) =40/(0+40) or 100 percent, and R(A) =
0/(60+0) = 0, and R(X) for the remaining candidates is given by 0/(100+0) =
0.

So the score totals (over all ballots) are T(A) = 40*100% + 60*0, T(E) =
40*0 + 60* 100%, and T(X) = 40*60% +60*0,  (for each of the clones of A).

In sum, E wins with a total of 60, followed by A with a total score of 40,
and finally the (near) clone candidates that are Pareto dominated by A,
with 24 points.each.

(I say "near" clones because in this context where equal first and equal
bottom are allowed, if a candidate falls into one of those extremes on a
ballot and a clone doesn't, then that clone is only a near clone IMHO.)

In my next messaage I'll fix the other problems with my first attempt at a
generalized frameworrk for adapting single winner methods to multiwinner
elections satisfying Proportional Representation.



Kristofer Munsterhjelm <km_elmet at t-online.de> wrote:

> On 1/22/20 12:05 AM, Forest Simmons wrote:
> >
> > The Multiwinner Method I have in mind chooses the winners sequentially.
> > It is based on the idea that ballots have an initial weight of one, and
> > that as candidates supported by a ballot are added to the winners'
> > circle, the weight is reduced according to some rule designed to
> > diminish the influence of the voters who have already achieved some
> > level of satisfaction.
> >
> > At each stage in the election the new seat is filled by the candidate
> > picked by the single winner method applied to the entire ballot set with
> > the current ballot weights in force.
> >
> > How, in general, do we diminish the weight of a ballot? Perhaps the
> > simplest way is to make the current weight 1/(1+S) where S is the
> > current satisfaction obtained by comparing the ballot preferences
> > (whether ratings or rankings) with the winners elected so far. As long
> > as the current satisfaction is zero, the weight remains at one since
> > 1/(1+0) is just one.
>
> A quick reply (been a bit busy lately): Approval methods need to pass a
> weaker proportionality criterion than ranked methods. For Approval, you
> just need to give X a seat if enough voters approve X, but Droop
> proportionality is nested: a vote can contribute to multiple solid
> coalitions at once.
>
> Thus I'm not sure basing a ranked proportional method on Approval will
> lead to a good outcome, at least not if that's not explicitly taken into
> account.
>
> E.g. consider the "D'Hondt without lists" proposal from 2002. It
> combined reweighting with pairwise matrices, but I'm pretty sure it
> fails the DPC.
>
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