[EM] Is there a standard way of defining "runner-up" in the context of single winner elections?
Greg Dennis
greg.dennis at voterchoicema.org
Sun Jan 26 14:40:26 PST 2020
I always thought the "standard" way of defining runner-up was that it lived
in the definition of the voting rule itself. In Arrow's definition, a
social welfare function determines a single social preference ordering from
a set of individual preference orderings, so it's whoever's second in that
social ordering. Maybe that's begging the question, but the answer for most
single-winner methods is pretty obvious / well-defined.
On Sun, Jan 26, 2020 at 4:27 PM Forest Simmons <fsimmons at pcc.edu> wrote:
> Richard,
>
> Thanks for your enthusiastic reply. I think it is a very good idea in the
> general context of "how do we define second choice?"
>
> But what I'm looking for is in the limited context of a single winner
> election how o we decide who came closest to beating the actual winner? In
> other words, who turned out to be the greatest rival of the winner for the
> single seat of a single winner election?
>
> We're not saying that this greatest rival should be the next candidate to
> be seated in a multi-winner election.
>
> For example. in an approval election the candidate with the second
> greatest approval would be the chief rival of the approval winner by any
> reasonable standard, but would probably not be the winner of the next round
> in the multi-winner context because voters who approved this runner-up
> would have the weight of their ballots cut in half for the second round.
>
> So it's not exactly what I was looking for, but very good related
> information!
>
> On Sun, Jan 26, 2020 at 10:57 AM VoteFair <electionmethods at votefair.org>
> wrote:
>
>> On 1/25/2020 3:43 PM, Forest Simmons wrote:
>> > We need a standard way of defining a second place candidate or
>> > "runner-up" for single winner elections.
>> > 1. One way is ....
>> > 2. ....
>> > 3. ....
>> > 4. Any other ideas?
>>
>> Yes to number 4.
>>
>> Please take a look at:
>>
>> https://electowiki.org/wiki/VoteFair_representation_ranking
>>
>> I'm not sure what your word "standard" means. But hopefully you intend
>> to mean "fair." If so, that's what VoteFair representation ranking is
>> all about.
>>
>> Specifically "VoteFair representation ranking" looks deeply into the
>> ballot info to correctly identify which candidate/choice is most popular
>> among the voters who are not well-represented by the winner of the first
>> seat.
>>
>> And it does so in a way that appropriately reduces the influence of
>> well-represented voters to the extent they exceed a 50% threshold.
>>
>> Forest, thanks for asking this important question.
>>
>> Richard Fobes
>>
>>
>> On 1/25/2020 3:43 PM, Forest Simmons wrote:
>> > Now that we have "de-cloned Borda" by changing the rankings to what we
>> > could call "pseudo ratings," we need just one one more ingredient for
>> > our PR framework:
>> >
>> > We need a standard way of defining a second place candidate or
>> > "runner-up" for single winner elections.
>> >
>> > 1. One way is to re-run the election with the winner removed from the
>> > ballots to see who the new winner is.
>> >
>> > 2. Another is to see which loser gave the winner the greatest pairwise
>> > opposition.
>> >
>> > 3. Another is to see which candidate needs the fewest "plump" votes to
>> > become the winner.
>> >
>> > 4. Any other ideas?
>> >
>> >
>> >
>> > Kristofer,
>> >
>> > 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
>> > <mailto: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.
>> >
>> >
>> >
>> > ----
>> > Election-Methods mailing list - see https://electorama.com/em for list
>> info
>> >
>>
> ----
> Election-Methods mailing list - see https://electorama.com/em for list
> info
>
--
*Greg Dennis, Ph.D. :: Policy Director*
Voter Choice Massachusetts
e :: greg.dennis at voterchoicema.org
p :: 617.863.0746
w :: voterchoicema.org
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