[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 15:38:56 PST 2020
In IRV, I think it's the last to be eliminated. Because that meaning of
"runner-up" is rarely if ever defined explicitly in advance, it's in part a
question of perception, and the last candidate to be eliminated is the one
who everyone thinks of as the runner-up in practice.
As you pointed out, that doesn't answer the question of who should be the
second elected in multi-winner race. Using the reverse elimination order is
a semi-proportional system often called "bottoms-up" IRV. Determining who
would have won if that winner had withdrawn is a winner-take-all method
known as "repeated IRV" or "block preferential." And STV is STV.
On Sun, Jan 26, 2020 at 6:02 PM Forest Simmons <fsimmons at pcc.edu> wrote:
> Greg,
>
> Thanks for your contribution.
>
> Take IRV/STV/Hare for example. Is the runner-up the last to be
> eliminated? Or is it the one that would have won if the winner had
> withdrawn?
>
> The second option can be used with any method, but the first option only
> with sequential elimination methods.
>
>
>
> On Sun, Jan 26, 2020 at 2:40 PM Greg Dennis <greg.dennis at voterchoicema.org>
> wrote:
>
>> 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
>>
>> :: Follow us on Facebook <https://www.facebook.com/voterchoicema> and
>> Twitter <https://twitter.com/voterchoicema> ::
>>
>
--
*Greg Dennis, Ph.D. :: Policy Director*
Voter Choice Massachusetts
e :: greg.dennis at voterchoicema.org
p :: 617.863.0746
w :: voterchoicema.org
:: Follow us on Facebook <https://www.facebook.com/voterchoicema> and
Twitter <https://twitter.com/voterchoicema> ::
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