# [EM] Proportional Representation via Approval Voting (fwd)

Forest Simmons fsimmons at pcc.edu
Fri Jan 19 07:51:51 PST 2001

```Bart, sorry to take so long in reply.

Let's do your first example with the two factions having relative sizes of
67% and 33% respectively.

The combinations of two and their respective scores are:

{A,B} 67%*(1 + 1/2) + 33%*(0) = 1.005
{C,D} 67%*(0) + 33%*(1 + 1/2) = 0.495

{A,C}, {A,D}, {B,C}, {B,D} all get scores of
67%*1 + 33%*1 = 1.000

So in this case {A,B} barely wins.

If the percentages were 66% and 34% respectively, {A,B} would get
66%*(1 + 1/2) = .99 , and would barely lose to the four tied combinations
having scores of 1.00 .

The (1 + 1/2) factor is because of two layers.  If a combination had three
layers of representation for a voter or faction of voters, the multiplying
factor would be (1 + 1/2 + 1/3).

On Wed, 17 Jan 2001, Bart Ingles wrote:

> I'm afraid I still don't grasp the procedure.  Maybe a simple example
> with two seats to fill, and two voter factions, would help illustrate
> the effects & strategy, as well a the actual counting procedure, of the
> method.
>
>
> Pct. of     Acceptable
> -----------------------
>   67%          A, B
>   33%          C, D
>
> How should the voters vote in this case, and how are the votes counted?
> What if the percentages were 66%/34%?  You can assume ties go to
> candidates in alphabetical order.
>
>  . . .
>
> I proposed a proportional approval method a couple of years ago which
> worked much like STV, except that the quotas were recalculated every
> time a winner was declared (this latter would also help STV work with
> less-than-full ranking).  I doubt that I was the first to propose it
> though -- I think E. M. Bolger wrote a couple of papers on unranked
> multi-winner methods, including one similar to this.  I can post the
> algorithm if anyone is interested, and if it isn't already obvious from
> the above description.
>
> I used the same name (but PA instead of PAV), but would rather see it
> used to denote any unranked proportional method in which the voter
> suffers no penalty for approving excess candidates, other than a
> reduction in utility if one of his compromise choices wins instead of
> his true favorite (similar to single-winner approval voting).
>
> Bart
>
>
>
> Forest Simmons wrote:
> >
> > Michael Welford has independently hit upon the same method as mine for
> > Proportional Representation via Approval Voting.
> >
> > I'm forwarding his brief explanation, since I still haven't had time to
> > get around to the "inexorable" logic that leads to it, and some of you are
> > still waiting for a simple explanation.
> >
> > ---------- Forwarded message ----------
> > Date: Tue, 16 Jan 2001 23:42:53 +0000
> > From: Michael Welford <welfordm at earthlink.net>
> > To: Forest Simmons <fsimmons at pcc.edu>
> > Subject: Re: [EM] Proportional Representation via Approval Voting
> >
> > Wow! This sounds like a PR system I've been playing with for a few weeks
> > now.  The key to proportional approval voting (PAV) as I conceptualize it is
> > to assign to each voter a kind imputed utility that I call fair satisfaction,
> > and to maximize that sum of all voters satisfaction scores.
> > Here is my description if PAV. If one of the candidates chosen by a voter is
> > elected the voter gets one point. If a second candidate chosen by that voter
> > is elected an the voter gets an additional half point. The n-th candidate
> > selected by that voter adds 1/n to that voters satisfaction score. With a
> > computer it's easy to find the set of cnandidates that maximizes the sum of
> > fair satisfaction scores over all the voters. If you insist on being able to
> > do a hand count you need to restrict each voter to 3 or maybe 4 choices.
> > It's easy to see why increments of 1/n assure proportionality. Suppose we
> > want to elect 4 and we have a large faction with about 80%  of the voters and
> > a small faction with about 20%. The large faction has gets 3 candidates
> > elected and is in contention for a fourth. If the small faction has more that
> > 20% of the voters then the gain in their total satisfaction score for having
> > their candidate elected is more 1 point times 20% while the large faction
> > gains less than 1/4 points times 80% on having their fourth candidate
> > elected. So to maximize total satisfaction the fourth candidate elected is
> > the one favored by the smaller faction. The readers can verify for themselves
> > that no strategy for the large faction can keep the small faction from
> > getting a representitive if the small faction has more than the 20% quota.
> > It turns out that PAV generalizes the method of Jefferson and d'Hondt. In
> > fact any of the divisor methods of Balinsky and Young can be used as basis
> > for a form of PAV. For instance, if the satisfaction increments are 1, 1/3,
> > 1/5, 1/7,.. We have a generalization of Websters method.
> > Some STV advocates have suggested, that if a representative elected under
> > that system cannot serve, a substitute can be found by using IRV on those
> > ballots that were considered as supporting the candidate to be replaced. A
> > replacement candidate can be found under PAV, by recounting the votes with
> > the constraint that all the other candidates originally elected are elected
> > again. So that, for example, someone who got one guy in on the original vote
> > ( not counting the guy to be replaced )  has his vote count only 1/2 for the
> > replacemant. While a voter who didn't have any of their choices elected, has
> > their votes count at full strength.
> >
> > and some of it's advantages a little clearer to the others on this list.
> >
> > Forest Simmons wrote:
> >
> > > Just as the STV system of proportional representation uses the front end
> > > of the Borda Count (ballots with the candidates ranked), so can the front
> > > end of Approval Voting (ballots marked with approved candidates) be used
> > > to achieve another system of proportional representation.
> > >
> > > Think in terms of layers.  If you are bundling up for extreme cold, the
> > > more layers the better, but the first one or two layers count the most; it
> > > doesn't do much good to add a third layer of sweaters for the torso if the
> > > legs are still uncovered. An extra pair of mittens is good, but that first
> > > pair of ear muffs is more urgently needed.
> > >
> > > The layers of protection for various body parts are analogous to layers of
> > > representation for the various parts of the body politic. (And just as
> > > long johns cover arms, legs, and torso, so some candidates may represent
> > > more than one segment of the electorate.)
> > >
> > > Various multiwinner systems of voting can be specified by assigning
> > > various weights to the various layers.
> > >
> > > It turns out that for proportional representation, the nth layer must
> > > receive 100/n percent of the weight of the first layer.
> > >
> > > For example, in any multiwinner election having six or more positions to
> > > be filled, the sixth layer should count only one sixth as much as the
> > > first layer.
> > >
> > > In general we have n positions to be filled from among N candidates. Let C
> > > be one of the N!/n!/(N-n)! possible combinations (coalitions) of n
> > > candidates, and let B be one of the approval ballots cast in the election.
> > > Suppose that ballot B approves exactly k of the members of the coalition
> > > C.  Then this ballot contributes support for this coalition in the amount
> > > 1 + 1/2 +...+ 1/k .
> > >
> > > The coalition with the highest total wins the election (unless side
> > > conditions are not met).
> > >
> > > A side condition might be, for example, that no one approved by fewer than
> > > 1/(n+1) of the voters could be elected. This reduces to requiring majority
> > > approval in a single winner race. This condition is easily incorporated by
> > > eliminating such candidates before examining the various possible
> > > coalitions.
> > >
> > > Another side condition might be that the candidate with the greatest
> > > over-all approval should be in the winning coalition. (This is not
> > > automatic.) This side condition would be appropriate if the winning
> > > coalition is to be the entire government, and one member of the coalition
> > > is to be the moderator or president.
> > >
> > > This condition can be incorporated by automatically including him in all
> > > of the coalitions to be evaluated. In other words, every admissible
> > > coalition is made up of the most popular candidate and n-1 other
> > > candidates.
> > >
> > > If there are three positions of preeminence, say a triumvirate, the method
> > > can be used to choose the coalition of three first, and then another
> > > application of the method in which every admissible coalition contains the
> > > winning three and n-3 other candidates.
> > >
> > > If the triumvirate is to have a distinguished member, three successive
> > > applications of the method to the voted ballots would do the trick; the
> > > first round picks the prez by restricting all coalitions to one member.
> > > The second round restricts coalitions to three members, one of which is
> > > the prez.  The third round considers only coalitions containing the
> > > triumvirate.
> > >
> > > Note that the same ballots (without alteration) are used in all three
> > > rounds.
> > >
> > > Well, that pretty well describes the voting procedure.
> > >
> > > If you want the gory details of some examples and a proof of the
> > > proportionality property, I'll post another message next time.
> > >
> > > I hope you like it.
> > >
> > > Forest
>
>

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