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

Forest Simmons fsimmons at pcc.edu
Wed Jan 17 17:23:47 PST 2001

```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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