# [EM] average over time proportionality election method

raphfrk at netscape.net raphfrk at netscape.net
Sat Mar 4 10:27:57 PST 2006

```The method works as follows:

For the first election, the candidate elected would be just the
plurality winner.

However, the winning candidate would get to keep votes equal to the
excess over the 2nd highest candidate.

All other candidates would get to keep all of their votes.

In subsequent elections, the candidate with the most votes would again
win.  However, the vote total for each candidate would be the votes
cast for the candidate in the election plus any they kept from the
previous election.  This leads to the long term average number of seats
for the candidate matching the candidate's proportion of votes.

There would also probably be need to be a rule to handle a candidate
who retires/dies so that his excess can be passed to someone else.  I
think a candidate should not be allowed to pass on excess unless the
candidate is retiring to minimise voters ending up voting for someone
they didn't really support and to prevent the party machines from
controlling excess transfers.

This gives the benefits of a random ballot while being deterministic (
I think ? ).

The disadvantage is that the averages are over time, and people may
prefer to vote for a candidate for the current election rather than
support a candidate so they can win in a later election.

On the other hand, voters might be happy with a musical chairs like
effect, especially if the elections are held often.  3rd party voters
would get to see their candidate gaining ground every election and then
getting the seat rather than having no hope of electing anyone.  If the
party wins a seat one third of the time in 100 districts, then it will
have a reasonably constant number of seats ranging from 30-36 ish
depending on which seats the party holds at a given time.

This could also be used for a multi seat district.  The cost of being
elected would be the number of votes held by the candidate who holds
the most votes who didn't get elected.

This would allow a single state district easily count the votes.  There
are no transfers, each counting location just announces how many votes
each candidate received.  The cost per seat in that election would be
the same statewide.

The main point is that wasted votes are "recycled" for use in the next
election so no votes are wasted in the long run.

There is a startup effect which favours the larger parties/candidates,
but then the cost to get elected closely matches the turnout and
candidates with any level of support will gets seats in proportion to
their level of support.  If retiring candidates are allowed transfer
their excess, then this will only have an effect for the first 3-4
elections after the system is started.

a candidate who is elected (scaled accordingly) and all votes for a
candidate who is eliminated, that don't specify a valid next choice
would passed to the candidate who is ranked first for use in the
subsequent election.  I am not sure how much of an improvement it would
be, but it would allow voters get a compromise candidate "now" rather
than their favorite later.

An example set of 10 elections with 4 candidates.  The turnout is 1000
+/- 50% and the candidates have supports of 10%, 20%, 30% and 40% of
the voters respectively.  The vote each candidate gets in each election
is +/- 50% nominal.

The results are:

(sorry about formatting, needs to be courier)

Election 1
Votes carried forward   0   0   0   0
Votes cast             58 125 182 352
Total                  58 125 182 352

Turnout               717
4 wins at cost 182

Election 2
Votes carried forward  58 125 182 170
Votes cast             57 234 479 500
Total                 115 359 661 670

Turnout               1270
4 wins at cost 661

Election 3
Votes carried forward 115 359 661   9
Votes cast            120 260 468 539
Total                 235 619 1129 548

Turnout               1387
3 wins at cost 619

Election 4
Votes carried forward 235 619 510 548
Votes cast            105 124 267 356
Total                 340 743 777 904

Turnout               852
4 wins at cost 777

Election 5
Votes carried forward 340 743 777 127
Votes cast             85 209 309 508
Total                 425 952 1086 635

Turnout               1111
3 wins at cost 952

Election 6
Votes carried forward 425 952 134 635
Votes cast             65 114 374 369
Total                 490 1066 508 1004

Turnout               922
2 wins at cost 1004

Election 7
Votes carried forward 490  62 508 1004
Votes cast             80 200 311 219
Total                 570 262 819 1223

Turnout               810
4 wins at cost 819

Election 8
Votes carried forward 570 262 819 404
Votes cast             89 184 174 435
Total                 659 446 993 839

Turnout               882
3 wins at cost 839

Election 9
Votes carried forward 659 446 154 839
Votes cast             92 244 468 508
Total                 751 690 622 1347

Turnout               1312
4 wins at cost 751

Election 10
Votes carried forward 751 690 622 596
Votes cast            155 169 322 237
Total                 906 859 944 833

Turnout               883
3 wins at cost 906

Summary
Percentage of votes: 0.09 0.18 0.33 0.40
Percentage of seats: 0.00 0.10 0.40 0.50

The result slightly favours the more popular candidates.  This is
because at the start they get "cheap" seats.

In the first election, candidate 4 gets the seat and it only costs 182
votes.  Near the end, the cost per seat approaches the average turnout
of 1000.

Running 13 elections and excluding the first 3 gives a more balanced
result.  The results after all 13 elections are run are shown for a few
sets of 13 elections.

Summary
Percentage of votes: 0.10 0.22 0.33 0.35
Percentage of seats: 0.00 0.20 0.40 0.40

Summary
Percentage of votes: 0.09 0.20 0.32 0.39
Percentage of seats: 0.10 0.20 0.30 0.40

Summary
Percentage of votes: 0.09 0.22 0.31 0.38
Percentage of seats: 0.10 0.30 0.30 0.30

Summary
Percentage of votes: 0.09 0.23 0.27 0.41
Percentage of seats: 0.10 0.20 0.30 0.40

Summary
Percentage of votes: 0.09 0.18 0.33 0.40
Percentage of seats: 0.10 0.20 0.30 0.40

Candidate 1 got his fair share for 4 of the 5 sets of elections and -1
for the other
Candidate 2 got his fair share for 4 of the 5 sets of elections and +1
for the other
Candidate 3 got his fair share for 4 of the 5 sets of elections and +1
for the other
Candidate 4 got his fair share for 4 of the 5 sets of elections and -1
for the other

The results show that excluding "startup" effects, the seats are fair
to all candidates.

Running 110 elections and ignoring the first 10 gives even better
accuracy:

Summary
Percentage of votes: 0.10 0.20 0.29 0.41
Percentage of seats: 0.10 0.21 0.30 0.39

Each candidate gets within 1 seat of the correct amount except
candidate 4, who gets 2 less than expected.
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