[EM] Free riding

[Raph Frank]

On 9/5/08, Kristofer Munsterhjelm <km-elmet at broadpark.no> wrote:
>  Not really. The vote transfers happen indirectly through reduced
> satisfication scores. If a voter doesn't vote for a party and instead votes
> for a group of personal candidates, his satisfication score will be lowered
> for the potential assemblies with only party members in them, and increased
> for the assemblies with independents in them, and so the chance of the
> method picking one of the latter would increase, also.

A voter who ends up with a full strength vote at the end is not represented.

A party with 20% of the vote in a five seater would likely tell its
supporters to vote just for their party.  This effectively wastes
around 3.3% of the vote as they only needed 16.7% of the votes.

Once their candidate is elected, they have no further effect on the
election (unless they also voted for non-party memebers).  The party's
2nd candidate will hover at 10% of the vote until the last count and
unless another party also does badly, will not be elected.

Larger parties have the advantage that they can more easily transfer
votes between their candidates.  All you do is tell your supporters to
approve all your candidates.

For example, assume voters approve only their party's candidate.

(5 seater)

A: 12
B: 12
C: 12
D1,D2,D3:  64

D gets first 4 seats.

A: 12
B: 12
C: 12
D1...D5:  64/5 = 12.80

D also gets 5th seat as 12.8 > 12

Under actual PAV, the results would be

D1,D2,D3,D4,D5
(1 + 1/2 + 1/3 + 1/4 + 1/5 ) * 64 = 146

D1,D2,D3,D4,A
(1 + 1/2 + 1/3 + 1/4 ) * 64 = 133
(1)*12 = 12
Total: 145

So, D still wins all seats even under full PAV.

Now if A,B and C were to form a coalition

A/B/C: 36
D1,D2,D3:  64

D gets 1st, 3rd and 4th seat
A/B/C gets 2nd and 5th seat

>  As for the D'Hondt problem, I suppose it could be generalized to (C *
> v)/(C+s) instead of V/(s+1). I would then pick C = 0.5. However, this may
> still lead to a problem in the left-right-center edge case.

Right.

>  One could also have a range version of ordinary (nonsequential) PAV, where
> v is replaced by the sum of the ratings for those that have already been
> elected.

I think there is a slight issue.  In PAV, the satisfaction of each
voter is determined by

S(N) = sum(1 + 1/2 + 1/3 + 1/4 + .... + 1/N )

Where N is equal to the number of candidates elected.

An approx function could be created that gives S(N) for non-integer N.
 The easiest would be just linear interpolation.  However, log(N) is
pretty close, but has a slight offset.

So maybe:

C(N) = log(N) - S(N)
Creal(x) = linear interpolate C(N)

Sreal(x) = log(x) - C(x)

This gives the right answer at the integers and a smooth curve between
them (maybe to much detail here :) ).

N would then be set equal to the sum of the ratings divided by the max
rating.  E.g. if I rate A as 100 and B as 30, and both are in the
result, then that counts as 1.3 candidates elected, so would take 1.3
terms of the above sum.  This would work out as a satisfaction of
around 1.2 with the above formula, which is between a satisfaction of
1.0 for 1 candidate and 1.5 for 2 candidates.

>  The hard part is finding a way of tallying group support. As mentioned,
> first preference wouldn't work, because then you'd get return of Woodall
> free-riding: parties would say "just vote for a friend as a write-in, then
> the order we give you".

Wouldn't parties want to have as many FPV as possible?

>  I think that option is, like AERLO, ATLO, and approval cutoffs, too complex
> for the ordinary voter to understand; at least unless it was camouflaged as
> a rated ballot, but that would limit the voter's expression since they may
> want "A > B1 > B2" with B1 and B2 both at max transfer value 1, something a
> rated ballot with max transfer derived from the ratings couldn't support.

Another option would be to auto-generate the threshold.  If the first
and 2nd choice are different parties (or maybe a direct mark on the
ballot), then it is assumed to be a personal vote.

A standard max value would then be used.

In theory, if there was some way to measure vote management, then a
party could be punished for vote management by being assigned a lower
value in a later election, but that is probably to complex.

I think 0.75 would be reasonable in most cases, especially if it had
an override.  It might even depend on first preference vote.  If a
candidate gets less than 75% of a quota first choice, then it doesn't
kick in.  Ofc, that allows parties do more complex management, but
convincing a popular candidate that he should aim for less than 75% of
a quota would be a hard sell.