[EM] Free riding

[Raph Frank]

On 9/1/08, Markus Schulze <markus.schulze at alumni.tu-berlin.de> wrote:
> Dear Raph Frank,
>
>  you wrote (31 Aug 2008):
>
>  > Btw, is there a simplified explanation of
>  > your PR-STV method somewhere?
>  >
>  > From what I can see, it compares possible
>  > outcomes pairwise like CPO-STV (but only
>  > compares outcomes that differ by 1 member).
>  >
>  > Ideally, the simplified version would just
>  > need to explain the way to perform that
>  > comparison (if each comparison doesn't
>  > depend on any of the others)
>
>  Suppose that M is the number of seats. Suppose
>  that _A_ and _B_ are two sets of M candidates
>  each. Suppose that sets _A_ and set _B_ differ
>  in exactly one candidate. Suppose that
>  candidate B is that candidate who is in
>  set _B_ but not in set _A_.
>
>  Suppose that each voter casts a complete
>  ranking of all candidates. Then the strength
>  of the win of set _A_ against set _B_ is the
>  maximum value X such that each candidate in
>  _A_ has a separate quota of X votes against
>  candidate B.
>
>  In mathematical terms:
>
>  X is the maximum value such that the voters
>  can be partitioned into M+1 disjoint
>  sets T(1),...,T(M+1) such that:
>
>  1. For all i = 1,...,M: |T(i)| >= X.
>
>  2. For all i = 1,...,M: Each voter in T(i)
>  prefers candidate A(i) to candidate B.
>
>  *********

Thanks, I have created an example indicating my understanding of your
description.

Assume the following

2 Seats to be filled and 3 candidates

A1: Popular candidate (party A)
A2: Other candidate (party A)
B: Party B candidate

Honest rankings are

A1's personal supporters


               20: A1>B>A2
15: A1>A2>B

Party A's supporters
15: A1>A2>B
15: A2>A1>B

Party B's supporters
25: B

Standard PR-STV

Quota = 30

Round 1
A1: 50
A2: 15
B: 25

A1 elected with 20 surplus (8 for B and 12 for A2)

Round 2
A1: 30(-20) Elected
A2: 27(+12)
B: 33(+8)

B elected

Result: (A1,B) wins

Assuming vote management.  Party A tells all supporters to vote A2>A1>B

Round 1
A1: 35
A2: 30
B: 25

A1 elected (5 surplus)

Round 2
A1: 30(-5)
A2: 32(+2)
B: 28(+3)

A2 is elected

Result: (A1,A2)

Vote management has paid off

Schulze's method

Comparing outcomes:

Divide the voters into groups with one group for each member of first
outcome such that the members of each group prefer the group's
candidate to the 'test' candidate (candidate in 2nd outcome not in
first outcome).

Do this in such a way as to make the smallest group as large as
possible.  The strength of the comparison is the the size of that
group.

Assume that the 2 results to be compared are (A1,A2) and (A1,B).  This
is reasonable as A1 has more than a quota of first preferences.

Compare (A1,A2) against (A1,B*)

B is the test candidate.

20 prefer A1 to B (Assign to A1's group)
0 prefer A2 to B
45 prefer both to B (Assign 12.5 to A1 and 32.5 to A2)

All groups are the same size of 32.5

Compare (A1,B) against (A1,A2*)

A2 is the test candidate

30 prefer A1 to A2 (assign to A1)
25 prefer B to A2 (assign to B)
20 prefer both to A2 (assign 7.5 to A1 and 12.5 to B)

All groups are the same size of 37.5

This means that (A1,B) beats (A1,A2) by 37.5 votes to 32.5 votes.

This matches the non-vote management option.  Ofc, the voters voted honestly.

If voters had voted under vote management, then

(A1,A2) vs (A1,B*) would be unaffected as the ordering of A1 and A2
wouldn't matter.  Party A supporters and A1's personal supporters just
spread out evenly in that case.

1) A1's personal vote (35)
2) party A's supporters (30)

Average group size must be 32.5

However,

(A1,B) vs (A1,A2*) would change as the ordering of A2 does matter as
A2 is the test result.

In effect, 'Party' A1 and B consists of
1) A1's personal supporters (35)
2) Party A's supporters who like A1 better (15 but under party A control)
3) B supporters (25)

Party A can remove group 2 from consideration without affecting the
(A1,A2) vs (A1,B) result.

This gives a result of 30 votes per group.

Thus A1,A2 wins under vote management.

I have being trying to find an example that will give (A1,A2) under
standard PR-STV and (A1,B) under Schulze's method.  It might require
more seats, two mightn't be enough or just a different arrangement of
the votes.

In any case, the example does should some of the resistance to vote
management.  The A1 party cannot change the (A1,A2) vs (A1,B*) result.