# [EM] Comparing ranked versus unranked methods

Adam Tarr atarr at ecn.purdue.edu
Tue Feb 12 15:52:33 PST 2002

```Forest Wrote:

> n A and C
> n B and C
> 51-n A only
> 49-n B only
>
> As long as 2*n is greater than 51, the winning
> combination is {A,C} in sequential PAV.
>
> But when 2*n is not too much greater than 51 [up until 65],
> the winning combination is still {A,B} in regular PAV.

So I was wrong... good thing I admitted that possibility!

This is a really interesting result, since this preference structure
could be easily produced by the classic "left, middle, right" three
candidate race.  C, the middle candidate, approval winner, and likely
Condorcet winner, is NOT elected in PAV.  This actually makes sense,
as it strikes a better balance between extremes to elect left and
right.  Only when middle has much stronger support than the weaker
extreme (65 vs. 49) does the popularity of middle override the desire
for a balanced council.

As you mentioned, d'Hondt was the rule used for discounting ballots
that had already elected one candidate.  I have observed that using
Webster's rule can create some strategic problems.  Specifically, in
a perfect-information election, a faction can do better for itself by
arranging to split its votes up among candidates, rather than having
the entire faction vote the entire slate of candidats from that
faction.  For example, say there are 4 seats to be filled.  One
faction has 1203 voters, and the other faction has 1200 voters.
Faction 1 votes for all four of its candidates with every vote.
Faction B devises a scheme where the voters split into three groups.
So the votes go,

1200 A,B,C,D
401 E,F
401 G,H
401 I,J

The first candidate elected is one of A,B,C, and D.  But the next
three candidates elected are one out of each of the sub-factions of
the second group.  The same outcome occurs if you apply PAV with
Webster's rule in the non-sequential fashion.  (Side note: I'm almost
sure sequential and non-sequential PAV are equivalent if there is no
overlap in the votes between various voting factions.)  Using
d'Hondt's rule, this vote-splitting technique nearly backfires, but
produces the proper 2-2 council split.

Even if we try to combat this by using modified Webster's
(1.5,3,5,... in stead of 1,3,5,...), these manipulations are still
possible, although more difficult.  Here's an example, with a 6-seat
race, where one faction has 1202 votes and the other has 1000.

1000 A,B,C,D,E,F
601 G,H,I
601 J,K,L

In this scenario, two candidates are elected out of each sub-faction,
which gives four seats to the strategically split faction where three
seats for each faction is more fair (it is what would be given by
either Webster's OR d'Hondt if the voting were simply 1202 to 1000).
With d'Hondt's rule, the election has the proper 3-3 split regardless
of whether the larger faction splits up or not.

Using d'Hondt's rule, this sort of offensive strategic manipulation
by clever vote-splitting appears to be impossible... it seems obvious
from playing with examples, although I'm having trouble coming up
with a clean way to explain it.  So, it looks like d'Hondt might be
the better choice for PAV for strategic reasons.  I consider this to
be a shame, since Webster's is the most proportional method we could
choose.  But when it comes to campaigning for the methods, d'Hondt
has two advantages: it seems more intuitive, and it can be called
Jefferson's method.  And no good American can disagree with our
illustrious third president.