On Jul 25, 2010, at 4:51 PM, Jameson Quinn wrote:
2010/7/23 Kristofer Munsterhjelm <[email protected]>
Jameson Quinn wrote:
I've been thinking recently about systems which enforce chiral
symmetry, making condorcet ties impossible. While it is possible to
"solve" the truncation/burial problem (eg, between two near-clones
who split a weak majority) in this way, I have not been able to come
up with an acceptably simple system. The closest I know of is
"tournament seeding"-style, condorcet-compliant (though not
necessarily condorcet-based) systems, where only certain pairwise
races are considered. In such systems, burial/truncation is a
nonstrategy, period.
I think I read somewhere that elimination tournament methods must
fail monotonicity. I do know that runoff-type elimination based on
weighted positional methods (e.g. IRV being based on Plurality, and
Borda-elimination being based on Borda) must fail monotonicity.
That seems wrong to me. A simple random seeding and binary-tree (as
in world cup elimination) pairwise tournament method is, as far as I
can see, monotone. However, if you base the seeding on random
ballot, then it probably does fail monotone.
If parties are a factor, you could seed based on last-election
performance. For instance, make a matrix of average-number-of-
candidates-in-between, with tied candidates counting as half, and
seed one round at a time so that the closest candidates face each
other. New parties would be seeded randomly. This method would be
very easy to explain to any sports fan (even though my seeding,
unlike world cup seeding, would pair off near-clones - even two
strong near-clones - in the first round, so that mutual burial
between those two groups becomes counterproductive in later rounds).
The problem is, while this method avoids the near-clone burial
problem (when the seeding works), it is vulnerable to a more-general
turkey-raising burial strategy, which could lead to DH3-type
pathology. Oh well.
JQ
There can be many kind of seeding / predetermined ways to order the
comparisons. Different cases offer different kind of strategic
opportunities.
1) purely random
2) derived from the election results
3) given by candidates / parties (parties, trees)
Here's one way to arrange the candidates in category 2.
Start top down. Split the candidates in two categories. Out of all
possible divisions pick the one that divides the candidates in most
clone like groups. Then proceed down in the same way in both branches
until the size of all the undivided subgroups is 1.
One could make the split as follows. (Ties will not be covered.)
Candidates A, B, C, D, E, and F will be divided in two groups. One
possible division is ABC - DEF. If A, B and C are clones and D, E and
F are clones, then this division is perfect. If they are not, then
count points that break this clone property. If someone voted
A>D>B>C>E>F, then this fails being a clone vote. The strength of that
failure is 1 since one can divide this vote e.g. as ADBC - EF and only
one candidate does not follow the given clone division. In vote
A>D>B>E>C>F one can make only divisions that violate the clone
division in 2 places. Then one sums up all the (smallest) violations
in all votes (for all possible divisions). The division with lowest
score is the best division.
Category 2 methods are not be fully strategy free but maybe the
strategies are at least more difficult to apply due to the increased
complexity :-).
Juho
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