The ordering need not be random. One can use also a tree structure as
given by the candidates. Clones and near clones will form branches.
The simplest approach is to consider defeats within a branch to be
weaker than defeats between branches.
Juho
On Jun 26, 2010, at 4:01 AM, Jameson Quinn wrote:
2010/6/25 <[email protected]>
>1a. Probabilistic dilemma. If truncation causes a cycle, then there
is some
probabilistic tiebreaker which always includes some chance of C
winning.
This can act as a goad to A and B voters to cooperate. I suspect
that some
system like this might be the theoretical optimum response if voters
were
pure rational agents; however, real people tend not to like
probabilistic
election systems.
Sports fans don't object to the use of a certain amount of
randomization in deciding the order of contests
in a tournament.
True, and good point.
Still, sports aren't elections. Sports are intended to be exciting,
and so an element of chance can be a positive advantage. Also, since
only a limited number of sequential two-way contests are possible,
there is really no alternative to a seed order, unlike with election
systems which have (too) many alternatives.
A single elimination tournament, for example, needs a "seed" order.
If a random order is not used, then
(if I remember correctly) the contestants with the better records
are seeded near the end of the
tournament, to avoid anticlimatic contests at the end of the
tournament.
I suggest the following way of picking a seed order SO for an
election: use the order of a random ballot
refined by the orders of additional ballots until the order is
complete.
One way to use the seed order SO is by single elimination, starting
at the bottom of the list and working
up.
Another (distinct!) way is to elect the lowest alternative on the
list that pairwise beats every alternative
listed above it.
Here's my favorite: initialize X as the highest alternative in the
SO. While X is covered, replace X with the
highest alternative on the SO that covers X. Then elect the final
value of X.
When the seed order SO is the refined random ballot order as given
above or any other social order that
is monotone and clone free, these methods will pick from the Smith
set, while preserving the clone
independence and monotonicity. Furthermore, the last of these (my
favorite), satisfies Independence
from both Smith and Pareto Dominated Alternatives, and will elect
from the uncovered set.
Do these methods solve the truncation dilemma?
Mostly. With a given seed ordering, if you are a B>A>C voter, a
B>A=C vote cannot change the winner from A to B, unless it causes C
to cover B. This is only possible if, with honest ballots, C beats B
and B beats A. Neither of these are consistent with a truncation
dilemma scenario.
There's still a possibility that your ballot is one of the random
ballots that helps define the seed, and so your strategic vote
causes C to come first in the seed order, AND causes A not to cover
C, so that B is then elected. So, your "refined by additional random
ballots until the order is complete" could break the truncation
resistance. I suspect - but am not sure - that a "single random
ballot refined by random choices" seed order would not have a
truncation-dilemma.
... On a separate note, perhaps the "covering" concept is too hard
to explain. How much better is that than simply a single bubble sort
pass up from the bottom of the seed order? That would also guarantee
Smith set.
JQ
----
Election-Methods mailing list - see http://electorama.com/em for
list info
----
Election-Methods mailing list - see http://electorama.com/em for list info
