[EM] truncation dilemma

[Jameson Quinn]

2010/6/25 <fsimmons at pcc.edu>

> >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
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