At 01:11 PM 4/12/2010, Warren Smith wrote:
I am not sure what the Nash equilibrium (or equilibria?) are, but I
am sure that
honest voting is not it, because each individual voter finds burial to
be an "improvement." Presumably the Nash strategy in that scenario
will be a probability-mixture of honest and strategic votes.
You are probably right, but you are shooting yourself in the foot by
implying that von Neumann-Morganstern utilities are not "honest." The
reason is the knee=jerk response to "strategic," which, if we look at
how it functions in Range Voting, simply means not voting stupidly, like this:
Hey, I can write in anyone I like? I think professor Egghead is the
absolute best candidate. In fact, Egghead is so much better than
everyone else, that I'll write the name in, then honestly rate
Egghead at 100%. Everyone else is below 10%....
Well, if Egghead has a chance, maybe! But if not, the "honest" votes
cast with the write-in are stupid, not "strategic." Indeed, the
choice to not even bother writing in the name of a truly preferred
eligible winner is strategic. The set of all possible winners
sometimes is far larger than the set of all candidates on the ballot.
"Strategic" originally meant not voting sincere preference order, and
the concept of suppressing preference because it is moot wasn't
thought of. It is better to think of "sincere" votes, which allows
the sincere categorization of candidates into sets, even if one has
some preference within the set.
You have demonstrated that a common decision-making process has a
stupid Nash equilibrium, and the idea of changing the voting system
to avoid this is nuts, because it's enough that the voting system
allows true preference to be expressed without harm. The basic
problem is that in voting, the probability of the outcome changing
because of a single vote is miniscule, and Nash equilibrium is
defined based on individual choices, not collective ones.
I don't agree, by the way, that the Nash equilibrium in a situation
where it is thought that everyone will vote for Hitler is nothing
other than the vote for Hitler.
Consider this: if I think a certain way, I am a human being and it is
likely that other human beings think like me. Therefore the choices
that I make in an election are likely to be repeated by others. Many
others, in fact. I can consider myself as deciding on behalf of an
entire set of voters, and so I must consider the possibility that my
choice is, in fact, controlling.
Indeed, good zero-knowledge voting strategy would assume this.
Indeed, as long as there is some non-zero probability that the
outcome would switch, then the value of voting for Gandhi is
non-zero, and therefore voting for Hitler (or abstaining from voting)
is less than the value of voting. On the other hand, if one is
convinced that everyone else will vote for Hitler, time to buy those
travel tickets....
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