[EM] How to fix the flawed "Nash equilibrium" concept for voting-theory purposes

[Abd ul-Rahman Lomax]

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....