"I'm a
little curious, since you seem to talk about multiple voters switching their
vote together....maybe this really represents a situation where there are
multiple equilibriums, as opposed to no
equilibriums?"
On the surface, "multiple equilibria" is kind of an
oxymoron, but the notion may be made precise.
A system that does not form an equilibirium but alternates
between two end-points is an "oscillatory" system, not "two equilibria". It can
be finitely oscillatory (which is almost always the case in discrete systems
such as we're discussing here) or "infinitely oscillatory".
Anyway, as we approach the end of another Western calendar
year, may I take this opportunity to wish everyone well.
On 12/23/05, Jan Kok <[EMAIL PROTECTED]> wrote:
From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] On Behalf Of rob brown
Sent: Saturday, December 24, 2005 2:50 PM
To: Jan Kok
Cc: [email protected]
Subject: Re: [EM] Approval Voting elections don't always have an equilibriumIn Rob Brown's "Movie Night" introduction to election methods, Rob
suggests that allowing people to watch the current vote results and
change their votes as often as they like would lead to a stable
situation where no one would feel a need to change their vote. (I
believe that situation is called a Nash equilibrium, is that right?)
Yes that is a Nash equilibrium. No individual can improve their outcome given all other individuals' actions stay fixed.
Here, I am defining "improve one's outcome" to mean "change one's ballot such that it now approves all candidates that one prefers to the leader among the other candidates". Even though doing this won't generally change who wins, it can be seen as "narrowing the gap" to one's preferred choices, and therefore we can consider it an improvement in outcome.Here is a situation where there apparently is no such equilibrium.
I'm a little curious, since you seem to talk about multiple voters switching their vote together....maybe this really represents a situation where there are multiple equilibriums, as opposed to no equilibriums?
Also, is it possible that this is a true tie? (that is, a situation whose likelihood of occurring would tend to be inversely proportional to the number of voters)
-rob
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