[EM] Approval Voting elections don't always have an equilibrium

[Paul Kislanko]

"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. 


  _____  

From: election-methods-bounces at electorama.com
[mailto:election-methods-bounces at electorama.com] On Behalf Of rob brown
Sent: Saturday, December 24, 2005 2:50 PM
To: Jan Kok
Cc: election-methods at electorama.com
Subject: Re: [EM] Approval Voting elections don't always have an equilibrium


On 12/23/05, Jan Kok <jan.kok.5y at gmail.com> wrote: 


In 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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