<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<HTML><HEAD>
<META http-equiv=Content-Type content="text/html; charset=us-ascii">
<META content="MSHTML 6.00.2900.2802" name=GENERATOR></HEAD>
<BODY>
<DIV dir=ltr align=left><SPAN class=946005520-24122005><FONT face=Arial
color=#0000ff size=2>"<FONT face="Times New Roman" color=#000000 size=3>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?"</FONT></FONT></SPAN></DIV>
<DIV dir=ltr align=left><SPAN class=946005520-24122005></SPAN> </DIV>
<DIV dir=ltr align=left><SPAN class=946005520-24122005><FONT face=Arial
color=#0000ff size=2>On the surface, "multiple equilibria" is kind of an
oxymoron, but the notion may be made precise. </FONT></SPAN></DIV>
<DIV dir=ltr align=left><SPAN class=946005520-24122005><FONT face=Arial
color=#0000ff size=2></FONT></SPAN> </DIV>
<DIV dir=ltr align=left><SPAN class=946005520-24122005><FONT face=Arial
color=#0000ff size=2>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".</FONT></SPAN></DIV>
<DIV dir=ltr align=left><SPAN class=946005520-24122005><FONT face=Arial
color=#0000ff size=2></FONT></SPAN> </DIV>
<DIV dir=ltr align=left><SPAN class=946005520-24122005><FONT face=Arial
color=#0000ff size=2>Anyway, as we approach the end of another Western calendar
year, may I take this opportunity to wish everyone well.
</FONT></SPAN></DIV><BR>
<BLOCKQUOTE
style="PADDING-LEFT: 5px; MARGIN-LEFT: 5px; BORDER-LEFT: #0000ff 2px solid; MARGIN-RIGHT: 0px">
<DIV class=OutlookMessageHeader lang=en-us dir=ltr align=left>
<HR tabIndex=-1>
<FONT face=Tahoma size=2><B>From:</B> election-methods-bounces@electorama.com
[mailto:election-methods-bounces@electorama.com] <B>On Behalf Of </B>rob
brown<BR><B>Sent:</B> Saturday, December 24, 2005 2:50 PM<BR><B>To:</B> Jan
Kok<BR><B>Cc:</B> election-methods@electorama.com<BR><B>Subject:</B> Re: [EM]
Approval Voting elections don't always have an
equilibrium<BR></FONT><BR></DIV>
<DIV></DIV>On 12/23/05, <B class=gmail_sendername>Jan Kok</B> <<A
href="mailto:jan.kok.5y@gmail.com">jan.kok.5y@gmail.com</A>> wrote:
<DIV><SPAN class=gmail_quote></SPAN>
<BLOCKQUOTE class=gmail_quote
style="PADDING-LEFT: 1ex; MARGIN: 0pt 0pt 0pt 0.8ex; BORDER-LEFT: rgb(204,204,204) 1px solid">In
Rob Brown's "Movie Night" introduction to election methods, Rob<BR>suggests
that allowing people to watch the current vote results and<BR>change their
votes as often as they like would lead to a stable<BR>situation where no one
would feel a need to change their vote. (I <BR>believe that
situation is called a Nash equilibrium, is that right?)</BLOCKQUOTE>
<DIV><BR>Yes that is a Nash equilibrium. No individual can improve their
outcome given all other individuals' actions stay fixed.<BR><BR>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. <BR><BR></DIV>
<BLOCKQUOTE class=gmail_quote
style="PADDING-LEFT: 1ex; MARGIN: 0pt 0pt 0pt 0.8ex; BORDER-LEFT: rgb(204,204,204) 1px solid">Here
is a situation where there apparently is no such equilibrium.</BLOCKQUOTE>
<DIV><BR>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?</DIV></DIV><BR>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)<BR><BR>-rob<BR></BLOCKQUOTE></BODY></HTML>