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<DIV dir=ltr align=left><SPAN class=224331922-24122005><FONT face=Arial
color=#0000ff size=2>I am definitely not intending an argument but once again
we've hit upon how slippery the language can be without proper
context.</FONT></SPAN></DIV>
<DIV dir=ltr align=left><SPAN class=224331922-24122005><FONT face=Arial
color=#0000ff size=2></FONT></SPAN> </DIV>
<DIV dir=ltr align=left><SPAN class=224331922-24122005><FONT face=Arial
color=#0000ff size=2>In physical sciences, "there are 11 equilibria" would be
expressed as there is no "equilibrium but there are 11 stable solutions to the
system." Perhaps "never the twain shall meet", but it would be nice if the
vocabulary were cross-discipline. </FONT></SPAN></DIV>
<DIV dir=ltr align=left><SPAN class=224331922-24122005><FONT face=Arial
color=#0000ff size=2></FONT></SPAN> </DIV>
<DIV dir=ltr align=left><SPAN class=224331922-24122005><FONT face=Arial
color=#0000ff size=2>The way Nash defined a "Nash equilibrium" there can be more
than one, which is part and parcel of the difficulties social scientists have.
Chemists faced with a reaction that beuatifully changes the container of a flask
from blue-on-top and red-on-bottom to red-on-top and blue-on-bottom describe an
"oscillating equilibrium". Mathemations would call it a
bifurcation.</FONT></SPAN></DIV>
<DIV dir=ltr align=left><SPAN class=224331922-24122005><FONT face=Arial
color=#0000ff size=2></FONT></SPAN> </DIV>
<DIV dir=ltr align=left><SPAN class=224331922-24122005><FONT face=Arial
color=#0000ff size=2>I think my point (if I had one) was that we should be
careful about how we use terms. A "Nash equilibrium" is defined in terms of
strategies and counter-strategies. It is not an inherent attribute of any voting
method. By Nash's definitions if there are well-defined game-players voting as
blocs then one or more Nash equilibria can be established regardless of the
counting method (I think it has been written here that someone has proven
that).</FONT></SPAN></DIV>
<DIV dir=ltr align=left><SPAN class=224331922-24122005><FONT face=Arial
color=#0000ff size=2></FONT></SPAN> </DIV>
<DIV dir=ltr align=left><SPAN class=224331922-24122005><FONT face=Arial
color=#0000ff size=2>Thanks for the wishes, and the same to
all!</FONT></SPAN></DIV><BR>
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<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 4:11 PM<BR><B>To:</B> Paul
Kislanko<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 12/24/05, <B class=gmail_sendername>Paul Kislanko</B> <<A
href="mailto:kislanko@airmail.net">kislanko@airmail.net</A>> wrote:
<DIV><SPAN class=gmail_quote></SPAN>
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<DIV dir=ltr align=left><SPAN><FONT face=Arial color=#0000ff size=2>Rob
Brown wrote: <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> <BR></DIV></SPAN></BLOCKQUOTE>
<DIV dir=ltr align=left><SPAN><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></BLOCKQUOTE>
<DIV><BR>Hmmm, aside from my glaring error in pluralizing "equilibrium" :)
.... I'm pretty sure that the concept of equilibrium allows there to be more
than one. <BR><BR>For instance Nash's famous proof is that there is *at least*
one Nash equilibrium for certain well defined types of games:<BR><BR>In this
article they give an example where there are 11 equilibria:<BR><A
href="http://en.wikipedia.org/wiki/Nash_equilibrium">http://en.wikipedia.org/wiki/Nash_equilibrium</A><BR><BR></DIV>
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style="PADDING-LEFT: 1ex; MARGIN: 0pt 0pt 0pt 0.8ex; BORDER-LEFT: rgb(204,204,204) 1px solid"> <SPAN><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.
<BR></FONT></SPAN></BLOCKQUOTE></DIV><BR>Likewise, and have a Merry Christmas
as well if you celebrate such a thing.
:)<BR><BR>-rob<BR></BLOCKQUOTE></BODY></HTML>