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

[Paul Kislanko]

I am definitely not intending an argument but once again we've hit upon how
slippery the language can be without proper context.
 
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. 
 
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.
 
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).
 
Thanks for the wishes, and the same to all!


  _____  

From: election-methods-bounces at electorama.com
[mailto:election-methods-bounces at electorama.com] On Behalf Of rob brown
Sent: Saturday, December 24, 2005 4:11 PM
To: Paul Kislanko
Cc: election-methods at electorama.com
Subject: Re: [EM] Approval Voting elections don't always have an equilibrium


On 12/24/05, Paul Kislanko <kislanko at airmail.net> wrote: 



Rob Brown wrote: 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. 


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. 

For instance Nash's famous proof is that there is *at least* one Nash
equilibrium for certain well defined types of games:

In this article they give an example where there are 11 equilibria:
http://en.wikipedia.org/wiki/Nash_equilibrium



  Anyway, as we approach the end of another Western calendar year, may I
take this opportunity to wish everyone well. 



Likewise, and have a Merry Christmas as well if you celebrate such a thing.
:)

-rob


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