[EM] Re: Alex: Nash Equilibrium

Alex Small alex_small2002 at yahoo.com
Wed Nov 24 09:34:01 PST 2004

Fair enough.  Your definition was not vague, simply less restrictive than mine.  The type of equilibrium that you were defining is simply a different concept from the one that I was defining.
I think the reason why I called it "vague" is that at some point in that thread the subject of debate was Nash equilibrium, a very particular type of equilibrium.  Due to the way the discussion had been going, I assumed that you were trying to define Nash equilibrium for voting, instead of some other type of equilibrium.  That was my mistake.
The great thing about Nash equilibrium is that if everything in your game is suitably specified (and the criteria aren't all that strict) then at least one Nash equilibrium is guaranteed to exist.  Your equilibrium obviously isn't a Nash equilibrium, since for some voting methods there will be situations where no equilibrium meeting your criterion exists.  (see my example with Approval Voting and a Condorcet cycle)

Also, I have acknowledged that my formulation of Nash equilibrium for voting is not the only valid way to define a Nash equilibrium.  I have always pointed out that you could treat individual voters as players instead of factions of voters.  I have gone on to argue, however, that such a treatment is not particularly useful, and that my definition of the players is the more useful one.  That's all.
Anyway, I apologize for misunderstanding and thinking that you were trying to define a Nash equilibrium.  I recognize that you were defining a different type of equilibrium.
Also, as to whether or not approval strategy depends on cardinal preferences in addition to ordinal preferences:

I completely agree that in the case of incomplete information it is necessary to factor in your utilities when deciding how to vote in an approval election.  The equilibrium that I was defining, however, referred only to hindsight.  I specified that no player _could_ have changed the outcome to something more preferable.
So, before an election a group of people with identical ordinal preferences A>B>C might disagree over whether it makes sense to approve B, because they don't yet know whether that will hurt A or C more, and some will put a higher premium on helping A while others will put a higher premium on defeating C.  But after an election, when the results are known, everybody in that faction will be able to agree in hindsight on what should have been done.
Admittedly, people don't vote with the benefit of hindsight, but (for good or for ill) equilibria are frequently (but admittedly not always) defined with regard to complete information.
I think this straightens things out.

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