[EM] group strategy equilibria

Steve Eppley seppley at alumni.caltech.edu
Mon Aug 23 17:38:21 PDT 2004

Alex S wrote:
> Group strategy equilibria are too common to be of much 
> interest to me. In many voting systems, including Condorcet 
> systems when there's a cycle, it will frequently be easy to 
> find a group strategy equilibrium where some group of people 
> acting in concert change the outcome. 

That's a disequilibrium, not an equilibrium, isn't it?
I presume Alex meant to write disequilibrium there.

I agree with James & Mike & others that group strategy eqs
are worth paying attention to.  Some of us have speculated
that some GSEs are easier for a majority to coordinate
than others, which is why we care about criteria like
minimal defense (which is nearly equivalent to Mike's 
Strong Defensive Strategy criterion, SDSC).

I don't mind disequilibrium when there's a cycle as long 
as the outcome will usually stay within the top cycle 
(in the election context we're talking about, where the 
top cycle will tend to be centrist compromises) since the 
candidates will have incentives to take centrist positions.

> By contrast, individual voter equilibria are too common
> to be of interest, since any election with a margin
> greater than one vote will be an equilbrium.
> The Nash equilibria that I defined in my previous message, 
> where the players are groups of voters with identical 
> preferences, are "just right." They're common enough that 
> you can always find one to study, but rare enough that they 
> aren't littering the place with trivial examples (any 
> election with margins of 2 or more votes). That balancing 
> act is one of the many reasons that Nash got his Nobel 
> prize. He found a phenomenon that's ubiquitous and hence 
> broadly applicable, but not so common as to be trivial.

That sounds wrong to me, on two counts.  First, it seems 
an overly bold assumption to say that voters with identical
preferences will behave as one coordinated group.  Second,
I don't believe Nash did that, and if I'm right about this 
it has nothing to do with why he received the Nobel prize.
Nash equilibria aren't useful when there are many voters, 
but they're very useful when the number of players in 
a game is small, and thus worthy of a prize.

There are other interesting refinements of the notion of 
equilibrium.  For instance, a strategy available to a 
player may be "dominated" by another strategy, in the 
sense that the other strategy will always work at least
as well for him and sometimes will work better.  A 
"rational" player will never choose a dominated strategy, 
so it's common to neglect all equilibria that incorporate 
dominated strategies.  And Myerson-Weber equilibria have 
been written about many times in this maillist.


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