[EM] equilibria

[James Green-Armytage]

Adam wrote:
>James, this argument has been advanced before.  Look up Nash Equilibria
>in 
>the archives.

	Okay, I did a quick search. I didn't read all of them, but I want to
reply to a few of the more recent entries that I found.
	Surely, I'm finding a similar concept, but I haven't seen exactly the
same systematic treatment of the subject which I gave more recently... and
some of the conclusions seem to differ from mine...

Mike wrote:
>As we on EM have been using the term for voting systems, a Nash
>equilibrium is an outcome, and the votes configuration that caused it,
>that no set of voters can improve on for themselves by voting
>differently. With IRV there will often by situations (configurations of
>candidates, and voters' preferences among candidates) in which the only
>Nash equilibria are ones in which some voters reverse a preference in
>order to save the win of a CW. With Approval every situation has at least
>one Nash equilibrium in which no one reverses a sincere preference. By
>the way, what I said about IRV above is also true of Plurality, Runoff,
>Borda, and margins Condorcet. 
Mike wrote:
>With Approval or wv Condorcet, every situation has at least one Nash
>equilibrium in which no one reverses a sincere preference. 

	I disagree. I think that if there is no sincere CW, there is no "Nash"
equilibrium for approval or for WV Condorcet. I'm happy for people to try
to prove me wrong, though. 
	Alex gave an example which he claimed was an equilibrium for an
electorate with no sincere CW, but I think I have contradicted him
successfully: See below.

Alex wrote:
>Even if there's no Condorcet Winner there can still be situations where
>at least some Nash equilibria involve all factions using pure strategies.
>e.g. 
>40 A>B>C----All 40 voters approve A and B 
>35 C>A>B----All 35 voters approve C and A 
>25 B>C>A----All 25 voters approve B only 
[A wins]
>(It's easy to verify that this is a Nash equilibrium.) 

	I'm not sure that it is. True, no single "faction" acting on its own can
get a preferable result, but if the C>A>B and B>C>A factions fully
cooperate, they can elect C instead of A. 
	For example, if all of the C>A>B voters stop approving A, the B>C>A
voters approve C... and at least 6 of the B>C>A voters stop approving B. I
know this last one is weird, but it makes sense as part of an agreement
with the C>A>B voters, in that the C>A>B voters refuse to withdraw support
from A unless some of the B>C>A voters withdraw support from B.
	This at least complies with Mike's definition of a Nash equilibrium
"As we on EM have been using the term for voting systems, a Nash
equilibrium is an outcome, and the votes configuration that caused it,
that no set of voters can improve on for themselves by voting differently."
	because there is a set of voters which includes the B>C>A and C>A>B
voters, who have improved the outcome for every member in the set by
electing C instead of A.
	Alex, are you convinced? That is, are you convinced that there can be no
"Nash" equilibrium in approval if there is no sincere CW?

Alex wrote:
>Are there any electorates for which ALL Nash equilibria involve at least
>one faction using a mixed strategy? 
 [in approval voting]

	According to my definition, no. Because if there is a CW, there are
equilibria where every voter approves the CW... these don't require mixed
strategies. And if there is no CW, there is no equilibrium.

Alex wrote:
>It has been proved on this list that if there's a Condorcet winner there
>will be at least one Nash equilibrium that elects the Condorcet winner,
>and all voters use pure strategies in that equilibrium. (see [
>http://groups.yahoo.com/group/election-methods-list/message/9907
>]http://groups.yahoo.com/group/election-methods-list/message/9907). 

	This link doesn't work, since the yahoo outlet for EM was discontinued.
I'd be very interested in finding this message, however.

Jobst wrote:
>Another question is: When can we be sure that equilibria exist? With
>plurality, it seems that the equilibria are exactly those voting
>situations where a majority votes for the Condorcet winner. In all other
>situations, some majority would have an incentive to vote for the
>Condorcet winner instead. Without a Condorcet winner, there would be no
>equilibria.

	What Jobst says here is very very close to what I said more recently. The
only thing I take issue with is "With plurality, it seems that the
equilibria are exactly those voting situations where a majority votes for
the Condorcet winner."
	I already gave an example of a situation where a majority votes for the
CW, but it is not an equilibrium. I will copy it here.  

Sincere preferences, and votes
46: A>B>C; vote for B
10: B>C>A; vote for B
44: C>B>A; vote for C
B is the winner, but this is not an equilibrium Vote, because the 46 A>B>C
voters can gain a preferable result by voting for A instead.

	Otherwise, I think that I agree with what Jobst wrote here. 
	Still, I think that it's worthwhile to make a succinct, systematic
statement of it.
	By the way, so far I've written that plurality can only reach equilibrium
when a sincere CW is elected. However, I think that the same may be true
of many other methods. Interesting to try to catalog them all. Approval, I
think. IRV, I think. Not sure about Borda, winning votes, or margins, but
probably... This is probably the next thing to figure out on this topic.

my best,
James