[EM] Some criteria

[MIKE OSSIPOFF]

Majority Co-operation Criterion (MCC):

Some time ago, James mentioned a co-operation/defection dilemma that can 
occur with Approval. I mentioned that it can occur with Condorcet wv too, 
via offensive order-reversal or the threat of it. ATLO gets rid of the 
problem in Condorcet wv.

The situation was one in which a majority prefer two candidates, A & B,  to 
a third candidate, C.

If the B voters are sure that the A voters are going to responsibly vote for 
A & B, then the B voters can make B win by voting only for B, in Approval. 
Of course how sure of that could they be, if they themselves are considering 
defecting.

When we discussed that subject before, I listed some reasons why I don´t 
consider it a serious problem, and so I won´t repeat those reasons here. 
But, in case I didn´t mention it before, here´s another: If I´m an A voter, 
and maybe the B voters want B to win so badly that they´re willing to risk 
C, and if having A win over B isn´t as important to me as having B win over 
A is to the B voters, then maybe I´ll vote for A & B, even if the B voters 
are likely to vote only for B.

Of course I and other A voters might make it publicly clear that if the B 
voters defect then we won´t ever vote for B or his party´s or faction´s 
candidates again, and that reduces the problem.

The successful defection doesn´t involve a violation of majority rule. Often 
one of A or B will be obviously CW, or more popular, or more the natural 
compromise, and then his/her voters would be in a position to say that 
they´re voting only for him/her because s/he´s the rightful winner.
In Condorcet they could vote C over B, to make a strategic cycle that B 
would win. But, in Condorcet, if the A voters make their ATLO line below A, 
then, when A loses, and there´s a cycle with members above & below that 
line, the A voters, via ATLO, will drop B from their ranking. So, if it´s 
known that they´re doing that, the B voters have nothing to gain by the 
order-reversal. In fact, in general, ATLO deters offensive order-reversal.

Of course after the defection has occurred, and it´s clear that the voters 
on the wrong end of it aren´t going to co/operate anymore, then the problem 
disappears, because the defectors know that it won´t work again.

Anyway, it seems to me that that situation is worth defining a criterion 
about. Eventually I´ll do that, but for now I´ll just say that there should 
be a criterion that a method passes if it doesn´t have that 
co-operation/defection dilemma. I´ll call it the Majority Co-operation 
Criterion (MCC), unless someone insists that another name is better, or if 
more than one person suggests that a particular other name would be better. 
Later I´ll suggest a precise wording for that criterion, though we all know 
what it means already.

Sincere Equilibrium Criterion (SEC):

A few years ago I suggested that we evaluate voting systems according to  
whether or not they have Nash equilibria in which no one reverses a 
preference, where Nash equilibrium is extended many-voter elections.

I suggested a way of extending Nash equilibrium to voting situations. 
Someone else later suggested another way, and suggested that it should be 
considered the only way that could be heir to the name Nash equilibrium, 
because it was more in the spirit of Nash´s equilibrium,  which discusses 
strategies of individual players.

I replied that both approaches, in different ways, are more in the spirit of 
Nash equilibrium. Also, if there are more than one way to extend Nash 
equilibrium to voting, then it doesn´t seem unreasonable for all of them to 
use the name Nash equilibrium, since they´re all voting extensions of Nash 
equilibrium, and because all of them are equally not about strategy-changes 
by one person.

Anyway, there have been maybe 3 such extensions defined on EM: The one that 
I defined when I first suggested the use of such Nash equilibrium extensions 
in EM discussion a few years ago, the similar one that that other person 
defined later, and my most recent simpler definition.

Myself, I don´t find anything problematic about having 3 voting extensions 
of Nash equilibrium, as long as they´re named so that it´s clear which one 
someone is referring to.

Here´s my most recent simpler one:

An outcome (with its ballot configuration) that no set of voters can improve 
on by changing their votes. I´ll call that the "any voters" , or "a", 
version.

My first definition was one of the following two:

An outcome that no same-voting set of voters can improve on by changing 
their votes. I´ll call that the "same once", or s1,  version.

OR

An outcome that no same-voting set of voters can improve on by changing 
their votes in the same way. I´ll call that the "same twice", or "s2" 
version.

I don´t know if that other person´s version was the same as s1 or s2, or was 
just similar to them.

So there are three or four extensions of Nash equilibrium to voting that 
have been defined on EM.

Anyway, I´d like to suggest a criterion about that:

Unfalsified Equilibrium Criterion (UFEC):

If there´s a CW, there should always a voting Nash equilibrium in which no 
one reverses a preference.

[end of definition]

Completely Expressive Equilibrium Criterion (CEEC)

If there´s a CW, there should always be a voting equilibrium in which no one 
reverses a preference, or votes two candidates equal (as I define that 
term).

[end of definition]

Of course the names of those criteria should be followed by the abbreviation 
of the equilibrium version in parentheses.

e.g. UFEC(a), or CEEC(s1)

Mike Ossipoff

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