[EM] Re: Utilities?

[Jobst Heitzig]

Dear Steve!

you wrote:
> Well, first a minor point... I believe the word 
> "linear" is not used when there are equivalences.
> A linear ordering means the same as a "strict" 
> ordering.  When there are equivalences, the term 
> in the literature is usually "weak" ordering. 
> (I sometimes call a weak ordering a nonstrict 
> ordering, or just an ordering.)  

You are right here of course!

However, order theorists like me frequently use "order" or "ordering" as
a synonym of *partial* order instead of weak ordering. Also, the term
"weak ordering" is often avoided by order theorists since it
misleadingly suggests that it is something weaker than a (partial)
order, which it is not. Instead, we most often call that class of
relations "total quasi-orders", "total" being the order-theoretic name
of what others call "completeness" (while "completeness" refers to the
existence of suprema and infima in ordern theory). So much about
terminology -- it always gets confusing when people with different
background talk, but this should be no real problem :-)

> 
> 
> It may be misleading to say that because voters 
> tend to have transitive preferences that it's 
> reasonable to assume when A?C is changed to A=C 
> that it will also lead to A>D and C>B.  So, 
> to avoid confusing the issues when studying 
> the difference between undecidedeness and 
> equivalence on voter behavior, I think we
> should prefer an example such as this:
> 
> 
>   undecidedness      equivalence
>      A    C              A=C
>      |\  /|               |
>      | \/ |               |
>      | /\ |               |
>      |/  \|               |
>      B    D              B=D
> 
> 
> Would a voter behave differently given the 
> undecidedness preferences on the left instead 
> of the equivalence preferences on the right?
> How would the undecided voter behave if allowed 
> to express equivalence but not undecidedness 
> when voting?  Is the distinction significant 
> enough to make it worth complicating the voting 
> method?  I'm sorry I didn't make my questions 
> clear earlier and I hope they are clearer now.
> 

You made it perfectly clear, I think. As for your example, I indeed
think that it will probably make no significant difference for most voters.

What I suggested however in my old paper was that the expression of
"undecidedness" between A and B (A?B) may be interpreted as the wish to
*delegate* the decision about A,B to the other voters in order to
improve the result in view of incomplete information or lack of
expertise, while expression of "equivalence" of A and B (A=B) may be
interpreted as the statement that they should have the same probability
to be chosen at the end, so that a coin would decide between A and B if
they were the only feasible alternatives.

Hence, the natural decision rule for the case of just two alternatives
A,B under this interpretation would...
   choose A if #(A>B) > #(B>A),#(A=B),
   choose B if #(B>A) > #(A>B),#(A=B),
   throw a coin if either #(A=B) >= #(A>B),#(B>A) or #(A>B)=#(B>A).
When I was thinking about this differnce, however, I mostly had
competitions with a small jury in mind such as dancing or design
competitions.

Instead, the main reason I insist on not misinterpreting undecidedness
as equivalence is the fact that in contrast to me many people on this
list do require preferences to be *transitive*. And with this additional
provision it is well important to distinguish the two notions, as my
original example shows: The "correct" preference relation A>B,C>D *is*
transitive (and will thus be deemed "rational" by most people here). But
the "incorrect" one with 4 equivalences instead of 4 undecidednesses is
A>B,C>D,A=C=B=D=A which is *not* transitive since A >= B >= C without A
>= C (it is only *quasi*-transitive)!

So, the problem is, when we allow equivalence and undecidedness but
require transitivity, our class of possible preference relations is that
of all *partial* orders, while if we only allow equivalence it is the
smaller class of all *weak* orderings.

Yours, Jobst