[EM] Re: Utilities?

[Jobst Heitzig]

Dear Steve!

Thanks for the clarification!

> Here's a bit more detail.  Suppose a voter not only has the
> preference order A > B > C, she also is indifferent between the
> following two lotteries:
> 
> 1. A has a 1/3 chance of being elected, and C a 2/3 chance.
> 
> 2. B is elected with certainty. (A "degenerate" lottery.)
> 
> How should that be interpreted?  It's not what I would call a utility
> since it's not something that can be compared between voters, which
> is needed to identify the classical utilitarian social choice.  It
> does give an idea of the voter's relative preference intensities-- 
> her utility difference between A and B is apparently twice her
> utility difference between B and C--but I don't know that this
> information, even if sincere, has as much value for making social
> choices as some people believe it does.  For one voter, the
> difference between A and C could mean life or death.  For another
> voter, it might merely mean a tax refund of $400 or not.

Now I see what the lotteries interpretation of utilities is all about. I
find that quite compelling and would agree that *when* someone feels
able to express a preference or indifference between such lotteries,
that can well be transformed in a quantitative measure of preference
strength. Still, as you say, these numbers cannot be compared between
different voters. However, it seems that in order to make James' June
19th proposal work, the lottery interpretation of the numbers will
absolutely suffice, so that we can proceed to work on that proposal!


You asked:
> Jobst, does a person behave differently when forced to choose between
> two alternatives about which he is undecided than when forced to
> choose between two he believes are equivalent?  Why should the
> distinction affect the design of the voting method?
> 
> I don't see anything clearly wrong with allowing the voter to express
> intransitive and/or incomplete preferences, as long as the default
> ballot interface makes it as easy as possible for voters to express 
> orderings.  For the few "irrational" voters who want to express
> intransitive or incomplete preferences, let them select an alternate
> interface that'll accept whatever they need to express.

Well, again my favourite example: Sincere preferences A>B, C>D, no
further preferences. So here I'm undecided between A and C, A and D, B
and C, and B and D. If I would consider A and C equivalent instead, I
would most probably also have the preferences A>D and C>B since usually
my preferences are transitive. The difference shows up easily in the
Hasse diagrams of the corresponding preference quasi-orders:

	undecidedness:		equivalence:
	A	C		  A=C
	|	|		  / \
	|	|		 /   \
	B	D		B     D

When a voters has no undecidedness and his preferences are transitive,
she has a linear ordering in which the possible equivalences are the
ties. When she has some undecided pairs, then she does not have a linear
 ordering but only a partial ordering (when there are no equivalences)
or a quasi-order (when there are also equivalences). That's all basic
order theory, isn't it?

Yours, Jobst