Date: Nov 13, 2012 11:07 PM
Subject: "Strategically equivalent utility functions" involve two lotteries
My apologies if this appears twice. Cross-posting to a moderated
group seems to have held up the appearance of this post.
I'm following the definition of "strategically equivalent utility
functions" in Keeney et al, "Decisions with Multiple Objectives...":
It says that "u1~u2" if they imply the same preference ranking for any
two lotteries. "Lotteries" is not defined, but the definition I've
found on the web is that a lottery is a complete set of mutually
exclusive outcomes (or "consequences"), along with associated
probabilities that add to 1. Usually, the implication is that there
are different lotteries for the same set of consequences depending
from a decision or action i.e. the probabilities of the consequences
depend on the decision/action, but (I assume) the set of consequences
are the same for the two lotteries.
"Preference ranking" is not defined. In the above definition for
strategically equivalent utitility functions, I assume that the
preference ranking is the ranking of the set of outcomes by the
decision maker. This is determined solely by the utility function,
and not by the probabilities of the consequences. Hence, utilities
that are monotonically related should yield the same rank, regardless
of what specific lottery is being considered (since the lottery
differs from the set of consequences only in that probabilities are
associated with the consequences). Therefore, I am confused by the
the specification of "any two lotteries" in the above definition.
Can anyone please clarify this?