On 2012-11-14, Paul <email@example.com> wrote: > 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...": > http://tinyurl.com/anrxytj
> 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.
If one specifies rankings between non-randomized actions only, the utility function cannot be essentially defined. If one introduces lotteries, and adds reasonable assumptions, one can get "a" utility function if the preferences satisfy the assumptions.
I put the "a" in quotes, because multiplying the utility function by a positive number and adding any real number provides an equivalent utility function. The lotteries are needed to provide enough input to specify the class of functions.
> Can anyone please clarify this?
-- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University firstname.lastname@example.org Phone: (765)494-6054 FAX: (765)494-0558