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Re: "Strategically equivalent utility functions" involve two lotteries
Posted:
Nov 14, 2012 12:51 PM


On 20121114, Paul <paul.domaskis@gmail.com> wrote: > My apologies if this appears twice. Crossposting 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 nonrandomized 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 hrubin@stat.purdue.edu Phone: (765)4946054 FAX: (765)4940558



