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Re: "Strategically equivalent utility functions" involve two lotteries
Posted:
Nov 14, 2012 12:51 PM
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On 2012-11-14, Paul <paul.domaskis@gmail.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
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
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