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Topic: "Strategically equivalent utility functions" involve two lotteries
Replies: 7   Last Post: Nov 14, 2012 1:37 PM

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Paul

Posts: 263
Registered: 2/23/10
Re: "Strategically equivalent utility functions" involve two lotteries
Posted: Nov 14, 2012 1:55 AM
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On Nov 13, 11:19 pm, Ray Vickson <RGVick...@shaw.ca> wrote:
> On Tuesday, November 13, 2012 8:07:48 PM UTC-8, Paul wrote:
>> 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.
>>
>> Can anyone please clarify this?

>
> I don't have access to the book, but I doubt that its presentation
> is very different from the standard. The outcome set is, typically,
> fixed at the start, at least until the utility function has been
> determined; then one can introduce other outcomes because one will
> then be able to compute their utility values.
>
> The preferences are for *lotteries*, not for the outcomes (although
> the ranking of the outcomes influences the ranking of the
> lotteries). Suppose, for example, the two outcomes under discussion
> are: O1 = lose $10, O2 = gain/gain $0 and outcome O3 = gain $100.
> Lottery A might be to win O1 with probability 2/3 and win O2 with
> probability 1/3, while Lottery B might be to win O1 with probability
> 1/4, win O2 with probability 1/2 and win O3 with probability 1/4.
> Mr. Smith might prefer prefer Lottery A over Lottery B, while Mr.
> Jones might prefer B over A.
>
> I am surprised that all this is either not explained in the book, or
> that references to it are not offered.


Thank you sir for that clarification.

The link I provided shows just the definition of strategically
equivalent utility functions, but a link in the upper left of the
image of the printed page will allow users to access an online version
of the book.

I will re-read the appropriate sections with your clarification in
mind.

P.S. If my web browsings are right, I believe you are associated with
a campus from where I got an engineering "option" in MSci a lifetime
ago. I hear the town has changed a lot. Hope it still retains some
of its down-to-earth character.



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