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.