On Nov 14, 12:00 pm, Ray Vickson <RGVick...@shaw.ca> wrote: > On Tuesday, November 13, 2012 10:55:44 PM UTC-8, Paul wrote: > > 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. > > Right: I was in the Man. Sci. Department of the Engineering Faculty at the University of Waterloo, and taught Operations Research courses for more than 30 years, before retiring and moving to the West Coast (specifically, Victoria). > > I have made a few trips back to Waterloo recently. It has changed a lot. There is RIM, of course (the Blackberry makers---taking a pasting in the market lately). Mike Lazaridis (the founder of RIM) gave about $150 million of his own money to found the Perimeter Institute on the banks of Silver Lake, just on the fringe of Waterloo Park. It works on problems similar to those at the Santa Fe Institute, although it perhaps concentrates more on quantum computing, quantum gravity and the like.