Paul
Posts:
493
Registered:
2/23/10


Re: "Strategically equivalent utility functions" involve two lotteries
Posted:
Nov 14, 2012 2:12 AM


On Nov 14, 1:55 am, Paul <paul.domas...@gmail.com> wrote: >On Nov 13, 11:19 pm, Ray Vickson <RGVick...@shaw.ca> wrote: >>On Tuesday, November 13, 2012 8:07:48 PM UTC8, 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 reread the appropriate sections with your clarification in > mind.
It just occurred to me that an unobvious assumption is needed in order for the clarification to make sense. For each possible pair of lotteries, if utility functions u1 & u2 prefer the same one lottery of the two regardless of the probabilities assigned to their consequences, then we have to make an assumption about what is meant for one lottery to be preferred over the other. The only definition I can fathom is that preferrence is based on the "certainty equivalents" of each lottery, as defined in the preceding page. The preferred lottery is the one with a higher certainty equivalent. Is this correct? (This again is not explained in the text).

