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.