Representative Sets for Stochastic Dominance Rules
Advising someone as to his best course of conduct is always treacherous and trying to steer his economic course can be perilous to personal relationships. If we knew his utility function, economists might say, we could confidently and with courage make optimal selections for anyone. But economic counselors would likely know, at most, only some salient characteristics of the advisee. Depending on what and how much we know about a client’s utility function we could more or less sharply delineate the options which are inferior for him. This idea is at the root of all studies under the topic of “stochastic dominance rules.” Given incomplete information about a person’s utility function the best that we can do is to classify him accordingly to one or more sets of utility functions. He may be one of any number of people whose utility functions are members of some given set. We know no more or less about his utility function than any other in the set. Any choice between two uncertain prospects appropriate to him would likewise be appropriate to anyone else whose utility function is in the same set because this set defines the limits of information about the utility functions we consider. It is equivalent then to ask; “If we know Tom’s utility function can be characterized as thus and so which choice should he make?” or, “If everyone we consider has utility functions which can be characterized as thus and so, which choice should they unanimously make?” The latter question is in the spirit of stochastic dominance research. Primarily, stochastic dominance rules dictate procedures for discovering unanimous orderings of uncer?tain prospects appropriate for utility functions within specified sets.
KeywordsUtility Function Concave Function Stochastic Dominance Lebesgue Dominate Convergence Theorem Dominance Rule
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