Rubinstein2005-page118 - inFnite space Z requires...

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October 21, 2005 12:18 master Sheet number 116 Page number 100 LECTURE 9 Risk Aversion Lotteries with Monetary Prizes We proceed to a discussion of a decision maker satisfying vNM as- sumptions where the space of prizes Z is a set of real numbers and a Z is interpreted as “receiving $ a .” Note that in Lecture 8 we as- sumed the set Z is Fnite; here, in contrast, we apply the expected utility approach to a set that is inFnite. ±or simplicity we will still only consider lotteries with Fnite support. In other words, in this lecture, a lottery p is a real function on Z such that p ( z ) 0 for all z Z , and there is a Fnite set Y such that z Y p ( z ) = 1. We will follow our general methodology and make special assump- tions that Ft the interpretation of the members of Z as sums of money. Let [ x ] be the lottery that yields the prize x with certainty. We will say that % satisFes monotonicity if a > b implies [ a ]Â[ b ] . Thus, if u is a vNM utility function representing a monotonic pref- erence relation, then u is a strictly increasing function. An axiomatization (not presented here) of vNM preferences on an
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Unformatted text preview: inFnite space Z requires strengthening of the continuity assump-tion so that if p  q , then small changes in the prizes, and not just in probabilities, leave the preferences unchanged. ±rom here on we focus the discussion on preference relations over the space of lotter-ies for which there is a continuous function u (referred to as a vNM utility function), such that the preference relation is represented by the function Eu ( p ) = ∑ z ∈ Z p ( z ) u ( z ) . This function assigns to the lot-tery p the expectation of the random variable that receives the value u ( x ) with a probability p ( x ) . The following argument, called the St. Petersburg Paradox , is some-times presented as a justiFcation for assuming that vNM utility func-tions are bounded. Assume that a decision maker has an unbounded vNM utility function u . Consider playing the following “trick” on him:...
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