8Decisions under Uncertainty

8Decisions under Uncertainty - X such that v = a + bu for...

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Choice Under (Objective) Uncertainty Primitives : X = { x 1 ,...,x n } , a set of outcomes. L = { p R n + | p i = 1 } the set of probability distributions (lotteries) on X . e i , the element of L that assigns probability 1 to x i X % , a binary relation on L (with ± and defined in the usual way). Axioms : (A0) n 3 and e n ± e n - 1 ± ... ± e 1 (A1) % is complete and transitive . (A2) % is continuous : For any p,q,r in P with p % q % r , there is an α in [0 , 1] such that αp + (1 - α ) r q. (A3) % satisfies independence : For any p,q,r in P we have q % p if and only if αq + (1 - α ) r % αp + (1 - α ) r for all α (0 , 1) . (A4) % is monotone : If q ± p and 0 α < β 1 then βq + (1 - β ) p ± αq + (1 - α ) p. 1 Theorem Assume (A0). The binary relation % on L satisfies (A1)-(A3) if and only if there is a real-valued function u on X such that for any p,q in L q % p if and only if n X i =1 u ( x i ) q i n X i =1 u ( x i ) p i . (1) Moreover, if v is a real-valued function on
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Unformatted text preview: X such that v = a + bu for real numbers a and b, with b &gt; , then (1) remains true with v in place of u. Exercise : Prove the last sentence of the Theorem. Write U ( p ) = u ( x i ) p i , so U ( e i ) = u ( x i ). Since p = p i e i , if follows that U ( p i e i ) = U ( e i ) p i , so the representation U of % is linear. The function U on L is called the von Neumann-Morgenstern (vN-M) utility . I will often abuse terminology and refer to the function u on X as the vN-M utility. 1 (A4) follows from (A0)-(A3) (proof?), but I assume it in my proof of the Theorem to save a few steps. 1...
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