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uncertainty - 12 Uncertainty 12.1 Want Expected utility...

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12 Uncertainty Lottery: ( π , ( x i ) n i =1 ) = p 1 x 1 p 2 x 2 . . . p n x n ( p i 0, P n i =1 p i = 1). Assumptions: L1. ( e j , ( x i ) n i =1 ) x j , L2. ( π , ( x i ) n i =1 ) ( σ [ π ] , σ [( x i ) n i =1 ]) , L3. q ( π 1 , ( x i ) n i =1 ) (1 q ) ( π 2 , ( x i ) n i =1 ) ( q π 1 + (1 q ) π 2 , ( x i ) n i =1 ) . here, σ is a permutation. In fact, L1-L3 could be required only for a two-outcomes lotteries. Then expand via L 1 and L 3 to more general lotteries. As a result any lottery can be described as ( π , ( x i ) n i =1 ), L –space of lotteries. Utility representation: ( π , ( x i ) n i =1 ) Â ( κ , ( z j ) m j =1 ) u ( π , ( x i ) n i =1 ) > u ( κ , ( z j ) m j =1 ). 12.1 Expected utility Want: u ( p x (1 p ) y ) = pu ( x ) + (1 p ) u ( y ) . (2) Theorem: If ( L , º ) satisfy U1-U4 , then there exists u that satis fi es (2). U1 (Continuity). { p | p x (1 p ) y º z } and { p | z º p x (1 p ) y } are closed sets for all x , y , z L . U2 . If x y , then p x (1 p ) z p y (1 p ) z . U3. Best, b , and worst, w , lotteries exist. U4. p b (1 p ) w is preferred to q b (1 q ) w i ff p > q . Proof: (start) u ( b ) = 1, u ( w ) = 0; for each z , set u ( z ) = p z de fi ned by p z b (1 p z ) w z .
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12.2 Risk aversion Consider money gambles for simplicity. Comparing expexted utility of a gamble to the utility of the expected payo
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