L061030ChoiceUnderUncertaintyWEB

# L061030ChoiceUnderUncertaintyWEB - S.Grant ECON501 2 CHOICE...

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S.Grant ECON501 2. CHOICE UNDER UNCERTAINTY Ref: MWG Chapter 6 Subjective Expected Utility Theory Elements of decision under uncertainty Under uncertainty, the DM is forced, in e f ect, to gamble. A right decision consists in the choice of the best possible bet, not simply in whether it is won or lost after the fact. Two essential characteristics: 1. A choice must be made among various possible courses of actions. 2. This choice or sequence of choices will ultimately lead to some consequence , but DM cannot be sure in advance what this consequence will be, because it depends not only on his or her choice or choices but on an unpredictable event . 1 S.Grant ECON501 Simple and Compound Lotteries X = ( f nite) set of outcomes (what DM cares about). •L set of simple lotteries (prob. distributions on X with f nite support). A lottery L in L is a fn L : X R ,thatsat is f es following 2 properties: 1. L ( x ) 0 for every x X. 2. P x X L ( x )=1 . Examples: Take X = { 1000 , 900 ,..., 100 , 0 , 100 , 200 900 , 1000 } 1. A ‘fair’ coin is F ipped and the individual wins \$100 if heads, wins nothing if tails L 1 ( x )= ½ 1 / 2 if x { 0 , 100 } 0 if x/ { 0 , 100 } 2

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S.Grant ECON501 2. Placing a bet of \$100 on black on a (European) roulette wheel L 2 ( x )= 18 / 37 if x = 100 19 / 37 if x = 100 0 if x/ { 100 , 100 } 3 .Ap a cko f52p l a y ingc a rd si sshu f ed. Win \$200 if the top card is an Ace, lose \$500 if the top card is the Queen of Spades otherwise no change in wealth. L 3 = 1 / 13 if x = 200 47 / 52 if x =0 1 / 52 if x = 500 0 if { 500 , 0 , 200 } 3 S.Grant ECON501 4. A ‘balanced’ die is rolled. Win \$100 if number on top is even & win nothing otherwise. L 4 ( x ) L 1 ( x ½ 1 / 2 if x { 0 , 100 } 0 if { 0 , 100 } A compound lottery is a two-stage lottery in which the outcomes from the f rst-stage randomization are themselves lotteries. Formally, a compound lottery is a fn C : L R ,thatsat is f es the following 2p ropert ies : 1. C ( L ) 0 for every L L , with strict inequality for only f nitely many lotteries L . 2. P L L C ( L )=1 . 4
S.Grant ECON501 Example: A‘ fa i r ’co ini s f ipped and the individual then plays out L 2 if heads and L 3 if tails. C 1 ( L )= ½ 1 / 2 if L { L 2 ,L 3 } 0 if L/ { L 2 3 } REDUCTION : ‘Multiply through’ 1 st -stage prob. to reduce a compound lottery to a one-stage lottery. I.e., if α 1 ,...,α n are the prob. of the possible 2 nd -stage lotteries L 1 ,...,L n then the reduction is the lottery α 1 L 1 + α 2 L 2 + ... + α n L n Example cont.: Reduction of C 1 ( L ) is lottery R 1 =(1 / 2) L 2 +(1 / 2) L 3 , i.e., R 1 ( x 1 / 26 if x =200 9 / 37 if x =100 47 / 104 if x =0 19 / 74 if x = 100 1 / 104 if x = 500 0 otherwise 5 S.Grant ECON501 Consequentialism: assume individual indi f erent between any compound lottery and the associated reduced lottery. We can also see that the set of lotteries is a ‘mixture space’.

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## This note was uploaded on 09/04/2010 for the course ECON 501 taught by Professor Grant during the Spring '10 term at Rice.

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L061030ChoiceUnderUncertaintyWEB - S.Grant ECON501 2 CHOICE...

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