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Paradoxes in Decision Making With a Solution

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Lottery 1 $3000 S1 $4000 $0 80% 20% R1 80%20%

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Lottery 2 $3000 $0 25% 75% S2 $4000 $0 20% 80% R2

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Lottery 2 $3000 $0 25% 75% S2 $4000 $0 20% 80% R2 35% 65%

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Lottery 3 $1,000,000 S3 $5,000,000 $1,000,000 $0 10% 89% 1% R3

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Lottery 4 $1,000,000 $0 11% 89% S4 $5,000,000 $0 10% 90% R4

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Lotteries 3 and 4 60% migration from S3 to R4 Is this a problem???

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Allais Paradox (1953) Violates “Independence of Irrelevant Alternatives” Hypothesis (or possibly reduction of compound lotteries) Example: §Offered in restaurant Chicken or Beef order Chicken. §Given additional option of Fish order Beef

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Restatement - Lottery 1 S1 oooo o $3000 R1 oooo o $4000 $0

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Restatement - Lottery 2 S2 oooo o $3000 oooo o $0 R2 oooo o $4000 $0 (80%) (20%) oooo o $0

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Restatement - Lottery 3 S4 oooooooooo ooooooooo $1,000,000 o $1,000,000 oooooooooo $1,000,000 R4 oooooooooo ooooooooo $1,000,000 o $0 oooooooooo $5,000,000

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Restatement - Lottery 4 S4 oooooooooo ooooooooo $0 o $1,000,000 oooooooooo $1,000,000 R4 oooooooooo ooooooooo $0 o $0 oooooooooo $5,000,000

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p3p3 p1p1 p2p2 Marschak-Machina Triangle 3 outcomes: Probabilities:

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4000 0 p2p2 p3p3 p1p1 3000 R1 (0.2, 0, 0.8) S1 R2 (0.8, 0, 0.2) S2 (0.75, 0.25, 0)

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p3p3 p1p1 P 2 =0 Reduce to two dimensions

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p3p3 p1p1 Subjective Expected Utility Theory (SEUT) Betweenness Axiom: If G 1 ~G 2 then [G 1, G 2 ; q, 1-q]~G 1 ~G 2 So, indifference curves linear! Independence Axiom: If G 1 ~G 2 then [G 1, G 3 ; q, 1-q]~ [G 2, G 3 ; q, 1-q] So, indifference curves are parallel!!

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Risk Neutrality: Along indifference curve p 1 x 1 +p 2 x 2 +p 3 x 3 =c p 1 x 1 +(1-p 1 -p 3 )x 2 +p 3 x 3 =c Linear and parallel Risk Averse: Along indifference curve p 1 u(x 1 )+p 2 u(x 2 )+p 3 u(x 3 )=c p 1 u(x 1 )+(1-p1-p 3 ) u(x 2 )+p 3 u(x 3 )=c Linear and parallel

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p3p3 p1p1 R1 S2 S1 R2 Common Ratio Problem

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p3p3 p1p1 R3 S4 S3 R4 Common Consequence Problem

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Prospect Theory Kahneman and Tversky (Econometrica 1979) §Certainty Effect §Reflection Effect §Isolation Effect

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Certainty Effect People place too much weight on certain events This can explain choices above

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Ellsberg Paradox Certainty Effect G1 $1000 if red G2 $1000 if black G3 $1000 if red or yellow G4 $1000 if black or yellow 33 67

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Ellsberg Paradox Most people choose G1 and G4. BUT: Yellow shouldn’t matter

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Reflection Effect All Results get turned around when discussing Losses instead of Gains

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Isolation Effect Manner of decomposition of a problem can have an effect. Example:2-stage game Stage 1: Toss two coins. If both heads, go to stage 2. If not, get $0. Stage 2: Can choose between $3000 with certainty, or 80% chance of $4000, and 20% chance of $0. This is identical to Game 2, yet people choose like in Game 1 (certainty), even if they must choose ahead of time!

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Example We give you $1000. Choose between: a) Toss coin. If heads get additional $1000, if tails gets $0. b) Get $500 with certainty.

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Example We give you $2000. Choose between: a) Toss coin. If heads lose $0, if tails lose $1000. b) Lose $500 with certainty.

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§84% choose +500, and 69% choose [-1000,0] §Very problematic, since outcomes identical! 50% of $1,000 and 50% chance of $2,000 or $1,500 with certainty §Prospect Theory explanation: isolation effect - isolate initial receipt of money from lottery reflection effect - treat gains differently from losses

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Preference Reversals (Grether and Plott) §Choose between two lotteries: ($4, 35/36; $-1 1/36) or ($16, 11/36; $-1.50, 25/36) §Also, ask price willing to sell lottery for. §Typically – choose more certain lottery (first one) but place higher price on risky bet. §Problem – prices meant to indicate value, and consumer should choose lottery with higher value.

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Wealth Effects §Problem: Subjects become richer as game proceeds, which may affect behavior §Solutions: l Ex-post analysis – analyze choices to see if changed l Induced preferences – lottery tickets l Between group design – pre-test l Random selection – one result selected for payment

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Measuring Preferences Administer a series of questions and then apply results. However, sometimes people contradict themselves – change their answers to identical questions

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