Paradoxes in Decision Making With a Solution

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

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

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

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

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

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

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

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

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

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

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

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

p2p2 p3p3 p1p R1 (0.2, 0, 0.8) S1 R2 (0.8, 0, 0.2) S2 (0.75, 0.25, 0)

p3p3 p1p1 P 2 =0 Reduce to two dimensions

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!!

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

p3p3 p1p1 R1 S2 S1 R2 Common Ratio Problem

p3p3 p1p1 R3 S4 S3 R4 Common Consequence Problem

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

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

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

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

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

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!

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

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

§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

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.

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

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