Paradoxes in Decision Making With a Solution Lottery

Paradoxes in Decision Making With a Solution

Lottery 1 $3000 $4000 $0 80% 20% 80% S 1 20% R 1

Lottery 2 $3000 25% $0 75% S 2 $4000 $0 20% 80% R 2

Lottery 2 $3000 $0 $4000 $0 35% 65% 25% 75% S 2 20% 80% R 2

Lottery 3 $1, 000 $5, 000 $1, 000 $0 10% S 3 89% R 3 1%

Lottery 4 $1, 000 $0 11% 89% S 4 $5, 000 10% $0 90% R 4

Lotteries 3 and 4 60% migration from S 3 to R 4 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 S 1 oooo $3000 R 1 o oooo o $4000 $0

Restatement - Lottery 2 S 2 oooo o $3000 R 2 oooo o $4000 $0 (80%) (20%) oooo $0 o oooo o $0

Restatement - Lottery 3 S 4 oooooooooo oooooooooo ooooo $1, 000, 000 ooooo $1, 000 R 4 oooooooooo oooooooooo ooooo $1, 000 o $0 ooooo $5, 000

Restatement - Lottery 4 S 4 oooooooooo oooooooooo ooooo $0 o $1, 000 ooooo $1, 000 R 4 oooooooooo oooooooooo ooooo $0 ooooo $5, 000

p 3 Marschak-Machina Triangle 3 outcomes: Probabilities: p 1 p 2

p 3 4000 R 1 (0. 2, 0, 0. 8) R 2 (0. 8, 0, 0. 2) 0 p 1 S 2 (0. 75, 0. 25, 0) 3000 S 1 p 2

p 3 Reduce to two dimensions P 2=0 p 1

p 3 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!! p 1

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

p 3 Common Ratio Problem R 1 R 2 S 1 S 2 p 1

p 3 Common Consequence Problem R 4 R 3 S 4 p 1

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 33 67 G 1 $1000 if red G 2 $1000 if black G 3 $1000 if red or yellow G 4 $1000 if black or yellow

Ellsberg Paradox Most people choose G 1 and G 4. 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! Ø Ø 50% of $1, 000 and 50% chance of $2, 000 or $1, 500 with certainty § Prospect Theory explanation: Ø Ø 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 l Ex-post analysis – analyze choices to see if changed Induced preferences – lottery tickets Between group design – pre-test 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