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Paradoxes in Decision Making

Paradoxes in Decision Making. With a Solution. $3000 S1. $4000 $0 80% 20% R1. Lottery 1. 80%. 20%. $3000 $0 25% 75% S2. $4000 $0 20% 80% R2. Lottery 2. $3000 $0

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Paradoxes in Decision Making

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  1. Paradoxes in Decision Making With a Solution

  2. $3000 S1 $4000 $0 80% 20% R1 Lottery 1 80% 20%

  3. $3000 $0 25% 75% S2 $4000 $0 20% 80% R2 Lottery 2

  4. $3000 $0 25% 75% S2 $4000 $0 20% 80% R2 Lottery 2 35% 65%

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

  6. $1,000,000 $0 11% 89% S4 $5,000,000 $0 10% 90% R4 Lottery 4

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

  8. 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

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

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

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

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

  13. p3 Marschak-Machina Triangle 3 outcomes: Probabilities: p1 p2

  14. p3 R1 (0.2, 0, 0.8) R2 (0.8, 0, 0.2) p1 S2 (0.75, 0.25, 0) S1 p2 4000 0 3000

  15. p3 Reduce to two dimensions P2=0 p1

  16. p3 p1 Subjective Expected Utility Theory (SEUT) Betweenness Axiom: If G1~G2 then [G1, G2; q, 1-q]~G1 ~G2 So, indifference curves linear! Independence Axiom: If G1~G2 then [G1, G3; q, 1-q]~ [G2, G3; q, 1-q] So, indifference curves are parallel!!

  17. Risk Neutrality: Along indifference curve p1x1+p2x2+p3x3=c p1x1+(1-p1-p3)x2+p3x3=c Linear and parallel Risk Averse: Along indifference curve p1u(x1)+p2u(x2)+p3u(x3)=c p1u(x1)+(1-p1-p3) u(x2)+p3u(x3)=c Linear and parallel

  18. p3 R1 R2 S1 S2 p1 Common Ratio Problem

  19. p3 R3 R4 S3 S4 p1 Common Consequence Problem

  20. Prospect TheoryKahneman and Tversky (Econometrica 1979) • Certainty Effect • Reflection Effect • Isolation Effect

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

  22. 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

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

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

  25. 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!

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

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

  28. 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

  29. 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.

  30. Wealth Effects • Problem: Subjects become richer as game proceeds, which may affect behavior • Solutions: • 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

  31. 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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