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Paradoxes in Decision Making

This document delves into various paradoxes in decision-making, particularly focusing on the Allais Paradox and its implications on the "Independence of Irrelevant Alternatives" hypothesis. It explores multiple lottery scenarios and their impact on choices, highlighting inconsistencies such as the Certainty Effect and Reflection Effect. The text analyzes how subjective expected utility theory and other decision-making frameworks can lead to counterintuitive behavior. Furthermore, it discusses preference reversals and potential solutions to these decision-making anomalies.

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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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