Paradoxes in Decision MakingWith 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 460% 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 Fishorder Beef

Restatement - Lottery 1

S1

oooo o

$3000

R1

oooo o

$4000 $0

Restatement - Lottery 2

S2

oooo o

$3000

oooo ooooo ooooo o

$0

R2

oooo o

$4000 $0 (80%) (20%)

oooo ooooo ooooo o $0

Restatement - Lottery 3S4

oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo ooooooooo$1,000,000 o$1,000,000oooooooooo$1,000,000R4

oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo ooooooooo$1,000,000 o$0oooooooooo$5,000,000

Restatement - Lottery 4S4

oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo ooooooooo$0 o$1,000,000oooooooooo$1,000,000R4

oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo ooooooooo$0 o$0oooooooooo$5,000,000

p3p1p2Marschak-Machina Triangle3 outcomes: Probabilities:

400003000

p3p1P2=0Reduce to two dimensions

Subjective Expected Utility Theory (SEUT)Betweenness Axiom:If G1~G2 then [G1, G2; q, 1-q]~G1 ~G2So, indifference curves linear!Independence Axiom:If G1~G2 then [G1, G3; q, 1-q]~ [G2, G3; q, 1-q]So, indifference curves are parallel!!

Risk Neutrality:Along indifference curve p1x1+p2x2+p3x3=cp1x1+(1-p1-p3)x2+p3x3=c

Linear and parallelRisk Averse:Along indifference curve p1u(x1)+p2u(x2)+p3u(x3)=cp1u(x1)+(1-p1-p3) u(x2)+p3u(x3)=c

Linear and parallel

Common Ratio Problem

Common Consequence Problem

Prospect TheoryKahneman and Tversky (Econometrica 1979)

Certainty EffectReflection EffectIsolation Effect

Certainty EffectPeople place too much weight on certain events

This can explain choices above

Ellsberg Paradox

Certainty Effect

G1 $1000 if red G2 $1000 if blackG3 $1000 if red or yellowG4 $1000 if black or yellow3367

Ellsberg ParadoxMost people choose G1 and G4.BUT: Yellow shouldnt matter

Red

Black

Yellow

G1

$1000

$0

$0

G2

$0

$1000

$0

G3

$1000

$0

$1000

G4

$0

$1000

$1000

Reflection EffectAll Results get turned around when discussing Losses instead of Gains

Isolation EffectManner of decomposition of a problem can have an effect.

Example:2-stage gameStage 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!

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

ExampleWe 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 $2,000or$1,500 with certaintyProspect Theory explanation:isolation effect - isolate initial receipt of money from lotteryreflection 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 EffectsProblem: Subjects become richer as game proceeds, which may affect behaviorSolutions:Ex-post analysis analyze choices to see if changedInduced preferences lottery ticketsBetween group design pre-testRandom 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