Prospect Theory

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Outline

• Review of expected utility

• Prospect theory

– Value function

• Experimental Evidence

• Expected utility vs Prospect theory

– Probability weighting function

• Experimental Evidence

• Expected utility vs Prospect theory

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St. Petersburg Paradox and Expected Utility

• How to define the utility/value of a lottery – mean?

• St. Petersburg paradox: You are faced with a sequence of tosses of a fair coin. The game will end for the first time the coin comes up Head. If this happens on the 𝑛 th trial, you get 2𝑛 dollars. – What is your willingness to pay to participate in

this game?

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St. Petersburg Paradox and Expected Utility

• Mean: U=1

2× 2 +

1

4× 4 +

1

8× 8 +⋯ =

1 + 1 + 1 +⋯ = +∞ – But you will not pay a large amount for the game

– Mean is not a good measure of utility

• Expected utility: 𝑢(𝑥) = 𝑙𝑛𝑥, U= 1

2× ln2 +

1

4× ln4 +

1

8× ln8 + …… =

[ (𝑛

2𝑛+∞𝑛=1 )] 𝑙𝑛2 = 2𝑙𝑛2

– The utility is finite with a concave 𝑢 𝑥

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Expected Utility

• A lottery specifies the probability for each prize. 𝑙 = (𝑥, 𝑝; 𝑦, 1 − 𝑝), 𝑥 occurs with probability 𝑝 and 𝑦 occurs with probability 1 − 𝑝.

• The expected utility that represents the decision maker - DM’s preference would be U 𝑙 =𝑝𝑢 𝑥 + 1 − 𝑝 𝑢 𝑦 , where 𝑢(𝑥) and 𝑢(𝑦) is utility for 𝑥 and 𝑦.

• Note that 𝐔 is the utility function over lottery, while 𝐮 is the utility over outcome/money. Thus, the preference over lottery solely depends on 𝑢 function

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Risk Aversion

• DM is risk averse if she prefers the mean of the gamble over the gamble.

• DM is risk averse if u is concave. – 𝑢(𝑎𝑥 + (1 − 𝑎)𝑦) ≥ 𝑎𝑢(𝑥) + (1 − 𝑎)𝑢(𝑦) for

any 𝑥, 𝑦, and 𝑎є[0,1]; OR 𝑢 ′′(x)≤0

• DM is risk neutral if she is indifferent between the mean of the gamble and the gamble.

• DM is risk seeking if she prefers the gamble over the mean of the gamble.

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An Example

• 𝑙 = ($100, 0.5; 0, 0.5)

– The mean of the lottery is $50

• If u(𝑥) = 𝑥0.5,

– The expected utility of the lottery is U(𝑙) =0.5(1000.5)+0.5(00.5)=5

– The expected utility of $50 is 𝑈 50 = 𝑢 50 = 500.5

– 𝑈 50 > U 𝑙 , so DM prefers $50 over the lottery

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Axioms of Expected Utility

• Completeness and transitivity

• Continuity

• Independence: For every lottery 𝑃, 𝑄, 𝑅, and every α ∈ 0,1 ,

𝑃 ≽ 𝑄 iff α𝑃 + (1 − α)𝑅 ≽ α𝑄 + (1 − α)𝑅

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An Example

• If 50 ≽ 100, 0.5; 0, 0.5 , independence implies 50, 0.02; 0, 0.98 ≽ 100, 0.01; 0, 0.99 .

• Why? 𝑃 = 50, 𝑄 = 100, 0.5; 0, 0.5 , R = 0, and α = 0.02 – α𝑃 + (1 − α)𝑅 = 50, 0.02; 0, 0.98

– α𝑄 + 1 − α 𝑅 = 100, 0.01; 0, 0.99 .

• Is your choice consistent with independence?

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Three Tenets of Expected Utility

• Asset Integration: (𝑥1, 𝑝1;⋯ ; 𝑥𝑛, 𝑝𝑛) is acceptable at asset w iff

𝑈(𝑤 + 𝑥1, 𝑝1;⋯ ;𝑤 + 𝑥𝑛, 𝑝𝑛) ≥ 𝑢(𝑤)

• Risk Aversion: 𝑢 is concave

• Expectation: 𝑈(𝑥1, 𝑝1;⋯ ; 𝑥𝑛, 𝑝𝑛) = 𝑝1𝑢(𝑥1) + ⋯+ 𝑝𝑛𝑢(𝑥𝑛)

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Asset Integration

• In addition to whatever you own, you have been given 1,000. You are now asked to choose between A1: (1,000, 0.5), and B1: (500)

– Most people would choose: 𝐵1 ≻ 𝐴1

• In addition to whatever you own, you have been given 2,000. You are now asked to choose between A2: (-1,000, 0.5), and B2: (-500)

– Most people would choose: 𝐴2 ≻ 𝐵2

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Asset Integration

• 𝑈(𝐴1) = 0.5𝑢(𝑤 + 1000 + 1000) + 0.5𝑢(𝑤 + 1000) = 0.5𝑢 𝑤 + 2000 + 0.5𝑢 𝑤 + 1000

• 𝑈 𝐵1 = 𝑢 𝑤 + 1000 + 500 = 𝑢 𝑤 + 1500

• 𝑈(𝐴2) = 0.5𝑢(𝑤 + 2000 − 1000) + 0.5𝑢(𝑤 + 2000) = 0.5𝑢 𝑤 + 1000 + 0.5𝑢 𝑤 + 2000

• 𝑈(𝐵2) = 𝑢(𝑤 + 2000 − 500) = 𝑢(𝑤 + 1500)

• Expected utility with asset integration could not account for the observation.

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Reference Dependence

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Reference Dependence

• Consider the following example

– Ice water in left hand bowl; hot water in right hand bowl; room temperature in the middle bowl.

– Immerse left hand in left bowl, and right hand in right bowl.

– And then dip both hand in the middle.

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Reference Dependence

• DM separates gain and loss relative to a reference point without asset integration

– 𝑈 𝐴1 = 0.5𝑢 1000 + 0.5𝑢 0 , and 𝑈(𝐵1) =𝑢(500)

– 𝑈 𝐴2 = 0.5𝑢 0 + 0.5𝑢 −1000 , and 𝑈 𝐵2 = 𝑢 −500

– We could have both 𝐵1 ≻ 𝐴1 and 𝐴2 ≻ 𝐵2 given some 𝑢 function.

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Risk Aversion

• You are asked to choose between A1: (3,000) and B1:(4,000,0.8). – Most people choose A1 ≻ B1, risk averse in the

gain domain

• You are asked to choose between A2: (-3,000) and B2: (-4,000,0.8). – Most people choose B2 ≻ A2, risk seeking in the

loss domain

– This is not consistent with the usual assumption of risk aversion

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Diminishing Sensitivity

• Principle of diminishing sensitivity applies to sensory dimensions (Weber-Fechner law).

– Turning on a weak light in a dark room versus turning on a weak light in a bright room

• Diminishing sensitivity in gain, u’’(x)<0 for x>0; diminishing sensitivity in loss, u’’(x)>0 for x<0.

– The difference between getting 900 and 1000 is smaller than the difference between getting 100 and 200.

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Loss Aversion

• You are asked to choose between A1: 0 and B1: −100, 0.5; 100, 0.5

– Most people choose A1 ≻ B1, risk averse across gain and loss domain

– 0.5𝑢(100) + 0.5𝑢(−100) < 𝑢(0) = 0 => 𝑢(100) < −𝑢(−100)

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Loss Aversion

• 0.5𝑢 𝑥 + 0.5𝑢 −𝑥 < 𝑢 0 => 𝑢(𝑥) < −𝑢(−𝑥)

• Loss looms larger than gain.

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Valuation Function

• Gain/loss relative to reference dependence

• Diminishing sensitivity towards gain and loss

• Loss aversion (Loss looms larger than gain)

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Reference

point

Sensitivity to probability

• Russian roulette: a potentially lethal game of chance in which a player places n bullets in a revolver (full with 6 bullets), spins the cylinder, places the muzzle against his or her head, and pulls the trigger. What is your willingness to pay for each of the reduction of bullets as follows.

– 6 to 5 (From sure to 5/6 chance)

– 3 to 2

– 1 to 0 (From 1/6 chance to impossible)

• People like to pay more for 6 to 5, and for 1 to 0.

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Allais Paradox: Common Ratio

• It is commonly observed that

– (3000, 0.90) ≻ (6000, 0.45)

– (6000, 0.001) ≻ (3000, 0.002)

• The expected utility of the choice:

– 0.9u(3000)> 0.45u(6000)

– 0.001u(6000)>0.002u(3000)

– Contradiction!

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Allais Paradox: Common Ratio

• Here we show that it violates the independence

• 𝑃 :(3000, 0.90) , 𝑄 :(6000, 0.45), R=0, α=2/900

• α𝑃 + 1 − α 𝑅 =(3000, 0.002).

• α𝑄 + 1 − α 𝑅 =(6000, 0.001).

• Hence (3000, 0.90) ≻ (6000, 0.45) and (6000, 0.001) ≻ (3000, 0.002) violates independence.

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Allais Paradox: Common Ratio

• When expected utility, U 𝑙 = 𝑝𝑢 𝑥 +1 − 𝑝 𝑢 𝑦 , could not account for the choice

behavior, we need to extend the theory further.

• We introduce probability weighting function 𝜋 𝑝 to have U 𝑙 = 𝜋(𝑝)𝑢 𝑥 + 𝜋 1 − 𝑝 𝑢 𝑦

• We examine the properties of 𝜋 𝑝

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Allais Paradox: Common Ratio

• We derive the condition for 𝜋 𝑝 , under which it can exhibit the choice patterns.

• 𝜋 0.90 𝑢(3000) > 𝜋 0.45 𝑢(6000)

𝜋 0.001 𝑢(6000) > 𝜋 0.002 𝑢(3000)

•𝜋 0.45

𝜋 0.90<𝑢 3000

𝑢 6000<𝜋 0.001

𝜋 0.002

Subproportionality: 𝜋 𝑝𝑞

𝜋 𝑝<𝜋 𝑝𝑞𝑟

𝜋 𝑝𝑟

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Allais Paradox: Common Consequence

• It is commonly observed that

– 2400 ≻ (2500, 0.33; 2400, 0.66)

– (2500, 0.33) ≻(2400, 0.34)

• The expected utility of the choice:

– 𝑢 2400 > 0.33𝑢 2500 + 0.66𝑢 2400

– 0.33𝑢 2500 > 0.34𝑢 2400

– Contradiction!

– Check violation of independence axiom!

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Allais Paradox: Common Consequence

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1-33 (0.33) 34 (0.01) 35-100 (0.66)

Gamble A1 2500 0 2400

Gamble B1 2400 2400 2400

Gamble A2 2500 0 0

Gamble B2 2400 2400 0

Suppose there are 100 balls numbered from 1 to 100. You randomly pick a ball, the number drawn determines your earning as follows.

Allais Paradox: Common Consequence

• Similarly, we derive the condition for 𝜋 𝑝

• 𝜋 0.33 𝑢 2500 > 𝜋 0.34 𝑢 2400

• 𝑢 2400 > 𝜋 0.66 𝑢 2400 + 𝜋 0.33 𝑢 2500

• (1 − 𝜋 0.66))𝑢 2400 > 𝜋 0.33 𝑢 2500 > 𝜋(0.34)𝑢(2400

• 𝜋 0.66 + 𝜋 0.34 < 1

Subcertainty: 𝜋 𝑝 + 𝜋 1 − 𝑝 < 1

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Gambling and Insurance

• Gambling: (5000, 0.001) ≻ (5)

• Expected utility: 0.001𝑢 5000 > 𝑢 5 ⇒𝑢 5

𝑢 5000< 0.001

– If 𝑢(𝑥) 𝑖𝑠 𝑐𝑜𝑛𝑐𝑎𝑣𝑒 𝑓𝑜𝑟 𝑥 > 0, 𝑢 5

𝑢 5000> 0.001,

contradiction!

• Insurance: (-5) ≻ (-5000, 0.0001) – Check whether it contradicts the expected utility!

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Gambling and Insurance

• Similarly, we derive the condition for 𝜋 𝑝

• 𝜋 0.001 𝑢 5000 > 𝑢 5

• 𝜋 0.001 >𝑢 5

𝑢 5000> 0.001

• Overweighting small p: 𝜋 𝑝 > 𝑝

• Check for insurance: (-5) ≻ (-5000, 0.0001)!

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Subadditivity for small p

• It is commonly observed that (0.001,6000) ≻ (0.002, 3000)

• Expected utility: 0.001𝑢 6000 >0.002𝑢 3000 ⇒ 𝑢(3000)/𝑢(6000) < 0.5

– If 𝑢(𝑥) 𝑖𝑠 𝑐𝑜𝑛𝑐𝑎𝑣𝑒 𝑓𝑜𝑟 𝑥 > 0, 𝑢 3000

𝑢 6000> 0.5,

contradiction!

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Subadditivity for small p

• Similarly, we derive the condition for 𝜋 𝑝

• 𝜋 0.001 𝑢(6000) > 𝜋 0.002 𝑢(3000)

• 𝜋(0.001)/𝜋(0.002) > 𝑢(3000)/𝑢(6000) >1/2

• Subadditivity: 𝜋(𝑟𝑝) > 𝑟𝜋(𝑝) for small probability

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Probability Weighting Function

• Subproportionality

• Subcertainty

• Overweighting for small p

• Subadditivity for small p

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Theory

• Editing Phase

– Coding

– Combination

– Cancellation

– Simplification

– Detection of Dominance

• Evaluation Phase

– Valuation

– Probability Weighting EC4394 L2 34

Theory

• 2 or 3 outcomes

–N-outcomes

–Violation of dominance

• What is reference point?

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Earlier Works

Markowitz (1952)

Karmarkar (1978)

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Earlier Works

“Columbus is viewed as the discoverer of America, even though every school child knows that the Americas were inhabited when he arrived, and that he was not even the first to have made a round trip, having been preceded by Vikings and perhaps others. What is important about Columbus’ discovery of America is not that it was the first, but that it was the last. After Columbus, America was never lost again.”

– Al Roth

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