Questions: Problem 3. (20 points) Recall the Allais paradox from lecture. Question 1: ( 1,1) or ( 1,0.89 ; 5,0.10 ; 0,0.01) Question 2: ( 1,0.11 ; 0,0.89) or ( 5,0.10 ; 0,0.90) Most people prefer the first lottery in Question 1 but the second lottery in Question 2. Suppose someone chooses according to prospect theory with value function v(x) = x^0.1 if x >= 0 2x if x<0 where all units are in billions. The probability weighing function they use is pi(p) = sqrt(p) / (sqrt(p) + sqrt(1-p))^2 Show that even without any editing, this can explain the Allais paradox (set reference point to 0 in both questions).

Problem 3. (20 points) Recall the Allais paradox from lecture.

Question 1: ( 1,1) or ( 1,0.89 ; 5,0.10 ; 0,0.01)
Question 2: ( 1,0.11 ; 0,0.89) or ( 5,0.10 ; 0,0.90)

Most people prefer the first lottery in Question 1 but the second lottery in Question 2. Suppose someone chooses according to prospect theory with value function

v(x) = x^0.1 if x >= 0
2x if x<0

where all units are in billions. The probability weighing function they use is

pi(p) = sqrt(p) / (sqrt(p) + sqrt(1-p))^2

Show that even without any editing, this can explain the Allais paradox (set reference point to 0 in both questions).

Transcript text: Problem 3. (20 points) Recall the Allais paradox from lecture. Question 1: $(\$ 1,1)$ or $(\$ 1,0.89 ; \$ 5,0.10 ; \$ 0,0.01)$ Question 2: $(\$ 1,0.11 ; \$ 0,0.89)$ or $(\$ 5,0.10 ; \$ 0,0.90)$ Most people prefer the first lottery in Question 1 but the second lottery in Question 2. Suppose someone chooses according to prospect theory with value function \[ v(x)=\left\{\begin{array}{ll} x^{0.1} & \text { if } x \geq 0 \\ 2 x & \text { if } x<0 \end{array}\right. \] where all units are in $\$$ billions. The probability weighing function they use is \[ \pi(p)=\frac{\sqrt{p}}{(\sqrt{p}+\sqrt{1-p})^{2}} \] Show that even without any editing, this can explain the Allais paradox (set reference point to $\$ 0$ in both questions).

Solution

Solution Steps

Step 1: Define the Lotteries

For Question 1, we have two lotteries:

Lottery 1: $ (1, 1) $
Lottery 2: $ (1, 0.89), (5, 0.10), (0, 0.01) $

For Question 2, the lotteries are:

Lottery 1: $ (1, 0.11), (0, 0.89) $
Lottery 2: $ (5, 0.10), (0, 0.90) $

Step 2: Calculate Expected Utility for Question 1

Using the value function $ v(x) = x^{0.1} $ for $ x \geq 0 $ and the probability weighting function $ \pi(p) = \frac{\sqrt{p}}{(\sqrt{p} + \sqrt{1-p})^2} $, we calculate the expected utility for each lottery in Question 1.

For Lottery 1: \[ EU(Lottery 1) = v(1) \cdot \pi(1) = 1^{0.1} \cdot 1 = 1.0 \]

For Lottery 2: \[ EU(Lottery 2) = v(1) \cdot \pi(0.89) + v(5) \cdot \pi(0.10) + v(0) \cdot \pi(0.01) \] Calculating each term:

$ v(1) = 1^{0.1} = 1 $
$ v(5) = 5^{0.1} $
$ v(0) = 0 $

Thus, \[ EU(Lottery 2) = 1 \cdot \pi(0.89) + 5^{0.1} \cdot \pi(0.10) + 0 \cdot \pi(0.01) \approx 0.8124287796435081 \]

Step 3: Calculate Expected Utility for Question 2

Next, we calculate the expected utility for each lottery in Question 2.

For Lottery 1: \[ EU(Lottery 1) = v(1) \cdot \pi(0.11) + v(0) \cdot \pi(0.89) \] Calculating each term:

$ v(1) = 1^{0.1} = 1 $
$ v(0) = 0 $

Thus, \[ EU(Lottery 1) = 1 \cdot \pi(0.11) + 0 \cdot \pi(0.89) \approx 0.20400212710436338 \]

For Lottery 2: \[ EU(Lottery 2) = v(5) \cdot \pi(0.10) + v(0) \cdot \pi(0.90) \] Calculating each term:

$ v(5) = 5^{0.1} $
$ v(0) = 0 $

Thus, \[ EU(Lottery 2) = 5^{0.1} \cdot \pi(0.10) + 0 \cdot \pi(0.90) \approx 0.23215445268361468 \]

Step 4: Compare Expected Utilities

From the calculations:

For Question 1: $ EU(Lottery 1) = 1.0 $ and $ EU(Lottery 2) \approx 0.8124287796435081 $
For Question 2: $ EU(Lottery 1) \approx 0.20400212710436338 $ and $ EU(Lottery 2) \approx 0.23215445268361468 $

This shows that the preference for Lottery 1 in Question 1 and Lottery 2 in Question 2 can be explained by the differences in expected utility calculated using prospect theory, illustrating the Allais paradox.