VIPSolutionsgive full answer and give final summary5) Consider the decision problem illustrated below. The probability for each state of the world is. 1. states Adios 131313 1 $53 S10 ss 2 ss ss 3 sss suo 2) You have a linear utility function for money. What action should you choose? b) Your utility function for money is u(x) = GE + 11), what action should you choose and why?
Question:
give full answer and give final summary
5) Consider the decision problem illustrated below. The probability for each state of the world is.
1.
states
Adios 131313
1 $53 S10 ss
2 ss ss
3 sss suo
2) You have a linear utility function for money. What action should you choose?
b) Your utility function for money is u(x) = GE + 11), what action should you choose and why?
This is AI generated Response.
Asked by: Dinil
Created at: 2025-12-13 13:49:08
Asked by: Dinil
Created at: 2025-12-13 13:49:08
Final Answer:
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# Decision Problem Analysis
Given:
- 3 actions, each with payoffs under 3 equally probable states (each with probability 1/3).
| Actions | State 1 | State 2 | State 3 |
|---------|---------|---------|---------|
| 1 | $53 | $110 | $5 |
| 2 | $89 | $70 | $5 |
| 3 | $5 | $5 | $110 |
---
## a) **Linear Utility Function**
**Linear utility** means:
\( u(x) = x \)
**Expected value for each action:**
\[
\text{Expected Value} = \frac{1}{3}(x_1 + x_2 + x_3)
\]
**Action 1:**
\[
\frac{1}{3}(53 + 110 + 5) = \frac{1}{3}(168) = 56
\]
**Action 2:**
\[
\frac{1}{3}(89 + 70 + 5) = \frac{1}{3}(164) \approx 54.67
\]
**Action 3:**
\[
\frac{1}{3}(5 + 5 + 110) = \frac{1}{3}(120) = 40
\]
### **Choice:**
- **Action 1** has the highest expected value ($56).
---
## b) **Utility function \( u(x) = \sqrt{x+11} \)**
Calculate expected utility for each action:
### **Action 1:**
\[
\text{Expected Utility} = \frac{1}{3}\left[\sqrt{53+11} + \sqrt{110+11} + \sqrt{5+11}\right]
\]
\[
= \frac{1}{3}\left[\sqrt{64} + \sqrt{121} + \sqrt{16}\right]
\]
\[
= \frac{1}{3}[8 + 11 + 4] = \frac{1}{3}(23) \approx 7.67
\]
### **Action 2:**
\[
\text{Expected Utility} = \frac{1}{3}\left[\sqrt{89+11} + \sqrt{70+11} + \sqrt{5+11}\right]
\]
\[
= \frac{1}{3}\left[\sqrt{100} + \sqrt{81} + \sqrt{16}\right]
\]
\[
= \frac{1}{3}[10 + 9 + 4] = \frac{1}{3}(23) \approx 7.67
\]
### **Action 3:**
\[
\text{Expected Utility} = \frac{1}{3}\left[\sqrt{5+11} + \sqrt{5+11} + \sqrt{110+11}\right]
\]
\[
= \frac{1}{3}\left[\sqrt{16} + \sqrt{16} + \sqrt{121}\right]
\]
\[
= \frac{1}{3}[4 + 4 + 11] = \frac{1}{3}(19) \approx 6.33
\]
### **Choice:**
- **Actions 1 and 2** both yield the highest expected utility (\(\approx 7.67\)). Either can be chosen.
---
# **Final Summary**
- **(a)** With a linear utility function, **choose Action 1** (highest expected monetary value).
- **(b)** With the utility \( u(x) = \sqrt{x+11} \), **choose either Action 1 or Action 2**, as both yield the highest expected utility.
Question:
give 2 sentence summary in each questions
This is AI generated Response.
Asked by: Dinil
Created at: 2025-12-13 13:49:44
Asked by: Dinil
Created at: 2025-12-13 13:49:44
Final Answer:
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# Summary of Decision Problem
## a) Linear Utility Function
With a linear utility function \( u(x) = x \), the optimal choice is **Action 1**, as it has the highest expected monetary value of $56. This action maximizes potential returns based on the probabilities of different states.
## b) Utility Function \( u(x) = \sqrt{x+11} \)
For the utility function \( u(x) = \sqrt{x+11} \), both **Action 1 and Action 2** yield the highest expected utility of approximately 7.67. Therefore, either action can be selected based on personal preference without loss of utility.
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