Question

Assume a lottery of payout $50 with a probability of 40% and $70 with a probability of 60%. Assume an agent with utility function u(p)=p ¹/³

1. What is the expected value of the lottery?

Answer 1

Using the utility function u(p) = p^1/3, the expected utility of the lottery is approximately 3.963. The expected monetary value of the lottery, however, is different and calculated to be $62.

Step-by-step explanation:

Calculation of Expected Value

To calculate the expected value of the given lottery, we need to take into account the utility function given for the agent, u(p) = p^1/3. The expected utility of the lottery is the sum of the utilities of each outcome multiplied by their respective probabilities.

First, we calculate the utility for both outcomes:

• 
  
• For the $50 payout: u(50) = 50^1/3 ≈ 3.684
• 
  
• For the $70 payout: u(70) = 70^1/3 ≈ 4.121

Next, we use these utilities to find the expected utility of the lottery:

• 
  
• (3.684 * 0.40) + (4.121 * 0.60) ≈ 3.963

Thus, the expected utility of the lottery is approximately 3.963. However, it is important to note that the expected value of the lottery in monetary terms is different and would be calculated as follows:

• 
  
• (50 * 0.40) + (70 * 0.60) = 20 + 42 = $62

The expected value of the lottery is therefore $62.

Answer 2

The expected value of the lottery with the given probabilities and utility function u(p)=p^1/3 is approximately 3.9464.

Step-by-step explanation:

To calculate the expected value of a lottery with a given probability distribution, we multiply each possible outcome by its corresponding probability and then sum these products. The expected value (EV) for the lottery in the question, given the utility function u(p)=p1/3, is:

• EV = ($501/3 × 0.40) + ($701/3 × 0.60)
• EV = (3.6840 × 0.40) + (4.1213 × 0.60)
• EV = 1.4736 + 2.4728
• EV = 3.9464

Therefore, the expected value of the lottery, considering the agent's utility function, is approximately 3.9464.