Question

Suppose Alex has utility function, where W is wealth in bitcoins: U (W ) = 12√W ∀ W ≥ 0 (1) = 3 4 W 3 ∀ W < 0 (2) Alex has just found a genie’s lamp and has wished for a memory stick of bitcoin since Alex has nothing at the moment but the clothes on their back. Alex knows the genie may try to trick them so Alex has assigned a 25% subjective probability the genie will find a way to take away their clothes which would cost 4 bitcoin to replace (the clothes are actually a BAPE NFT...) (W = −4, assume they can buy on credit and have negative wealth). In addition, Alex has assigned a 50% subjective probability to getting a medium amount of 16 bitcoin (W = 16), and a 25% probability of getting a large amount of 100 bitcoin (W = 100). A. What is the expected value of Aladdin’s gamble?

Answer 1

To find the expected value of Alex's wealth in bitcoins after the encounter with the genie, we calculate the sum of each wealth outcome multiplied by its probability, resulting in an expected value of 32 bitcoins.

Step-by-step explanation:

The student is asking about expected utility theory in economics, specifically in the context of a scenario with Alex who has certain probabilities attached to different levels of wealth in bitcoins.

To calculate the expected value of Alex's wealth, we need to multiply each potential amount of wealth by its corresponding probability and then sum these products. The calculation would look like this:

Expected Value (EV) = (Probability of losing clothes) * (Wealth if losing clothes) + (Probability of medium amount) * (Wealth of medium amount) + (Probability of large amount) * (Wealth of large amount)

Therefore:

EV = (0.25 * -4) + (0.50 * 16) + (0.25 * 100)

EV = (-1) + (8) + (25)

EV = 32

The expected value of Alex's wealth after the encounter with the genie, without considering the utility of that wealth, is 32 bitcoins.

To calculate the expected value of Alex's gamble, we multiply each possible outcome by its corresponding probability and sum them up. Let's denote the amount of bitcoins as W.

For the medium amount, which has a 50% probability, the weighted utility is: 0.5 * 12√(16) = 6√16 = 12. For the large amount, which has a 25% probability, the weighted utility is: 0.25 * 12√(100) = 3 √100 = 30.

Subtracting the 4 bitcoin cost for clothes, which has a 25% probability, the weighted utility is: -0.25 * 12√(-4) = -1.5. Adding up all the weighted utilities gives the expected value: 12 + 30 - 1.5 = 40.5.