91Ó°ÊÓ

To find out the most a risk-neutral person will pay, we calculate the expected monetary value of the game. This involves finding the probability of each outcome and multiplying it by its respective monetary gains, and then finding the sum of these values. The die can land on any number from 1 to 10. Therefore, there are 9 outcomes where the die lands on numbers 1 to 9, and 1 outcome where it lands on 10.1. Probability of rolling 1 to 9: \[ \text{Probability} = \frac{9}{10} \]2. Probability of rolling 10: \[ \text{Probability} = \frac{1}{10} \]Calculate the expected monetary value:\[ \text{EMV} = \left( \frac{9}{10} \times 10 \right) + \left( \frac{1}{10} \times 110 \right) \]\[ \text{EMV} = 9 + 11 = 20 \]Thus, the most a risk-neutral player would pay to play the game is $20.

To determine Risa's maximum payment, we need to consider her utility function, which is given by \( U = W^2 \). Her decision is based on expected utility rather than expected monetary value.Suppose she pays \( X \) to play the game. Her wealth after the payment is \( W - X \).For rolling numbers 1 to 9, her wealth becomes \( W - X + 10 \) with probability \( \frac{9}{10} \), and for rolling 10, her wealth becomes \( W - X + 110 \) with probability \( \frac{1}{10} \).Calculate the expected utility:\[ EU = \left( \frac{9}{10} \times (W - X + 10)^2 \right) + \left( \frac{1}{10} \times (W - X + 110)^2 \right) \]

Determine the maximum \( X \) such that Risa's expected utility from playing the game equals her current utility, \( W^2 \).\[ (W - X + 10)^2 = W^2 \]\[ (W - X + 110)^2 = W^2 \]Solving these equations for \( X \) requires setting up the inequality so that Risa's expected utility matches the utility without the game (i.e., \( W^2 \)), which would yield:Upon simplification, her maximum payment can be solved numerically to ensure: \[ X = 0 \] This means Risa is only willing to play if it doesn't cost her anything, indicating she is risk-averse or indifferent.