Consider the following decision problem of Alice:Sunny Cloudy RainyBeach 1 2 1Park 2 2 1Mall 1 2 3Suppose the probability of Sunny is 0.4 the probability of Cloudy is p ∈ [0, 0.6] and the proba-bility of Rainy is 0.6 − p. For what values of p is Mall an expected utility maximizer?[Write your answer as an interval; e.g.: [0.2, 0.4] or (0.3, 0.5], etc.]
Question
Consider the following decision problem of Alice:Sunny Cloudy RainyBeach 1 2 1Park 2 2 1Mall 1 2 3Suppose the probability of Sunny is 0.4 the probability of Cloudy is p ∈ [0, 0.6] and the proba-bility of Rainy is 0.6 − p. For what values of p is Mall an expected utility maximizer?[Write your answer as an interval; e.g.: [0.2, 0.4] or (0.3, 0.5], etc.]
Solution
The expected utility of each location can be calculated by multiplying the utility of each weather condition by the probability of that weather condition, and then summing these products.
For the beach, the expected utility is: E(Beach) = 10.4 + 2p + 1*(0.6-p) = 0.4 + 2p + 0.6 - p = 1 + p
For the park, the expected utility is: E(Park) = 20.4 + 2p + 1*(0.6-p) = 0.8 + 2p + 0.6 - p = 1.4 + p
For the mall, the expected utility is: E(Mall) = 10.4 + 2p + 3*(0.6-p) = 0.4 + 2p + 1.8 - 3p = 2.2 - p
The mall is an expected utility maximizer when its expected utility is greater than the expected utility of both the beach and the park. This gives us the following inequalities:
2.2 - p > 1 + p 2.2 - p > 1.4 + p
Solving these inequalities gives us the interval for p where the mall is an expected utility maximizer:
[0.4, 0.6]
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