Consider Bob's decision problem: Sunny Cloudy RainyBeach 2 3 2Park 3 3 2Mall -1 2 xSuppose the probability of Sunny is 0.25, the probability of Cloudy is 0.25, and the probability of Rainy is 0.5. What is the smallest value of x for which Mall is an expected utility maximiser (that is, Mall is at least the equal-best choice)?
Question
Consider Bob's decision problem: Sunny Cloudy RainyBeach 2 3 2Park 3 3 2Mall -1 2 xSuppose the probability of Sunny is 0.25, the probability of Cloudy is 0.25, and the probability of Rainy is 0.5. What is the smallest value of x for which Mall is an expected utility maximiser (that is, Mall is at least the equal-best choice)?
Solution
To find the smallest value of x for which Mall is an expected utility maximizer, we need to calculate the expected utility for each option (Beach, Park, and Mall) and compare them.
First, let's calculate the expected utility for each option based on the given probabilities:
For Beach: Expected utility = (2 * 0.25) + (3 * 0.25) + (2 * 0.5) = 0.5 + 0.75 + 1 = 2.25
For Park: Expected utility = (3 * 0.25) + (3 * 0.25) + (2 * 0.5) = 0.75 + 0.75 + 1 = 2.5
For Mall: Expected utility = (-1 * 0.25) + (2 * 0.25) + (x * 0.5) = -0.25 + 0.5 + 0.5x = 0.25 + 0.5x
To find the smallest value of x for which Mall is an expected utility maximizer, we need to compare the expected utility of Mall with the expected utilities of Beach and Park.
Since the expected utility of Mall should be at least equal to the expected utilities of Beach and Park, we can set up the following inequality:
0.25 + 0.5x ≥ 2.25 and 0.25 + 0.5x ≥ 2.5
Simplifying the inequalities, we have:
0.5x ≥ 2 and 0.5x ≥ 2.25
Dividing both sides of the inequalities by 0.5, we get:
x ≥ 4 and x ≥ 4.5
The smallest value of x that satisfies both inequalities is x = 4.5.
Therefore, the smallest value of x for which Mall is an expected utility maximizer is 4.5.
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