Consider Bob's decision problem: Sunny Cloudy RainyBeach 2 3 2Park 3 3 2Mall -2 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)? Round your answer to one decimal place (e.g. 0.5).
Question
Consider Bob's decision problem: Sunny Cloudy RainyBeach 2 3 2Park 3 3 2Mall -2 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)? Round your answer to one decimal place (e.g. 0.5).
Solution
To find the smallest value of x for which Mall is an expected utility maximiser, we need to calculate the expected utility for each location (Beach, Park, Mall) and then compare them.
The expected utility for each location is calculated by multiplying the utility of each weather condition by the probability of that weather condition, and then summing these products.
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For the Beach, the expected utility is: (2 * 0.25) + (3 * 0.25) + (2 * 0.5) = 0.5 + 0.75 + 1 = 2.25
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For the Park, the expected utility is: (3 * 0.25) + (3 * 0.25) + (2 * 0.5) = 0.75 + 0.75 + 1 = 2.5
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For the Mall, the expected utility is: (-2 * 0.25) + (2 * 0.25) + (x * 0.5)
For the Mall to be an expected utility maximiser, its expected utility must be at least equal to the highest expected utility of the other locations. In this case, the highest expected utility is 2.5 (Park).
So, we set the expected utility of the Mall equal to 2.5 and solve for x:
-0.5 + 0.5 + 0.5x = 2.5 0.5x = 2.5 + 0.5 0.5x = 3 x = 3 / 0.5 x = 6
So, the smallest value of x for which Mall is an expected utility maximiser is 6.
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