 Solution

ID: past-exam-of-the-msc-in-advanced-computer-science-course-of-the-university-of-oxford/2022/computational-game-theory/1/c/solution

First evaluate the inner lottery:

U (ℓ_{1}) = 12 p + 3 (1 - p) = 3 + 9 p

U (ℓ_{2}) = 2 p + (1 - p) (3 + 9 p) = 3 + 8 p - 9 p^{2} = \frac{43}{9} - 9 (p - \frac{4}{9})^{2} .

Since

4/9

lies in the permitted real interval, the square is minimized there. The unique maximizer and maximum expected utility are $p = 4/9$ and $U (ℓ_{2}) = 43/9$ . Directly flattening the lottery gives probabilities

p (1 - p)

(1 - p)^{2}

and

p

for the three outcomes; these sum to one and give the same expression.

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