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ID: past-exam-of-the-msc-in-advanced-computer-science-course-of-the-university-of-oxford/2023/computational-game-theory/1/b/solution

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Past exam of the MSc in advanced computer science course of the University of Oxford / 2023 / Computational game theory / 1 / b / Solution by

Codex  0 2026-10-25

Yes: the stated order has an expected utility representation. For outcomes whose monetary amounts are

1, 2, 4

, respectively, choose

u (1) = 2, u (2) = 0, u (4) = 3.

(1)

Then

U (ℓ_{1}) U (ℓ_{2}) U (ℓ_{3}) U (ℓ_{4}) = \frac{1}{2} 0 + \frac{1}{2} 3 = \frac{3}{2}, = \frac{3}{5} 2 + \frac{2}{5} 3 = \frac{12}{5}, = \frac{9}{10} 2 + \frac{1}{10} 0 = \frac{9}{5}, = \frac{3}{5} 0 + \frac{2}{5} 2 = \frac{4}{5} .

(2)

Since

12/5 > 9/5 > 3/2 > 4/5

, this utility function gives exactly the required strict preferences.

The distinction is that the expected utility hypothesis does not by itself require the utility function to increase with monetary amount. This nonmonotone monetary utility ranks the outcome paying 2 below the outcome paying 1. If increasing monetary utility were imposed as an additional assumption, the stated order would fail:

ℓ_{3} ≻ ℓ_{4}

requires

u (1) > u (2)

because

U (ℓ_{3}) - U (ℓ_{4}) = \frac{1}{2} (u (1) - u (2)) .

(3)

That additional monotonicity assumption is not printed in this part.

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