Past exam of the MSc in advanced computer science course of the University of Oxford / 2022 / Computational game theory / 1

Let $X$ denote the monetary reward. Its expected value and expected utility are different quantities:Thus indifference to a sure nonnegative reward requires $k^2=10.89$, giving $k=3.30$ dollars. This is the certainty equivalent, hence the lottery's monetary value in the question's comparison with a sure reward. The utility function $u(x)=x^2$ is increasing and strictly convex on nonnegative monetary amounts. Its certainty equivalent exceeds its expected value, $3.30>1.89$, demonstrating risk seeking; the same conclusion follows from the Jensen inequality.

The wording about paying an entry fee needs a qualification. A certainty equivalent compares a gross lottery with a sure gross reward; a literal utility indifference price also accounts for the fee's effect on wealth. If $u$ instead describes final wealth, with initial wealth $w$ and fee $c$, the comparison is $\mathbb{E}[(w-c+X{)}^2]=w^2$, on a domain where the utility function is increasing. Here $\mathrm{Var}(X)=7.3179$, so the nonnegative-wealth branch gives $c=w+1.89-\sqrt{w^2-7.3179}$ whenever that branch is admissible. An initial wealth and an admissible payment domain are not specified. Therefore $3.30$ is the determined certainty equivalent, while a literal wealth-dependent maximum fee is not uniquely determined. This is the distinction in certainty equivalent and willingness to pay. If negative rewards were allowed without requiring an increasing utility function, the equation for $k$ also has the algebraic root $-3.30$; the usual nonnegative-reward interpretation selects the positive root.

Write $A$ and $B$ for the two deterministic outcomes. Under the expected utility hypothesis, indifference between them implies $u(A)=u(B)=c$. The expected utility of the fair lottery is consequentlyIt must therefore be indifferent to each deterministic outcome, contradicting the asserted strict preference. These preferences do not admit an expected utility representation over the stated outcomes. Equivalently, affine utility on probability measures preserves this indifference under mixing, as required by the independence axiom for lottery preferences.

A concern for the fairness of the procedure can be perfectly intelligible. It would require enlarging the outcome description to include the procedure; it cannot be represented by an expected utility over only the two final gift recipients while retaining the stated strict preference.

First evaluate the inner lottery: $U({\ell }_1)=12p+3(1-p)=3+9p$. By reduction of compound lotteries, the outer expected utility isSince $4/9$ lies in the permitted real interval, the square is minimized there. The unique maximizer and maximum expected utility are $p=4/9$ and $U({\ell }_2)=43/9$. 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.