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

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.