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

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.