Past exam of the computer science course of the University of Oxford / 2018 / Computational game theory / 1 / c / Solution

Yes: choose $Y=4/X$. The equilibrium expected payoffs are opposite because this is a zero-sum game. They are equal exactly when both are zero. Part (b)'s value vanishes exactly when $XY=4$, and $4/X>0$ for every allowed $X>0$. This establishes both existence and the unique suitable positive choice of $Y$.