Past exam of the MSc in advanced computer science course of the University of Oxford / 2021 / Computational game theory / 2 / c / Solution

The bold black curve is now the pointwise maximum of the two pure-row expected payoff functions:It measures the largest expected payoff that the row player can achieve against column mixed strategy $q$. The column player minimizes this quantity because its own payoff has the opposite sign. Equality of the two lines gives $5q-3=1-2q$, or $q=4/7$. The envelope decreases before that point and increases afterwards, givingThe endpoint values $h(0)=1$ and $h(1)=2$ are both larger. A row mixed strategy cannot exceed the greatest pure-row expected payoff, because its expected payoff is a convex combination of those two values.