Past exam of the Probability, statistics and options II course of the University of Oxford / 2008 / T / 2 / b

The proposed vectors are nonnegative, and direct multiplication givesThus $Ax\le 1$ and $A^{\mathsf{T}}y\ge 1$, proving primal and dual feasibility.

Both objectives equal $1/4$: $1^{\mathsf{T}}x=1/12+1/6=1/4$ and $1^{\mathsf{T}}y=1/6+1/12=1/4$. Weak duality bounds the value of every primal feasible point above by every dual feasible objective. Equality for this feasible pair therefore proves both are optimal, without needing to solve either problem further.

Normalize each optimal vector by its sum $1/4$. Player I's row strategy and Player II's column strategy are respectivelyAs a direct check $Aq=(4,4{)}^{\mathsf{T}}$ and $p^{\mathsf{T}}A=(4,5,4)$, so Player II guarantees a loss of at most four and Player I guarantees a gain of at least four.