= Solution
Let $U=\sum_i u_i$. The candidate strategies are
$$
\boxed{x_i^*=X\frac{u_i}{U},\qquad y_i^*=Y\frac{u_i}{U}.}
$$
At this pair, A's partial derivatives $2u_iy_i/(x_i+y_i)^2$ are independent of $i$, as are the negatives of B's partial derivatives. Since $F$ is concave in $x$ for fixed $y$ and convex in $y$ for fixed $x$, the <Lagrange sufficiency theorem> proves
$$
F(x,y^*)\leq F(x^*,y^*)\leq F(x^*,y)
$$
for all feasible $x,y$. Hence $(x^*,y^*)$ is a <saddle point> and gives the optimal strategies in this <zero-sum game>. Its game value is
$$
\boxed{F(x^*,y^*)=U\frac{X-Y}{X+Y}.}
$$
Back to article page
