Gibbs lemma

In game theory and in particular the study of Blotto games and operational research, the Gibbs lemma is a result that is useful in maximization problems. It is named for Josiah Willard Gibbs.

Consider $$\phi=\sum_{i=1}^n f_i(x_i)$$. Suppose $$\phi$$ is maximized, subject to $$\sum x_i=X$$ and $$x_i\geq 0$$, at $$x^0=(x_1^0,\ldots,x_n^0)$$. If the $$f_i$$ are differentiable, then the Gibbs lemma states that there exists a $$\lambda$$ such that


 * $$\begin{align}

f'_i(x_i^0)&=\lambda \mbox{ if } x_i^0>0\\ &\leq\lambda\mbox { if }x_i^0=0. \end{align} $$