Gibbs lemma

Josiah Willard Gibbs

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

${\begin{aligned}f'_{i}(x_{i}^{0})&=\lambda {\mbox{ if }}x_{i}^{0}>0\\&\leq \lambda {\mbox{ if }}x_{i}^{0}=0.\end{aligned}}$