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.^[1] It is named for Josiah Willard Gibbs.

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

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

Notes

1. J. M. Danskin (6 December 2012). The Theory of Max-Min and its Application to Weapons Allocation Problems. Springer Science & Business Media. ISBN 978-3-642-46092-0. ... problems in which one side must make his move knowing that the other side will then learn what the move is and optimally counter. They are fundamental in particular to military weapons-selection problems involving large systems...