Gibbs lemma explained

thumb|200px|Josiah Willard GibbsIn 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

\phi = \sum ^{n} _{i =1} f _{i(x} _{i)} . Suppose \phi is maximized, subject to \sum x _{i=X} and x _{i\geq} 0 , at x ^{0 )} _{n} . If the f _{i} are differentiable, then the Gibbs lemma states that there exists a λ such that \begin{align} f' _{i(x} ^{0 )& =λ} ^{i} if x ^{0=0 . \end{align}} _{i}

Notes and References

1. Book: J. M. Danskin. The Theory of Max-Min and its Application to Weapons Allocation Problems. 6 December 2012. Springer Science & Business Media. 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... .