Gibbs lemma

FILE:Josiah Willard Gibbs -from MMS-.jpg

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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.{{cite book|author=J. M. Danskin|title=The Theory of Max-Min and its Application to Weapons Allocation Problems|date=6 December 2012|publisher=Springer Science & Business Media|isbn=978-3-642-46092-0|quote= ... 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... }} It is named for Josiah Willard Gibbs.

Consider $\backslash phi=\backslash sum\_\{i=1\}^n\; f\_i(x\_i)$. Suppose $\backslash phi$ is maximized, subject to $\backslash sum\; x\_i=X$ and $x\_i\backslash geq\; 0$, at $x^0=(x\_1^0,\backslash ldots,x\_n^0)$. If the $f\_i$ are differentiable, then the Gibbs lemma states that there exists a $\backslash lambda$ such that

:$\backslash begin\{align\}$

f'_i(x_i^0)&=\lambda \mbox{ if } x_i^0>0\\

&\leq\lambda\mbox { if }x_i^0=0.

\end{align}