The Gibbs lemma

The Gibbs lemma
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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. It is named for Josiah Willard Gibbs.

Consider $\phi=\sum_{i=1}^n f_i(x_i)$ . Suppose $φ$ 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 $λ$ such that

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

References

J. M. Danskin 1967. The Theory of Max-Min, Springer-Verlag; page 10.

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