Gibbs lemma

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 $x\_igeq\; 0$, 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

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References

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