Gibbs lemma

thumb|200px|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. It is named for Josiah Willard Gibbs.

Consider [math]\displaystyle{ \phi=\sum_{i=1}^n f_i(x_i) }[/math]. Suppose [math]\displaystyle{ \phi }[/math] is maximized, subject to [math]\displaystyle{ \sum x_i=X }[/math] and [math]\displaystyle{ x_i\geq 0 }[/math], at [math]\displaystyle{ x^0=(x_1^0,\ldots,x_n^0) }[/math]. If the [math]\displaystyle{ f_i }[/math] are differentiable, then the Gibbs lemma states that there exists a [math]\displaystyle{ \lambda }[/math] such that

[math]\displaystyle{ \begin{align} f'_i(x_i^0)&=\lambda \mbox{ if } x_i^0\gt 0\\ &\leq\lambda\mbox { if }x_i^0=0. \end{align} }[/math]