Gibbs lemma
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 <math>\phi=\sum_{i=1}^n f_i(x_i)</math>. Suppose <math>\phi</math> is maximized, subject to <math>\sum x_i=X</math> and <math>x_i\geq 0</math>, at <math>x^0=(x_1^0,\ldots,x_n^0)</math>. If the <math>f_i</math> are differentiable, then the Gibbs lemma states that there exists a <math>\lambda</math> such that
:<math>\begin{align} f'_i(x_i^0)&=\lambda \mbox{ if } x_i^0>0\\ &\leq\lambda\mbox { if }x_i^0=0. \end{align} </math>