Gibbs lemma

File:Josiah Willard Gibbs -from MMS-.jpg

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 $\varphi =\sum _{i=1}^n{f}_i(x_i)$. Suppose $\varphi$ is maximized, subject to $\sum x_i=X$ and $x_i\ge 0$, at $x^0=(x_1^0,\dots ,x_n^0)$. If the $f_i$ are differentiable, then the Gibbs lemma states that there exists a $\lambda$ such that

$\begin{array}{rl}f{\text{'}}_i(x_i^0)& =\lambda \text{ if }{x}_i^0>0\\ & \le \lambda \text{ if }{x}_i^0=0.\end{array}$