Gibbs lemma

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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_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 \lambda such that

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


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