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 ϕ = ∑ i = 1 n f i ( x i ) . Suppose ϕ is maximized, subject to ∑ x i = X and x i ≥ 0 , at x 0 = ( x 1 0 , … , x n 0 ) . If the f i are differentiable, then the Gibbs lemma states that there exists a λ such that