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 ) {\displaystyle \phi =\sum _{i=1}^{n}f_{i}(x_{i})} . Suppose ϕ {\displaystyle \phi } is maximized, subject to ∑ x i = X {\displaystyle \sum x_{i}=X} and x i ≥ 0 {\displaystyle x_{i}\geq 0} , at x 0 = ( x 1 0 , … , x n 0 ) {\displaystyle x^{0}=(x_{1}^{0},\ldots ,x_{n}^{0})} . If the f i {\displaystyle f_{i}} are differentiable, then the Gibbs lemma states that there exists a λ {\displaystyle \lambda } such that
f i ′ ( x i 0 ) = λ if x i 0 > 0 ≤ λ if x i 0 = 0. {\displaystyle {\begin{aligned}f'_{i}(x_{i}^{0})&=\lambda {\mbox{ if }}x_{i}^{0}>0\\&\leq \lambda {\mbox{ if }}x_{i}^{0}=0.\end{aligned}}}
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