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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 {\displaystyle \phi =\sum _{i=1}^{n}f_{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 = {\displaystyle x^{0}=}. If the f i {\displaystyle f_{i}} are differentiable, then the Gibbs lemma states that there exists a λ {\displaystyle \lambda } such that