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

)

{\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.