200px, Josiah Willard Gibbs In game theory and in particular the study of

Blotto games A Colonel Blotto game is a type of two-person constant-sum game in which the players (officers) are tasked to simultaneously distribute limited resources over several objects (battlefields). In the classic version of the game, the player devoting ...

and

operational research Operations research ( en-GB, operational research) (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a discipline that deals with the development and application of analytical methods to improve deci ...

, the Gibbs lemma is a result that is useful in maximization problems. It is named for

Josiah Willard Gibbs Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in t ...

. Consider

\phi=\sum_^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 In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...

, then the Gibbs lemma states that there exists a

\lambda

such that :

\begin
f'_i(x_i^0)&=\lambda \mbox x_i^0>0\\
&\leq\lambda\mbox x_i^0=0.
\end

Notes

References

Game theory {{gametheory-stub