Gibbs lemma

Last updated April 04, 2019

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.^[1] It is named for Josiah Willard Gibbs.

Game theory is the study of mathematical models of strategic interaction between rational decision-makers. It has applications in all fields of social science, as well as in logic and computer science. Originally, it addressed zero-sum games, in which one person's gains result in losses for the other participants. Today, game theory applies to a wide range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals, and computers.

Josiah Willard Gibbs was an American scientist who made important theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in transforming physical chemistry into a rigorous inductive science. Together with James Clerk Maxwell and Ludwig Boltzmann, he created statistical mechanics, explaining the laws of thermodynamics as consequences of the statistical properties of ensembles of the possible states of a physical system composed of many particles. Gibbs also worked on the application of Maxwell's equations to problems in physical optics. As a mathematician, he invented modern vector calculus.

Consider $\phi =\sum _{i=1}^{n}f_{i}(x_{i})$ . 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, then the Gibbs lemma states that there exists a $\lambda$ such that

Differentiable function function whose derivative exists at each point in its domain

In calculus, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. As a result, the graph of a differentiable function must have a (non-vertical) tangent line at each point in its domain, be relatively smooth, and cannot contain any breaks, bends, or cusps.

{\begin{aligned}f'_{i}(x_{i}^{0})&=\lambda {\mbox{ if }}x_{i}^{0}>0\\&\leq \lambda {\mbox{ if }}x_{i}^{0}=0.\end{aligned}}

Notes

↑ J. M. Danskin (6 December 2012). The Theory of Max-Min and its Application to Weapons Allocation Problems. Springer Science & Business Media. ISBN 978-3-642-46092-0. ... problems in which one side must make his move knowing that the other side will then learn what the move is and optimally counter. They are fundamental in particular to military weapons-selection problems involving large systems...

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[Danskin2012-1] J. M. Danskin (6 December 2012). The Theory of Max-Min and its Application to Weapons Allocation Problems. Springer Science & Business Media. ISBN 978-3-642-46092-0. ... problems in which one side must make his move knowing that the other side will then learn what the move is and optimally counter. They are fundamental in particular to military weapons-selection problems involving large systems...