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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]]. |
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| | Consider <math>\phi=\sum_{i=1}^n f_i(x_i)</math>. Suppose <math>\phi</math> is maximized, subject to <math>\sum x_i=X</math> and <math>x_i\geq 0</math>, at <math>x^0=(x_1^0,\ldots,x_n^0)</math>. If the <math>f_i</math> are [[differentiable]], then the Gibbs Lemma states that there exists a <math>\lambda</math> such that |
| | |
| | :<math>\begin{align} |
| | f'_i(x_i^0)&=\lambda \mbox{ if } x_i^0>0\\ |
| | &\leq\lambda\mbox { if }x_i^0=0. |
| | \end{align} |
| | </math> |
| | |
| | ==References== |
| | * J. M. Danskin 1967. ''The Theory of Max-Min'', Springer-Verlag; page 10. |
| | |
| | [[Category:Game theory]] |
| | |
| | {{mathapplied-stub}} |
Revision as of 12:39, 19 March 2013
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 . Suppose is maximized, subject to and , at . If the are differentiable, then the Gibbs Lemma states that there exists a such that
References
- J. M. Danskin 1967. The Theory of Max-Min, Springer-Verlag; page 10.
Template:Mathapplied-stub