NashEquilibrium 2002-07-24 00:15:22 2002-09-30 13:48:54 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here A Nash equilibrium of a game is a set of (possibly mixed) strategies $\sigma=(\sigma_1,\ldots,\sigma_n)$ such that, if each player $i$ believes that that every other player $j$ will play $\sigma_j$, then $i$ should play $\sigma_i$. That is, when $u_i$ is the utility function for the $i$-th player: \begin{displaymath} \sigma_i\neq \sigma^\prime_i\rightarrow u_i(\sigma_i,\sigma_{-1})>u_i(\sigma^\prime_i,\sigma_{-1}) \end{displaymath} \begin{displaymath} \forall i\leq n \text{ and } \forall \sigma^\prime_i\in \Sigma_i \end{displaymath} Translated, this says that if any player plays any strategy other than the one in the Nash equilibrium then that player would do worse than playing the Nash equilibrium.