strategy Strategy2 2002-07-24 21:33:23 2002-07-28 14:49:20 Definition strategy pure strategy mixed strategy strategy space % 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 \emph{pure strategy} provides a \PMlinkescapetext{complete} definition for a way a player can play a game. In particular, it defines, for every possible choice a player might have to make, which option the player picks. A player's strategy space is the set of pure strategies available to that player. A \emph{mixed strategy} is an assignment of a probability to each pure strategy. It defines a probability over the strategies, and reflect that, rather than choosing a particular pure strategy, the player will randomly select a pure strategy based on the distribution given by their mixed strategy. Of course, every pure strategy is a mixed strategy (the function which takes that strategy to $1$ and every other one to $0$). The following notation is often used: \begin{itemize} \item $S_i$ for the strategy space of the $i$-th player \item $s_i$ for a particular element of $S_i$; that is, a particular pure strategy \item $\sigma_i$ for a mixed strategy. Note that $\sigma_i\in S_i\rightarrow [0,1]$ and $\sum_{s_i\in S_i} \sigma_i(s_i)=1$. \item $\Sigma_i$ for the set of all possible mixed strategies for the $i$-th player \item $S$ for $\prod_i S_i$, the set of all possible \PMlinkescapetext{combinations} of pure strategies (essentially the possible outcomes of the game) \item $\Sigma$ for $\prod_i \Sigma_i$ \item $\sigma$ for a \emph{strategy profile}, a single element of $\Sigma$ \item $S_{-i}$ for $\prod_{j\neq i} S_j$ and $\Sigma_{-i}$ for $\prod_{j\neq i} \Sigma_j$, the sets of possible pure and mixed strategies for all players other than $i$. \item $s_{-i}$ for an element of $S_{-i}$ and $\sigma_{-i}$ for an element of $\Sigma_{-i}$. \end{itemize}