# strategy

A pure strategy provides a 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 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:

• $S_{i}$ for the strategy space of the $i$-th player

• $s_{i}$ for a particular element of $S_{i}$; that is, a particular pure strategy

• $\sigma_{i}$ for a mixed strategy. Note that $\sigma_{i}\in S_{i}\rightarrow[0,1]$ and $\sum_{s_{i}\in S_{i}}\sigma_{i}(s_{i})=1$.

• $\Sigma_{i}$ for the set of all possible mixed strategies for the $i$-th player

• $S$ for $\prod_{i}S_{i}$, the set of all possible of pure strategies (essentially the possible outcomes of the game)

• $\Sigma$ for $\prod_{i}\Sigma_{i}$

• $\sigma$ for a strategy profile, a single element of $\Sigma$

• $S_{-i}$ 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 $i$.

• $s_{-i}$ for an element of $S_{-i}$ and $\sigma_{-i}$ for an element of $\Sigma_{-i}$.

Title strategy Strategy 2013-03-22 12:52:02 2013-03-22 12:52:02 Henry (455) Henry (455) 7 Henry (455) Definition msc 91A99 Game strategy pure strategy mixed strategy strategy space