strategy

A pure strategy provides a 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 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 combinations 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 $\Sigma_{-i}$ .