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 ${\sigma }_i\in S_i\to [0,1]$ and ${\sum }_{s_i\in S_i}{\sigma }_i(s_i)=1$.

• ${\mathrm{\Sigma }}_i$ 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)

• $\mathrm{\Sigma }$ for ${\prod }_i{\mathrm{\Sigma }}_i$

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

• $S_{-i}$ for ${\prod }_{j\ne i}{S}_j$ and ${\mathrm{\Sigma }}_{-i}$ for ${\prod }_{j\ne i}{\mathrm{\Sigma }}_j$, 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 ${\mathrm{\Sigma }}_{-i}$.