## Strategies

### Pure Strategies

A (pure) strategy specifies how a player will react in all possible circumstances in which s/he may be called to act. The strategy $s_i$ maps from the set of information sets, $\mathbf{\mathcal{H}}$ to the set of actions $\mathbf{\mathcal{A}}$
$s_i : \mathbf{\mathcal{H}} \rightarrow \mathbf{\mathcal{A}}$
such that $s_i(H) \in C(H) \;\forall\; H \in \mathbf{\mathcal{H}}$
requiring the strategy to specify only feasible actions at each information set.

#### Cardinality of the Strategy Space

A player with $m$ information sets choosing from $b_k$ actions at each information set $H$, the number of possible

#### Strategy Profiles

A strategy profile $s= (s_1,\dots,s_I)$ specifies a collection of strategies for each player and may also be written $s=(s_i,s_{-i})$

Throughout the following discussion the set $\mathbf{\mathbb{S}}_i$ represents the set of all pure strategies available to player $i$ and the set
$\mathbf{\mathbb{S}}=\times_{i=1}^I\mathbf{\mathbb{S}}_i$ be the set of pure strategy profiles

### Randomized Strategies

#### Mixed Strategies

The mixed strategy $\sigma$ assigns each pure strategy $s_i \in \mathbf{\mathbb{S}}$ a probability it will be played,
$\sigma_i : \mathbf{\mathbb{S}}_i \rightarrow [0,1]$
such that $\sum_{s_i\in\mathbf{\mathbb{S}}_i}\sigma_i(s_i)=1$
requiring the probabilities assigned to the elements of $\mathbf{\mathbb{S}}$ sum to one, $\sigma$ is a probability distribution function over $\mathbf{\mathbb{S}}$

#### Mixed Extension

A mixed extension, the simplex $\Delta(\mathbf{\mathbb{S}}_i)$, denotes the space of all mixed strategies over a pure strategy set $\mathbf{\mathbb{S}}_i$.
$\Delta(\mathbf{\mathbb{S}}_i) = \left\{(\sigma_{1,i},\dots,\sigma_{M,i}): \sigma_{m,i} \geq 0 \;\forall\; m = 1,\dots,M\mbox{ and } \sum_{m=1}^M\sigma_{m,i} = 1\right\}$

#### Expected Utility

Given a mixed strategy profile $\sigma$ the expected utility $E_{\sigma}[u_i(s)]$ maps from all possible outcomes to the real line. Intuitively, calculating expected utility requires weighting the uvtility associated with each pure strategy profile $u_i(s)$ by the probability each profile will be played,
$E_{\sigma}[u_i(s)] = \sum_{s\in\mathbf{\mathbb{S}}}Pr(s)\cdot u_i(s)$
The mixed profile $\sigma$ assigns probabilities to each pure strategy $s$ implying
$Pr(s) \equiv [\sigma_1(s_1)\cdot\sigma_w(s_2)\dots\cdot\sigma_I(s_I)] = \prod_{i=1}^I\sigma_i(s_i)$
Thus, the expected utility from $\sigma$
$E_{\sigma}[u_i(s)]=\sum_{s\in\mathbb{\mathbf{S}}}\left[ \left( \prod_{i=1}^I\sigma_i(s_i)\right)u_i(s) \right]$

#### Behavioral Strategy

In lieu of randomizing over pure strategies, a randomized strategy may be written as a tuple of probability distributions over actions available at each information set. A behavioral strategy then, specifies
$\forall H\in\mathbf{\mathcal{H}} \mbox{ and action } a \in \mathbf{\mathcal{A}} \mbox{ a probability } \lambda_{i}(a, H) \geq 0$
such that $\sum_{a\in C(H)} \lambda(a,H) = 1 \;\forall\;H\in\mathbf{\mathcal{H}}$

#### Behavioral vs Mixed Strategies

A key distinction between behavioral and mixed strategies when the randomization occurs during the course of play. In the case of mixed strategies, players randomize over the set of pure strategies prior to play. For behavioral strategies, randomization occurs during the course of play. A behavior strategy mixture admits both types of randomization, allowing the specification of a mixed strategy over the space of all behavioral strategies, $\sigma_i$ which assigns positive probabilities to one or more (finite) behavioral strategies $(b_{1,i}, \dots, b_k)$.
Any game exhibiting perfect recall admits pairs of behavioral and mixed strategies which exhibit outcome (realization) equivalence, meaning each strategy produces the same probability distribution over outcomes. The probability distribution over outcomes implied by any mixed strategy also results from a (unique?) behavioral strategy.