Advanced Microeconomics/Strategies

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Pure Strategies[edit]

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[edit]

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

Strategy Profiles[edit]

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[edit]

Mixed Strategies[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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.