# Advanced Microeconomics/Strategies

## Strategies[edit | edit source]

### Pure Strategies[edit | edit source]

A (pure) strategy specifies how a player will react in all possible circumstances in which s/he may be called to act. The strategy maps from the set of information sets, to the set of actions

such that

requiring the strategy to specify only feasible actions at each information set.

#### Cardinality of the Strategy Space[edit | edit source]

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

#### Strategy Profiles[edit | edit source]

A *strategy profile* specifies a collection of strategies for each player and may also be written

Throughout the following discussion the set represents the set of all pure strategies available to player and the set

be the set of pure strategy profiles

### Randomized Strategies[edit | edit source]

#### Mixed Strategies[edit | edit source]

The mixed strategy assigns each pure strategy a probability it will be played,

such that

requiring the probabilities assigned to the elements of sum to one, is a probability distribution function over

#### Mixed Extension[edit | edit source]

A mixed extension, the simplex , denotes the space of all mixed strategies over a pure strategy set .

#### Expected Utility[edit | edit source]

Given a mixed strategy profile the expected utility maps from all possible outcomes to the real line. Intuitively, calculating expected utility requires weighting the uvtility associated with each pure strategy profile by the probability each profile will be played,

The mixed profile assigns probabilities to each pure strategy implying

Thus, the expected utility from

#### Behavioral Strategy[edit | edit source]

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

such that

#### Behavioral vs Mixed Strategies[edit | edit source]

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, which assigns positive probabilities to one or more (finite) behavioral strategies .

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.