Advanced Microeconomics/Strategies

StrategiesEdit

Pure StrategiesEdit

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 SpaceEdit

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

Strategy ProfilesEdit

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 StrategiesEdit

Mixed StrategiesEdit

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 ExtensionEdit

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

Expected UtilityEdit

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 StrategyEdit

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 StrategiesEdit

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.