Perfect recall

A player has perfect recall when the information sets never erase that player's previously available information or earlier chosen actions. With perfect recall, randomizing once over complete pure strategies and using suitable independent randomizations at information sets can induce the same outcome distributions.

Without perfect recall, one information set may be encountered twice on the same play path. A behavioural strategy then uses the same probabilities with fresh draws at both encounters; its outcome distribution need not be attainable by any mixed strategy over deterministic plans.

In a finite extensive-form game with perfect recall, each mixed strategy has a realization-equivalent behavioural strategy, and every behavioural strategy has a realization-equivalent mixed strategy. The latter direction samples all contingent actions independently in advance. For the former, condition the random complete plan on the player's own action and information history leading to an information set; perfect recall makes that own history common to its nodes. Conditional action probabilities then reproduce the terminal path factors. At an information set reached with zero probability under that player's own plan distribution, choose any action distribution. The result does not assert equivalence without perfect recall.^[ref]