action of stage m if, for every pair of infinite histories u = ( a$_{0}$, a$_{1}$, . . . ) and u ′ = ( a ′ $_{0}$, a ′ $_{1}$, . . . ), u ∼$_{n}$ u ′ implies a$_{m}$ = a ′ $_{m}$. 

1.1. Definition. The information partitions ( P$_{n}$ )$_{n}$$_{≥}$$_{0}$ satisfy perfect recall if the following conditions are satisfied: 

(1) Players know their own actions: at stage n the player observes the action of stage n − 2. 

(2) Players do not forget information: if u, u ′ ∈ A N and u ∼$_{n}$$_{+2}$ u ′ then u ∼$_{n}$ u $^{′}$. 

The setup of infinite games with perfect recall is general enough to subsume two cases which have been extensively studied: 

Gale-Stewart games. If, at every stage n , players observe previous actions of their op- ponents, then the game is called a Gale-Stewart game or a game with perfect information . Gale and Stewart [4] proved that such games are determined if the winning set W is closed (or open) and asked whether the result can be extended to more complex winning sets. In a seminal paper, Martin [8] proved that the game is determined for every Borel winning set W . Gale-Stewart games admit pure 0-optimal strategies, and the value is 0 or 1. Moreover, Gale-Stewart games with an infinite action set A and Borel winning set are also determined. Blackwell games. Assume that at even stages n = 2 k , player 1 observes the actions of stages 0 , 1 , . . . , 2 k − 1, and at odd stages n = 2 k + 1, player 2 observes the actions of stages 0 , 1 , . . . , 2 k − 1 (his own actions and all the previous actions of his opponent except for the last one), and that the information partitions are the roughest partitions that satisfy these conditions. This essentially means that the players choose actions simultaneously and independently at stages 2 k and 2 k + 1, and then both actions are made public. Such games are called Blackwell games . Blackwell [1, 2] proved the determinacy of Blackwell games (which he called “infinite games with imperfect information”) with a G$_{δ}$ winning set, and conjectured that every Blackwell game with a Borel winning set is determined. Vervoort [15] advanced higher in the Borel hierarchy, proving determinacy of games with G$_{δσ}$ winning sets. Blackwell’s conjecture was proved by Martin in 1998 [9]. 

