At CSL 2002, Jerzy Marcinkowsi and Tomasz Truderung presented the notions of positive games and persistent strategies [8]. A strategy is persistent if, given any finite or infinite run played on a game graph, each time the player visits some vertex already encountered, this player repeats the decision made when visiting this vertex for the first time. Such strategies require memory, but once a choice is made, it is made for ever. So, persistent strategies are a weakening of memoryless strategies.
The same authors established a direct relation between positive games and the existence of persistent winning strategies. We give a description of such games by means of their topological complexity. In games played on finite graphs, positive games are unexpectedly simple. On the contrary, infinite game graphs, as well as infinite alphabets, yield positive sets involved in non determined games.
Last, we discuss positive Muller winning conditions. Although they do not help to discriminate between memoryless and LAR winning strategies, they bear a strong topological characterization.