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The solution of parity games over pushdown graphs (Walukiewicz '96) was thefirst step towards an effective theory of infinite-state games. It was shownthat winning strategies for pushdown games can be implemented again as pushdownautomata. We continue this study and investigate the connection between gamepresentations and winning strategies in altogether six cases of game arenas,among them realtime pushdown systems, visibly pushdown systems, and countersystems. In four cases we show by a uniform proof method that we obtainstrategies implementable by the same type of pushdown machine as given in thegame arena. We prove that for the two remaining cases this correspondencefails. In the conclusion we address the question of an abstract criterion thatexplains the results.

Formats of Winning Strategies for Six Types of Pushdown Games