Controlling a Random Population is EXPTIME-hard
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Bertrand et al. [1] (LMCS 2019) describe two-player zero-sum games in which one player tries to achieve a reachability objective in n games (on the same finite arena) simultaneously by broadcasting actions, and where the opponent has full control of resolving non-deterministic choices. They show EXPTIME completeness for the question if such games can be won for every number n of games. We consider the almost-sure variant in which the opponent randomizes their actions, and where the player tries to achieve the reachability objective eventually with probability one. The lower bound construction in [1] does not directly carry over to this randomized setting. In this note we show EXPTIME hardness for the almost-sure problem by reduction from Countdown Games.
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