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How do we resolve initiative tie in combat between heroes? Especially in the last (Fall of the dark star) scenario, where there is no attacking player?
In normal PvP battles the hero who challenges another hero is the attacker. This is found in "Challenge" section of Rule Reference:
The hero who made the challenge is the attacker and wins ties for
As for final battle in Fall of the Dark Star I'm not sure. I don't have the RB box by me. It might have been that first player in turn order wins initiative.