Say, for Cruel Edict I have to choose which one of my opponent's heads sacrifices a creature. But I think that Sudden Disappearance makes all of their permanents go away, not just one person's. Can someone tell me why?
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Sudden Disappearance affects only one opponent, even in Two-Headed Giant.
Players do not share control of permanents in Two-Headed Giant.
Cards that refer to one opponent only affect one opponent. Cards that refer to multiple opponents affect both. The only change to what you'd expect from normal Magic is for certain cards that say "attacking player" or "defending player":
Cruel Edict and Sudden Disappearance both only affect a single opponent. In general, the only cards that would affect both heads of an opponent in two headed giant (2HG) are cards that say each or all within the rules text (example: Heartless Hidetsugu), but this would be have the same effect in non-2HG duels or multiplayer games.
The only differences in 2HG with respect to rules text, is that players attack/defend in teams and take their turn as a team. A card that referred to an attacking or defending player would be referring to both members of the teams, or cards that add/skip turns or phases. The MtG Comprehensive Rules state:
You are also "sharing" turns.
A card that reads in part "target player takes an extra turn" will cause both heads to take an additional turn.
The same is true for controling another player's Turn (Mindslaver, Sorin Markov)