Rules FAQ
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If multiple characters are tied for the lowest STR, the losing opponent chooses which one to kill.
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A character that cannot be killed (Balerion), cannot leave play (The Crone), or is immune to events (A Tourney for the King) cannot be chosen to be killed. If such a character is the only character with the lowest STR among characters the losing opponent controls, A Lannister's Honor cannot be played.
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You check for the character with the lowest STR at the time the event is played, not at the time the winner of the challenge is determined. In other words, if the character with the lowest STR at the point of winning the challenge is removed from play (for instance, by Jaqen H'ghar), it is not relevant in any way when determining valid targets for A Lannister's Honor.