I plan on building a deck using Fateful Showdown and Treasure Hunt so I was wondering if I did lethal but I would have to mill out myself would I win, lose, or tie?
If Fateful Showdown deals lethal damage to your opponent and causes you to draw from an empty library, the game will be a draw. This is because both the loss from being at 0 or less life and the loss from having had to draw from an empty library are state-based actions, which are checked right after Showdown is done resolving. Since you would both lose at the same time, that makes the game a draw.
You would tie. No player loses during the resolution of Fateful Showdown, only after it finished resolving are state based actions checked, two of which are losing for having 0 or less life and losing for having tried and failed to draw a card from an empty library since the last check of state based actions. all state based actions are performed simultaneously, so both players lose at the same time.
Worth to note here, that if you draw exactly the number of cards left in your library to kill your opponent, you win. You would only lose the game (or in this case let the game end in a tie), if you try to draw from an empty library.
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