Nash equilibrium and score

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I'm investigating a 3-player Nash game, and I need a sanity check.

Is it true that the outcome of a 3 player nash game need not be determined, even when the strategies are pure? The game is cubicup, where only one player can win, deterministic, finite actions/space. Specifically, it seems like one player is doomed but can 'choose who to lose to' with its nash-equilibrium strategies.