The player can force a strategy where they win 2⁄3 of the time (guess a door and never switch). The player never needs to accept worse
The host can force a strategy where the player loses 1⁄3 of the time (never let the player switch). The host never needs to accept worse.
Therefore, the equilibrium has 2⁄3 win for the player. The player can block this number from going lower and the host can block this number from going higher.
The player can force a strategy where they win 2⁄3 of the time (guess a door and never switch). The player never needs to accept worse
The host can force a strategy where the player loses 1⁄3 of the time (never let the player switch). The host never needs to accept worse.
Therefore, the equilibrium has 2⁄3 win for the player. The player can block this number from going lower and the host can block this number from going higher.