Backward induction with an -component utility vector. At a node controlled by player , evaluate each child, choose a child maximizing coordinate and copy its entire vector. A fixed tie rule makes the returned vector well defined. By mathematical induction from terminal positions, the selected continuation is a best response at every node of a finite perfect-information game, giving a pure strategy subgame perfect equilibrium. The vector must be copied intact; separate coordinatewise maxima need not describe any attainable outcome.
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