This paper investigates the constrained existence problem for subgame perfect equilibria (SPEs) in multiplayer graph games. In the proposed framework, each player has a preference relation over the set of plays, assumed to be $ω$-recognizable. Equivalently, he has a preference relation over a finite set of payoffs, and the set of plays with the same payoff is $ω$-regular, for each payoff. This generic framework avoids the need to focus on specific payoff functions. We show that the constrained SPE existence problem is EXPTIME-complete, as well as for Nash equilibria (NEs).

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