ABSTRACT. We study n-player turn-based games played on a finite directed graph. For each play, each player receives a gain that he wants to maximise. Instead of the well-known notions of Nash equilibrium (NE) and subgame perfect equilibrium (SPE), we focus on the recent notion of weak subgame perfect equilibrium (weak SPE), a refinement of SPE. In this setting, players who deviate cannot use the full class of strategies but only a subclass with a finite number of deviation steps. We are interested in the constraint problem: given an upper and a lower thresholds x, y in {0,1}^n, we want to determine if there exists a weak SPE such the gain of each player is componentwise between the two bounds. The goal of our ongoing research is to characterise the complexity of the constraint problem for the class of games with Boolean omega-regular objectives such that Muller, Büchi, Reachability ...