I have a multiple leader, multiple follower Cournot game below:
Sequence: In stage 1, leaders independently decide their supply quantities to each of the followers, and selling prices are determined as a function of the $q_{ij}$ and number of markets $j$ competes in stage 2, i.e. $P_j=function(q_{ij}, n_j)$.
In stage 2, followers take $P_j$ as the procurement cost, and independently decide their supply quantities to each of the markets, and their selling prices are determined with a linear inverse demand function $P_k=\alpha_k-\beta_kQ_k$. If the leaders' selling prices $P_j$ are so high that some follower find it non-profitable to supply to some market $k$, then we have $n_j:=n_j-1$. Then the upstream prices should change and we go back to stage 1 and resolve the equilibrium until we find an equilibrium where $n_j$ is stable.
I wonder: (1) If the above game admits a subgame perfect equilibrium? (2) How to describe the problem to make the equilibrium (whether a Nash equilibrium or a subgame perfect equilibrium) well-defined?