I am defining an equilibrium for a multiple-leader, multiple-follower Cournot game where the selling prices are functions of the equilibrium quantities. I require that, in the equilibrium, 1) each player should maximize its total profit; and 2) the total quantity provided by the leaders should be equal to the total quantity provided by the followers. However, the reviewer says that this is not a rigorously defined subgame perfect equilibrium.
My question is: 1) despite that this is a sequential game, can I say that this is a definition of the Nash equilibrium, rather than a subgame perfect equilibrium? 2) If I have to define a subgame perfect equilibrium, what should I do to make it rigorous?
The equilibrium quantity is defined as
(1) each leader i: $q^*_{ij}= argmax_{\boldsymbol{q}\geq0} \Pi_i(\boldsymbol{q})$, and $\Pi_i(\boldsymbol{q})= \sum_{j} (P_j-v_i)q_{ij}$;
(2) each follower j: $q^*_{jk}= argmax_{\boldsymbol{q}\geq0} \Pi_j(\boldsymbol{q})$, and $\Pi_j(\boldsymbol{q})= \sum_{k} (P_k- P_j-v_j)q_{jk}$.
(3) total quantity produced by leaders is equal to that produced by followers: $\sum_{ij} q_{ij} = \sum_{jk} q_{jk}$.
(4) leaders' selling prices are functions of quantities and also depend on number of markets in which each follower competes. $P_j=function(q_{ij}, n_j)$.