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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. $P_j=function(q_{ij}, n_j)$.

wenxiu0000
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  • Could you provide more details about the game you're defining? I suppose your reviewers are saying that "each player should maximise total profit" is not precise enough. – Gary Liang May 18 '23 at 03:31
  • Sure. I have included more information in the originally posted question. – wenxiu0000 May 18 '23 at 03:48
  • Thanks. What happens if the followers choose quantities which do not add up to the total quantity produced by the leaders? It's not clear why you would need to have this requirement, and why this requirement is the definition of a SPNE. The subgame perfect equilibrium already has a standardised definition, and you can use some form of backward induction to find it. – Gary Liang May 18 '23 at 04:18
  • Thank you. The reason for setting total quantities equal is that I assume selling price is endogenously derived from ∑q=∑q. Taking partial derivative on Π(q), we can sum the first order conditions, and then explicitly show that = a + b ∑q where a, b are convex combinations of the (exogenously given) market parameters where follower j competes. I wonder if this requirement will weaken the definition of subgame perfect equilibrium? – wenxiu0000 May 18 '23 at 04:37

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