#### Summary

Duopolies are commonly used when explaining sequential games, because they model the interdependence between two firms. We learn in this Learning Path how duopolists react to each other’s actions, how collusions work and how repeated sequential games may change the essence of a game.#### Basic definitions:

#### Duopolies and collusions:

- Cournot duopolies
**Stackelberg duopolies**- Collusion

#### Repeated games:

Stackelberg duopoly, also called Stackelberg competition, is a model of *imperfect competition* based on a non-cooperative game. It was developed in 1934 by Heinrich Stackelbelrg in his “Market Structure and Equilibrium” and represented a breaking point in the study of *market structure*, particularly the analysis of *duopolies*, since it was a model based on different starting assumptions and gave different conclusions to those of the *Cournot’s* and *Bertrand’s duopoly models*.

In *game theory*, a Stackelberg duopoly is a *sequential game* (not *simultaneous* as in Cournot’s model). There are two firms, which sell homogeneous products, and are subject to the same demand and cost functions. One firm, the leader, is perhaps better known or has greater brand equity, and is therefore better placed to decide first which quantity q_{1} to sell, and the other firm, the follower, observes this and decides on its production quantity q_{2}. To find the *Nash equilibrium* of the game we need to use backward induction, as in any sequential game. That is, start analyzing the decision of the follower.

For firm 2 (follower), the problem is similar to the Cournot’ model. The reaction as a function of q_{1 }(blue lines) is as follows:

Firm 1 (leader) anticipates the follower’s behavior and takes it into consideration to make the strategic choice of q_{1}:

Therefore, the quantities sold by each firm at equilibrium are:

The perfect equilibrium of the game is the Stackelberg equilibrium. In this game, the leader has decided not to behave as in the Cournot’s model, however, we cannot ensure that the leader is going to produce more and make more profits than the follower (*production* will be larger for the firm with lower *marginal costs*). Total production will be greater and prices lower, but player one will be better off than player two, which serves to highlight two things: the importance of accurate market information when defining a strategy, and the interdependence of each player’s strategies, especially when there is a market leader (with the benefit of moving first) and a follower.

When it comes to economic efficiency, the result is similar to Cournot’s duopoly model. The Nash equilibrium is not *Pareto efficient* (isoprofit curves, green curves, are not tangent to each other) and therefore, there is a loss in economic efficiency. Nevertheless, the loss is lower in the Stackelberg duopoly than in Cournot’s.

Stackelberg and Cournot equilibria are stable in a static model of just one period. In a dynamic context (*repeated games*), the models need to be reconsidered.

**Comparison with Cournot duopolies:**

-Stackelberg’s model is a sequential game, Cournot’s is a simultaneous game;

-In Stackelberg duopolies, the quantity sold by the leader is greater than the quantity sold by the follower, while in Cournot duopolies quantity is the same for both firms;

-When comparing each firm’s output and prices, we have:

Leader: q^{S}_{1} > q^{C}_{1} and π^{S}_{1} > π^{C}_{1}

Follower: q^{S}_{2} < q^{C}_{2} and π^{S}_{2} < π^{C}_{2}

-With regard to total output and prices we have the following:

Q_{M} < Q_{C} < Q_{S} < Q_{PC}

P_{M} > P_{C} > P_{S} > P_{PC} = MC

with:

Q_{C}: total Cournot output

Q_{S}: total Stackelberg output

Q_{PC}: total perfect competition output

Q_{M}: total monopoly output

P_{C}: Cournot price

P_{S}: Stackelberg price

P_{PC}: perfect competition price

P_{M}: monopoly price

MC: marginal cost