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 q₁ to sell, and the other firm, the follower, observes this and decides on its production quantity q₂. 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 q₁ (blue lines) is as follows:

Firm 1 (leader) anticipates the follower’s behavior and takes it into consideration to make the strategic choice of q₁:

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^S₁ > q^C₁ and π^S₁ > π^C₁

Follower: q^S₂ < q^C₂ and π^S₂ < π^C₂

-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

Cournot duopoly, also called Cournot competition, is a model of imperfect competition in which two firms with identical cost functions compete with homogeneous products in a static setting. It was developed by Antoine A. Cournot in his “Researches Into the Mathematical principles of the Theory of Wealth”, 1838. Cournot’s duopoly represented the creation of the study of oligopolies, more particularly duopolies, and expanded the analysis of market structures which, until then, had concentrated on the extremes: perfect competition and monopolies.

Cournot really invented the concept of game theory almost 100 years before John Nash, when he looked at the case of how businesses might behave in a duopoly. There are two firms operating in a limited market. Market production is: P(Q)=a-bQ, where Q=q₁+q₂ for two firms. Both companies will receive profits derived from a simultaneous decision made by both on how much to produce, and also based on their cost functions: TC_i=C-q_i.

So, algebraically:

In order to maximise, the first order condition will be:

And, if q_i=q_j, then both equal:

Therefore, the reaction functions (blue lines), where the key variable is the quantity set by the other firm, will take the following form:

What all this explains is a very basic principle. Both companies are vying for maximum benefits. These benefits are derived from both maximum sales volume (a larger share of the market) and higher prices (higher profitability). The problem stems from the fact that increasing profitability through higher prices can damage revenue by losing market share. What Cournot’s approach does is maximise both market share and profitability by defining optimum prices. This price will be the same for both companies, as otherwise the one with the lower price will obtain full market share, which makes this a Nash equilibrium, also known for this model the Cournot-Nash equilibrium.

If we consider isoprofit curves (those which show the combinations of quantities that will render the same profit to the firm, red curves) we can see that the equilibrium of the game is not Pareto efficient, since isoprofit curves are not tangent. The outcome is below that of perfect competition and therefore is not socially optimal, but it is better than the monopoly outcome.

Extending the model to more than two firms, we can observe that the equilibrium of the game gets closer to the perfect competition outcome as the number of firms increases, decreasing market concentration.

Comparison with Stackelberg duopolies:

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

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

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

Leader: q^S₁ > q^C₁ and π^S₁ > π^C₁

Follower: q^S₂ < q^C₂ and π^S₂ < π^C₂

-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

Folk theorems are used in Economics specially in the field of game theory and specifically to repeated games. This theorem is said to be satisfactorily fulfilled when the equilibrium outcome in a game that is repeated an infinity number of times, is the same as the feasible and strongly individually rational outcome in the one-shot game. The outcome is said to be in equilibrium, because any attempt of trying to increase a player’s individual outcome will imply a decrease in at least another player’s individual outcome.

The origin of this theorem is unknown but it appeared in the late fifties, and became quickly known and spread within the game theory academia, and thus the reason why it became known as the folk theorem.  James W. Friedman in his article “A Non-cooperative Equilibrium for Supergames”, 1971, was the first to write and publish a paper in which he thoroughly deals with the theorem.

The main issue that this theorem explains, is the strong relationship between repeated and cooperative games, revealing the enforcement mechanism characteristic of repetition. One of the first applications of the theorem was to collude when dealing with a Cournot duopoly:
 

The left hand side represents the payoff derived from collusion, which can be held infinitely over time, with δ being the discount factor to bring future benefits forward to the present. For our threats or offers to be credible, this left hand side must be greater than the right hand side, which represents the one off payoff to be gained from deviating and breaking our cartel. The higher δ is, the higher the value assigned to future benefits and therefore the greater the chances of collusion. It is worth reminding, that fair competition is regulated in almost all countries, with cartelsbeing banned, and so most markets that lend themselves to reduced competition and price fixing, are closely monitored.

In game theory, a subgame is a subset of any game that includes an initial node (which has to be independent from any information set) and all its successor nodes. It’s quite easy to understand how subgames work using the extensive form when describing the game. In the following game tree there are six separate subgames other than the game itself, two of them containing two subgames each.

A subgame-perfect equilibrium is an equilibrium not only overall, but also for each subgame, while Nash equilibria can be calculated for each subgame. Consider the following game: player 1 has to decide between going up or down (U/D), while player 2 has to decide between going left or right (L/R). In this case, we can represent this game using the strategic form by laying down all the possible strategies for player 2:

-go Right if player 1 goes Up, go Left otherwise;

-go Left if player 1 goes Up, go Right otherwise;

-go Right no matter what;

-go Left no matter what.

We can see how this game is described using the extensive form (game tree on the left) and using the strategic form (game matrix on the left). Since this is a sequential game, we must describe all possible outcomes depending on player 2 decisions, as seen in the game matrix. 

If we look for the equilibrium of this game, considered as a whole, we find that Up-Left is a Nash equilibrium (red). However, it’s not a perfect equilibrium. In order to find the subgame-perfect equilibrium, we must do a backwards induction, starting at the last move of the game, then proceed to the second to last move, and so on. In this particular case, we know that player 2 will choose Left if player 1 goes Up, and Right if player 1 goes Down, since these are the moves that maximise his utility. Because there is complete information (and therefore each player’s payoffs are known), player 1 knows these choices in advance, and will therefore choose to go Down, because the payoff will be greater. Therefore, Down-Right is the perfect subgame equilibrium (green).

In sequential games, a series of decisions are made, the outcome of each of which affects successive possibilities. In game theory, the analysis of sequential games is of great interest because they usually model reality better than simultaneous games: producers will usually observe demand before deciding how much output to produce, duopolists will observe each other’s decisions before dumping more goods in the market, etc.  For example, let’s examine the decision a company faces when trying to break a market which is currently a monopoly. Sequential games are represented through decision trees, with successive nodes at each decision point:

The game represented in this decision tree shows firm 1 choosing whether to compete in a monopolistic market or not. As there are no existing competitors in the market, should they choose not to enter, its payoff will be zero and the existing monopoly will have a payoff of two. However, if firm 1 chooses to enter we reach node two, where the second decision must be made by the challenged player (the monopoly), whether to accommodate the new competitor or fight back to try and block its entry. In this particular example, if they choose to fight, both players will have a payoff of -1, whereas if the monopoly decides to accommodate the new player, our monopoly becomes a duopoly and the resulting drop in prices creates larger market demand, meaning that the existing company will still receive a payoff of one and the new player a payoff of two.

So, what can we assume will happen? We assume that both players dispose of complete and perfect information. They have both carried out a thorough market study and know the consequences of each outcome. In that case, what they can do is work backwards to know what the competition will do. If firm 1 enters the market, player 2 has no reason to fight back, as its payoff will suffer by two points. This means that player 2 knows what the outcome will be if they choose to enter, so all they have to do is choose between a payoff of two if the new firm enters the market or zero if it doesn’t. In this case, it is pretty clear what the outcome will be. The game would change again, of course, if successive nodes were introduced, where fighting could again flip the tables.