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

Dominant strategies are considered as better than other strategies, no matter what other players might do. In game theory, there are two kinds of strategic dominance:

-a strictly dominant strategy is that strategy that always provides greater utility to a the player, no matter what the other player’s strategy is;

-a weakly dominant strategy is that strategy that provides at least the same utility for all the other player’s strategies, and strictly greater for some strategy.

A dominant strategy equilibrium is reached when each player chooses their own dominant strategy. In the prisoner’s dilemma, the dominant strategy for both players is to confess, which means that confess-confess is the dominant strategy equilibrium (underlined in red), even if this equilibrium is not a Pareto optimal equilibrium (underlined in green).

It must be noted that any dominant strategy equilibrium is always a Nash equilibrium. However, not all Nash equilibria are dominant strategy equilibria.

The elimination of dominated strategies is commonly used to simplify the analysis of any game. The way to proceed is to eliminate for each player every strategy that seems ‘unreasonable’, which will greatly reduce the number of equilibria. This method is quite easy to use when only strictly dominated strategies are in place, but the elimination of weakly dominated strategies can turn problematic, ending up with a game that does not resembles the original one from a strategic point of view.

A good example of elimination of dominated strategy is the analysis of the Battle of the Bismarck Sea. In this game, as depicted in the adjacent game matrix, Kenney has no dominant strategy (the sum of the payoffs of the first strategy equals the sum of the second strategy), but the Japanese do have a weakly dominating strategy, which is to go North (the payoffs are equal for one strategy but strictly better for the other). Since only one of them has a dominant strategy, there is no dominant strategy equilibrium. We must then proceed by eliminating dominated strategies. As we’ve already mentioned, for the Japanese strategy ‘go North’ weakly dominates strategy ‘go South’. Therefore, we eliminate the strategy ‘go South’ for the Japanese, who will go North. Now that we only consider the Japanese going North, Kenney’s strategy ‘go North’ is strictly dominant over strategy ‘go South’, which will be eliminated. Therefore, North-North is the weak-dominance equilibrium.

However, as it was mentioned before, it can be easily seen that the game has lost its strategic nature.