#### Summary

In this LP we learn everything there is about simultaneous games. These games, used when considering a game where players move or play their strategies simultaneously, are commonly used in many fields. From military strategies to collusion agreements, the analysis of these situations as simultaneous games can help us discover the best way to act.#### Nash equilibria and dominant strategies:

#### Mixed strategies:

#### Continuous strategies:

**Cournot duopoly**

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=q_{1}+q_{2} 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^{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