Editing Duopoly (section)

====Cournot model in game theory====
In 1838, [[Antoine Augustin Cournot]] published a book titled "Researches Into the Mathematical Principles of the Theory of Wealth" in which he introduced and developed this model for the first time. As an imperfect competition model, Cournot duopoly (also known as Cournot competition), in which two firms with identical cost functions compete with homogenous products in a static context, is also known as [[Cournot competition]].<ref>{{Cite journal |last1=Tremblay |first1=Carol Horton |last2=Tremblay |first2=Victor J. |date=June 2011 |title=The Cournot–Bertrand model and the degree of product differentiation |journal=[[Economics Letters]] |volume=111 |issue=3 |pages=233–235 |doi=10.1016/j.econlet.2011.02.011 |issn=0165-1765}}</ref> The Cournot model, shows that two firms assume each other's output and treat this as a fixed amount, and produce in their own firm according to this. The Cournot duopoly model relies on the following assumptions:<ref>{{Cite book |last=Dranove |first=David |title=Economics of Strategy |publisher=Hoboken:Wiley |year=2016 |edition=7th}}</ref>

* Each firm chooses a quantity to produce independently
* All firms make this choice simultaneously
* The cost structures of the firms are public information

In this model, two companies, each of which chooses its own quantity of output, compete against each other while facing constant marginal and average costs.<ref>{{Cite journal |last=Symeonidis |first=George |date=January 2003 |title=Comparing Cournot and Bertrand equilibria in a differentiated duopoly with product R&D |journal=International Journal of Industrial Organization |volume=21 |issue=1 |pages=39–55 |doi=10.1016/S0167-7187(02)00052-8 |issn=0167-7187}}</ref> The market price is determined by the sum of the output of two companies. <math>P(Q)=a-bQ</math> is the equation for the market demand function.<ref name=":0">{{cite book|title=Recherches surplus Principes Mathématiques de la Théorie des Richesses|trans-title=Researches Into the Mathematical Principles of the Theory of Wealth|last=Cournot|first=Antoine Augustin|author-link=Antoine Augustin Cournot|translator-last=Bacon|translator-first=Nathaniel T.|date=1897|orig-date=Originally published 1838|publisher=[[Macmillan Publishers|The Macmillan Company]]|location=[[New York City|New York]]|url=https://books.google.com/books?id=eGgPAAAAYAAJ|hdl=2027/hvd.32044024354821|access-date=January 18, 2023}}</ref>

* Market with two firms {{math|1= ''i'' = 1, 2}} with constant marginal cost {{mvar|c{{sub|i}}}}
* Inverse market demand for a homogeneous good: {{math|1= ''P''(''Q'') = ''a'' − ''bQ''}}
* Where {{mvar|Q}} is the sum of both firms' production levels: {{math|1= ''Q'' = ''q''{{sub|1}} + ''q''{{sub|2}}}}
* Firms choose their quantity simultaneously (static game)
* Firms maximize their profit: 
<math display="block">\begin{aligned}
\Pi_1(q_1,q_2) &= \left(P(q_1 + q_2) - c_1\right)*q_1\,, \\
\Pi_2(q_1,q_2) &= \left(P(q_1 + q_2) - c_2\right)*q_2
\end{aligned}</math>

The general process for obtaining a Nash equilibrium of a game using the [[best response]] functions is followed in order to discover a Nash equilibrium of Cournot's model for a specific cost function and demand function. A Nash Equilibrium of the Cournot model is a {{nowrap|(<math display=inline>q_1^*, q_2^*</math>)}} such that

For a given {{nowrap|<math display=inline>q_1^*</math>,}} <math display=inline>q_2^*</math> solves:

<math display="block>\begin{aligned}
\operatorname{MAX}_{q1} \Pi_1(q_1, q_2^*) &= (P(q_1 + q_2^*) - c_1)q_1\,, \\
\operatorname{MAX}_{q2} \Pi_2(q_1^*, q_2) &= (P(q_1^* + q_2) - c_1)q_2
\end{aligned}</math>

Given the other firm's optimal quantity, each firm maximizes its profit over the residual inverse demand. In equilibrium, no firm can increase profits by changing its output level Two first order conditions equal to zero are the [[best response]].<ref>{{Cite book |last=Motta |first=Massimo |title=Competition Policy: Theory and Practice |publisher=Cambridge University Press |year=2004}}</ref>

Cournot's duopoly marked the beginning of the study of oligopolies, and specifically duopolies, as well as the expansion of the research of market structures, which had previously focussed on the extremes of perfect competition and monopoly. In the Cournot duopoly model, firms choose the quantity of output they produce simultaneously, taking into consideration the quantity produced by their competitor. Each firm's profit depends on the total output produced by both firms, and the market price is determined by the sum of their outputs. The goal of each firm is to maximize its profit given the output produced by the other firm. This process continues until both firms reach a Nash equilibrium, where neither firm has an incentive to change its output level given the output of the other firm.