Editing Duopoly (section)

==Duopoly models in economics and game theory==

===Cournot duopoly===

====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.

===Bertrand duopoly===
====Bertrand model in game theory====

The [[Bertrand competition]] was developed by a French mathematician called [[Joseph Louis François Bertrand]] after investigating the claims of the Cournot model in "Researches into the mathematical principles of the theory of wealth, 1838".<ref name=":0" /> According to the Cournot model, firms in a duopoly would be able to keep prices above marginal cost and hence be extremely profitable.<ref>{{Cite journal |last=Vives |first=Xavier |author-link=Xavier Vives |date=October 1984 |title=Duopoly information equilibrium: Cournot and bertrand |journal=[[Journal of Economic Theory]] |volume=34 |issue=1 |pages=71–94 |doi=10.1016/0022-0531(84)90162-5 |issn=0022-0531}}</ref>  Bertrand took issue with this. In this market structure, each firm could only choose whole amounts and each firm receives zero payoffs when the aggregate demand exceeds the size of the amount that they share with each other. The market demand function is <math>Q(P)=a-bP</math>. The Bertrand model has similar assumptions to the Cournot model:

* Two firms
* Homogeneous products
* Both firms know the market demand curve
* However, unlike the Cournot model, it assumes that firms have the same MC. It also assumes that the MC is constant.

The Bertrand model, in which, in a [[Game theory|game]] of two firms, competes in price instead of output. Each one of them will assume that the other will not change prices in response to its price cuts. When both firms use this logic, they will reach a [[Nash equilibrium]].

* Consider price competition among two firms ({{math|1=''i'' = 1, 2}}) selling homogeneous good
* Downward sloping market demand {{math|''D''(''p'')}}, with {{math|''{{prime|D}}''(''p'') < 0}}
* Constant, symmetric marginal cost {{math|1= ''c''{{sub|1}} = ''c''{{sub|2}} = ''c''}}
* Static game: firms set prices simultaneously
* Rationing rule of demand:

# lowest priced firm wins all demand at its price
# if prices are tied, each firm gets half of market demand at this price

* Firm {{mvar|i}}{{'}}s profits: <math display=inline>\Pi_i = (p_i-c)D_i(p_i, p_j)</math>

Let {{mvar|p{{sup|m}}}} be the monopoly price, <math display=inline>p^m = \operatorname{argmax}_p(p-c)D(p)</math>

* Firm {{mvar|i}}{{'}}s best response {{math|''R{{sub|i}}''(''p{{sub|j}}'')}} is:

<math display="block>R_i(p_j) = \begin{cases}
p^m, & \text{if } p_j > p^m \\
p_j - c, & \text{if } c < p_j \le p^m \\
c, & \text{if } p_j \le c
\end{cases}</math>

For rival prices above cost, each firm has incentive to undercut rival to get the whole demand. If rival prices below cost, firms make losses when it attracts demand; firm better off charging at cost level. Nash equilibrium is {{math|1= ''p''{{sub|1}} = ''p''{{sub|2}} = ''c''}}.

====Bertrand paradox====
Under static price competition with homogenous products and constant, symmetric marginal cost, firms price at the level of marginal cost and make no [[Profit (economics)|economic profits]]. In contrast to the Cournot model, the Bertrand duopoly model assumes that firms compete on price rather than quantity. Each firm sets its price simultaneously, anticipating that the other firm will not change its price in response. When both firms use this logic, they will reach a Nash equilibrium, where neither firm has an incentive to change its price given the price set by the other firm. In this model, firms tend to price their products at the level of their marginal cost, resulting in zero economic profits, a phenomenon known as the [[Bertrand paradox (economics)|Bertrand paradox]].