A paper written by John Nash in 1950 surely is an innovation contribution to the economy world of that period, especially to the field of Game Theory and of common perception of strategic decision-making. The paper that is called, “Equilibrium points in N-person games”, introduced a fundamental idea which came to be known as Nash Equilibrium.
Based on John Nash’s theory, Nash equilibrium is a cornerstone concept in the game theory and the most extensively known as well as used approach of concluding the result of a strategic competition in the social sciences. According to International Encyclopedia of the Social Sciences, the game of Nash’s theory consists of three elements, which are: a set of actions, a set of players and a pay-off function for each player so it could be explained that a pure-strategy Nash equilibrium, is an action figure with the condition that no single player could obtain a greater payoff by deviating unilaterally from this profile.
The Nash equilibrium concept can be understood by explaining it in example. Consider first a game with two players, each of them has two actions, which we can call X and Y. If they choose different options, both of them would get 0. In a consequence, if they choose X, they will get 2, and if they both choose Y, they will get 1.
Consider a duopoly market in which two firms produce quantities Q_1 and Q_2 of a homogeneous good with increasing marginal cost schedules 〖MC〗_1(Q_1) and 〖MC〗_2 (Q_2), and face a decreasing demand function D(P). Consumers are price-takers with no market power, and price discrimination is not possible. The two firms offer their output for sale at prices P_1 and P_2. If the two prices are different, the low-price seller sells out first to the highest-demand consumers. Output is perishable, but any unsold output maybe disposed of at no cost.
As written above that in Bertrand competition, firms set prices and for Cournot competition, firms set quantities. In these both cases the equilibrium concept adopted from the non-cooperative Nash equilibrium. Accordingly, we have noticed that profits are symmetric in prices and quantities. So, in Bertrand and Cournot equilibria cases: Cournot competition firm 1 chooses q_1 to maximize (α_1-β_1 q_1-γq_2 〖)q〗_1, taking as given q_2, and in Bertrand case, the firm 1 chooses p_1 to maximize p_1 (α_1-b_1 p_1+cp_2), taking as given p_2. Both expressions above are perfectly dual. To sum up, both competition will have similar strategic properties and that we shall be able to derive the Bertrand reaction functions, equilibrium strategies, and profits from the Cournot ones (Singh, 1984).
His Nash equilibrium theory applied to the study of oligopoly in 19th century which are Cournot (1838) and Bertrand (1883). In that century, economists worked to build a deeper theory of the foundations of supply and demand in the market, based on models of rational competitive decision-making by commodity market players. So, as economic theorists learned how to think systematically about rational competitive decision-making, it went natural to apply such rational-choice analysis to social problems that period of time. Consequently, the search for such a general concept was undertaken by the early game theorists, giving that Nash’s theory of non-cooperative games was the critical breakthrough in this process of extending the scope of rational-choice analysis to general competitive situations (Myerson, 1999).
With the understanding, Cournot adopted a Nash equilibrium in a duopoly market when firms produce homogeneous goods and compete in quantity of output. On the other hand, Bertrand adopted a Nash equilibrium where firms compete in price of the homogeneous goods. This means that output competition predicts in an equilibrium price below the monopoly price but above marginal cost, yet for the price competition results in the competitive solution.
Kreps and Scheinkman (1983) argued that whether firms compete in output or price is definitely an empirical question. In the real life, Cournot and Bertrand behavior are observed. Nagurney (2013) extended the notion of Nash equilibrium, he built on the recent work on game theory ideas for a service-oriented Internet with the goal of extending the generality of applying game theory models that are also computable. The service associated with news, videos, music, which is cloud computing. The work of Nagurney inspired by the work of Zhang et al. (2010) who employed Cournot and Bertrand competitions to model games among service and network providers, with the former competing in a Cournot manner, and the latter in a Bertrand manner. In his framework, he does not restrict the number of service providers, nor the number of network providers, nor users. He employed the methodology that utilizing the integrated network economic model of the service-oriented Internet under Cournot-Nash-Bertrand equilibrium is variational inequality theory. In Nagurney’s paper, he proposed an algorithm, which yields a time-discretization of the continuous-time adjustment processes in service volumes, quality levels, and prices until an approximation of the stationary point is achieved.
Similar to Nagurney, Ledvina (2011) computed the Bertrand and Cournot competition under asymmetric costs in number of active firms in equilibrium. In her paper, he is focusing on both the Cournot and Bertrand static games, which allows for a clean interpretation of the results and his analysis is complementary to the existing literature on entry or exit decision of firms. Within each of these models, we have two cases: when the goods are homogeneous and differentianted. The methodology showed that, within a fized type of market, Cournot or Bertrand, differentiated goods result in more active firms in equilibrium than homogeneous goods, the results depended on cost asymmetries between the firms, as with symmetric costs from all active firms and the inactive ones. The results of Ledvina’s work, partially explains the relative differences to consumers between these two types of markets, as consumers have more choice in Cournot markets but with higher prices, whereas they have less choice in Bertrand markets but they are compensated for this fact reduced prices. Consequently, she found that the degree of product differentiation is critical for the determination of the natural market size. The assumption of asymmetric costs was critical as he has found that the results for equal costs across firms is very different than with asymmetric costs as equal costs lead to the maximum of active firms in a market, while asymmetric costs result to the possibility that some firms are inactive in equilibrium.
The duopoly game model of Cournot-Bertrand Nash equilibrium has also been examined by Wang (2013) with the addition of limited information. The gaming system can be described by non-linear difference equations, which modifies the results of Naimzada (2012), which considered the firms with static expectations by linear difference equations. The paper proposed a Cournot-Bertrand mixed game model, with the firms do not have extensive information of the market and opponent, and they make their decisions according to their own marginal profit. Giving that the boundary equilibrium is unstable and the presence of the Nash equilibrium are analyzed.