Quantum Model of Bertrand Duopoly

In economics, oligopoly refers to a market condition in which sellers are so few that action of each seller has a measurable impact on the price and other market factors [1]. If the number of firms competing on a commodity in the market is just two, the oligopoly is termed as duopoly. The competitive behavior of firms in oligopoly makes it suitable to be analyzed by using the techniques of game theory. Cournot and Bertrand models are the two oldest and famous oligopoly models [2,3]. In Cournot model of oligopoly firms put certain amount of homogeneous product simultaneously in the market and each firm tries to maximize its payoff by assuming that the opponent firms will keep their outputs constant. Later on Stackelberg introduced a modified form of Cournot oligopoly in which the oligopolistic firms supply their products in the market one after the other instead of their simultaneous moves. In Stackelberg duopoly the firm that moves first is called leader and the other firm is the follower [4]. In Bertrand model the oligopolistic firms compete on price of the commodity, that is, each firm tries to maximize its payoff by assuming that the opponent firms will not change the prices of their products. The output and price are related by the demand curve so the firms choose one of them to compete on leaving the other free. For a homogeneous product, if firms choose to compete on price rather than output, the firms reach a state of Nash equilibrium at which they charge a price equal to marginal cost. This result is usually termed as Bertrand paradox, because practically it takes many firms to ensure prices equal to marginal cost. One way to avoid this situation is to allow the firms to sell differentiated products [1].

For the last one decade quantum game theorists are attempting to study classical games in the domain of quantum mechanics [5][6][7][8][9][10][11][12][13][14]. Various quantum protocols have been introduced in this regard and interesting results have been obtained [15][16][17][18][19][20][21][22][23][24][25]. The first quantization scheme was presented by Meyer [15] in which he quantized a simple penny flip game and showed that a quantum player can always win against a classical player by utilizing quantum superposition.

In this letter, we extend the classical Bertrand duopoly with differentiated products to quantum domain by using the quantization scheme proposed by Marinatto and Weber [17]. Our results show that the classical game becomes a subgame of the quantum version. We found that entanglement in the initial state of the game makes the players better off. Before presenting the calculation of quantization scheme, we first review the classical model of the game.

Consider two firms A and B producing their products at a constant marginal cost c such that c < a, where a is a constant. Let p 1 and p 2 be the prices chosen by each firm for their products, respectively. The quantities q A and q B that each firm sells is given by the following key assumption of the classical Bertrand duopoly model

where the parameter 0 < b < 1 shows the amount of one firm’s product substituted for the other firm’s product. It can be seen from Eq. (1) that more quantity of the product is sold by the firm which has low price compare to the price chosen by his opponent. The profit function of the two firms are given by

In Bertrand duopoly the firms are allowed to change the quantity of their product to be put in the market and compete only in price. A firm changes the price of its product by assuming that the opponent will keep its price constant. Suppose that firm B has chosen p 2 as the price of his product, the optimal response of firm A to p 2 is obtained by maximizing its profit function with respect to its own product’s price, that is, ∂u A /∂p 1 = 0, this leads to

Firm B response to a fixed price p 1 of firm A is obtained in a similar way and is given by

Solution of Eqs.(3 and 4) lead to the following optimal price level that defines the Nash equilibrium of the game

The profit functions of the firms at the Nash equilibrium become

From Eq. ( 6), we see that both firms can be made better off if they choose higher prices, that is, the Nash equilibrium is Pareto inefficient.

To quantize the game, we consider that the game space of each firm is a two dimensional Hilbert space of basis vector |0 and |1 , that is, the game consists of two qubits, one for each firm. The composite Hilbert space H of the game is a four dimensional space which is formed as a tensor product of the individual Hilbert spaces of the firms, that is, H = H A ⊗H B , where H A and H B are the Hilbert spaces of firms A and B, respectively. To manipulate their respective qubits each firm can have only two strategies I and C. Where I is the identity operator and and C is the inversion operator also called Pauli spin flip operator. If x and 1 -x stand for the probabilities of I and C that firm A applies and y, 1 -y, are the probabilities that f

…(Full text truncated)…

This content is AI-processed based on ArXiv data.