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Quantum Model of Bertrand Duopoly
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CHIN. PHYS. LETT. Vol. 27, No. 8 (2010) 080302
Quantum Model of Bertrand Duopoly
*
Salman Khan** , M. Ramzan, M. K. Khan Department of Physics Quaid-i-Azam University, Islamabad 45320, Pakistan
(Received 26 January 2010) We present a quantum model of Bertrand duopoly and study the entanglement behavior on the profit functions of the firms. Using the concept of optimal response of each firm to the price of the opponent, we find only one Nash equilibirum point for the maximally entangled initial state. The presence of quantum entanglement in the initial state gives payoffs higher to the firms than the classical payoffs at the Nash equilibrium. As a result, the dilemma-like situation in the classical game is resolved.
PACS: 03. 65. Ta, 03. 65. βw, 03. 67. Lx
DOI: 10.1088/0256-307X/27/8/080302
In economics, oligopoly refers to a market condition in which sellers are so few that the 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. The Cournot and Bertrand models are the two oldest and famous oligopoly models.[2,3] In the Cournot model of oligopoly, firms put a 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, 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 the leader and the other firm is the follower.[4] In the 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 the Nash equilibrium at which they charge a price equal to marginal cost. This result is usually termed as the 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] In the past decade, quantum game theorists have been attempting to study classical games in the domain of quantum mechanics.[5β14] Various quantum protocols have been introduced in this regard and interesting results have been obtained.[15β25] 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 find 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 π΄ and π΅ producing their products at a constant marginal cost π such that π < π, where π is a constant. Let π1 and π2 be the prices chosen by firms for their products, respectively. The quantities ππ΄ and ππ΅ that each firm sells is given by the following key assumption of the classical Bertrand duopoly model ππ΄ = π β π1 + ππ2 ,
ππ΅ = π β π2 + ππ1 ,
(1)
where the parameter 0 < π < 1 shows the amount of one firmβs product substituted for the other firmβs product. It can be seen from Eq. (1) that more quantity of the product is sold by the firm which has a low price compared to the price chosen by his opponent. The profit function of the two firms are given by π’π΄ (π1 , π2 , π) = ππ΄ (π1 β π) = (π β π1 + ππ2 ) (π1 β π) , π’π΅ (π1 , π2 , π) = ππ΅ (π2 β π) = (π β π2 + ππ1 ) (π2 β π).
(2)
In the 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 π΅ has
* Supported
by the National Scholarship Program of Pakistan.
[email protected] c 2010 Chinese Physical Society and IOP Publishing Ltd β ** Email:
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CHIN. PHYS. LETT. Vol. 27, No. 8 (2010) 080302
chosen π2 as the price of his product, the optimal response of firm π΄ to π2 is obtained by maximizing its profit function with respect to its own productβs price, that is, ππ’π΄ /ππ1 = 0, this leads to π1 =
1 (ππ2 + π + π) . 2
(3)
Firm π΅ response to a fixed price π1 of firm π΄ is obtained in a similar way and is given by 1 π2 = (ππ1 + π + π) . 2
firm π΅. The moves (prices) of the firms and the probabilities π₯ and π¦ of using the operators can be related by 1 1 , π¦= , (9) π₯= 1 + π1 1 + π2 where the prices π1 and π2 β [0, β) and the probabilities π₯, π¦ β [0, 1]. By using Eqs. (7)β(9), the nonzero elements of the final density matrix are obtained,
(4)
π14
Solution to Eqs. (3) and (4) leads to the following optimal price level that defines the Nash equilibrium of the game, π+π π*1 = π*2 = . (5) 2βπ
ππ = π₯π¦πΌπ΄ β
β πΌπ΅ ππ πΌπ΄
β
π23
π1 cos2 πΎ + π2 sin2 πΎ , (1 + π1 ) (1 + π2 )
π44 =
π1 π2 cos2 πΎ + sin2 πΎ . (1 + π1 ) (1 + π2 )
(10)
π’π΄ (π1 , π2 , π) = Trace (ππ΄oper ππ ) , π’π΅ (π1 , π2 , π) = Trace (ππ΅oper ππ ) ,
(11)
where ππ΄oper and ππ΅oper are payoffs operators of the firms, defined as ππ΄ (ππ΅ π11 β π22 + π33 ) , π12 ππ΄ = (ππ΄ π11 + π22 β π33 ) , π12
ππ΄oper = ππ΅oper
(12)
1 . where ππ΄ = π1 βπ, ππ΅ = π2 βπ and π12 = (1+π1 )(1+π 2) By using Eqs. (10)β(12), the payoffs of the firms are obtained,
π’π΄ (π1 , π2 , π) = (π β π1 + ππ2 )[ππ΄ cos2 πΎ + {π2 + π1 (β1 β ππ2 + π22 )} sin2 πΎ], π’π΅ (π1 , π2 , π) = (π β π2 + ππ1 )[ππ΅ cos2 πΎ + {π1
β β + π₯(1 β π¦)πΌπ΄ β πΆπ΅ ππ πΌπ΄ β πΆπ΅
β π2 (1 + ππ1 β π21 )} sin2 πΎ].
β β + π¦(1 β π₯)πΆπ΄ β πΌπ΅ ππ πΆπ΄ β πΌπ΅
(7)
In Eq. (7) ππ = |ππ β©β¨ππ | is the initial density matrix with initial state |Ξ¨π β©, which is given by |ππ β© = cos πΎ|00β© + sin πΎ|11β©,
π33 =
The payoffs of the firms can be found by the following trace operations
β πΌπ΅
β β + (1 β π₯)(1 β π¦)πΆπ΄ β πΆπ΅ ππ πΆπ΄ β πΆπ΅ .
π2 cos2 πΎ + π1 sin2 πΎ , (1 + π1 ) (1 + π2 ) (π1 + π2 ) cos πΎ sin πΎ , = π32 = (1 + π1 ) (1 + π2 )
π22 =
The profit functions of the firms at the Nash equilibrium become ]οΈ2 [οΈ π+π * * βπ . π’π΄ = π’π΅ = (6) 2βπ From Eq. (6), we can 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 |0β© and |1β©, that is, the game consists of two qubits, one for each firm. The composite Hilbert space β 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, β = βπ βββ¬ , where βπ and ββ¬ are the Hilbert spaces of firms π΄ and π΅, respectively. To manipulate their respective qubits, each firm can have only two strategies πΌ and πΆ with πΌ being the identity operator and and πΆ the inversion operator also called the Pauli spin flip operator. If π₯ and 1 β π₯ stand for the probabilities of πΌ and πΆ that firm π΄ applies, π¦ and 1 β π¦ are the probabilities that firm π΅ applies, then the final state ππ of the game is given by[17]
(cos2 πΎ + π1 π2 sin2 πΎ) , (1 + π1 ) (1 + π2 ) (1 + π1 π2 ) cos πΎ sin πΎ = π41 = , (1 + π1 ) (1 + π2 )
π11 =
(8)
where πΎ β [0, π] represents the degree of entanglement of the initial state. In Eq. (8) the first qubit corresponds to firm π΄ and the second qubit corresponds to
(13)
One can easily see from Eq. (13) that the classical payoffs can be reproduced by setting πΎ = 0 in Eq. (13). We proceed, similarly to the classical Bertrand duopoly, to find the response of each firm to the price chosen by the opponent firm. For firm π΅βs price π2 , the optimal response of firm π΄ is obtained by maximizing its own payoff (Eq. (13)) with respect to π1 . Similarly, the reaction function of firm π΅ to a known π1 is obtained. These reaction functions can be writ-
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CHIN. PHYS. LETT. Vol. 27, No. 8 (2010) 080302
ππ΅ [β1 + π2 (π + ππ2 )] π1 = (2 β π2 ππ΅ ) cos 2πΎ β 2π2 ππ΅ [π + π2 + 2ππ2 β ππ22 ππ΅ + π{2 β π2 ππ΅ }] cos 2πΎ + , (2 β π2 ππ΅ ) cos 2πΎ β 2π2 ππ΅ ππ΄ [β1 + π1 (π + ππ1 )] π2 = (2 β π1 ππ΄ ) cos 2πΎ + 2π1 ππ΄ [π + π1 + 2ππ1 + ππ21 ππ΄ + π{2 + π1 ππ΄ }] cos 2πΎ + . (2 β π1 ππ΄ ) cos 2πΎ + 2π1 ππ΄ (14) The results of Eq. (14) reduce to the classical results given in Eqs. (3) and (4) for the initially unentangled state, which leads to the classical Nash equilibrium. This shows that the classical game is a subgame of the quantum game. Now, we discuss the behavior of entanglement in the initial state on the game dynamics. It can be seen from Eq. (14) that the optimal responses of the firms to a fixed price of the opponent firm, for a maximally entangled state, are given by ππ22 + ππ2 β 1 , 2π2 ππ2 + ππ1 β 1 π2 = 1 . 2π1
π1 =
(15)
Solving these equations, we can obtain the optimal price levels and the corresponding payoffs of each firm. In this case the following four points are obtained, βοΈ π + π2 + 4π½ * * , π1 (1) = π2 (1) = β2π½ 2 βοΈ π*1 (2) = π*2 (2) = , π + π2 + 4π½ 2π (οΈβ )οΈ , π*1 (3, 4) = β π 2+π 2+πΒ±πΎ [οΈ ]οΈ 1 πΎ * π2 (3, 4) = β πΒ± β , (16) 2π 2+π where the numbers in the parentheses correspond to the respective points (the symbols Β± correspond to points 3 and 4 respectively). To verify which point (points) defines the Nash equilibrium of the game, we use the second partial derivative condition. That is, for the Nash equilibrium, the strategy (point) must be the global maximum of the payoff function, that is, π 2 π’π΄(π΅) /ππ21(2) < 0 and the payoff function at the point must be higher than the payoff function at the boundary points. It can easily be verified that this condition is satisfied only at point 1. Hence point 1 defines the Nash equilibrium of the game. The payoffs of the firms at the Nash equilibrium become 1 4 π’π΄ (1) = π’π΅ (1) = [π + 2πΌ2 + 2π2 ππ½ + π3 ππ½ 4π½ 4 βοΈ β π{(π½ β 2)π½ β 3}ππ½ 2 + π2 + 4π½(π3 + 2ππΌ + ππΌ2 + π2 ππ½)].
(17)
The new parameters introduced in Eqs. (16) and (17) are defined as π½ = π β 2, πΌ = 2 β 3π + π2 . The payoffs of the firms at the Nash equilibrium must be real and positive for the entire range of substitution parameter π. This condition for marginal cost π < 1.4 is satisfied when π β₯ 3.5. The firmsβ payoffs at the other three points become π’π΄ (2) = π’π΅ (2) = β
4 βοΈ
π2
4π½)4
[οΈ 5 π π
(π + + βοΈ {οΈ }οΈ + π(β1 + π) (β9 + 5π)π β 2 π2 + 4π½ βοΈ }οΈ {οΈ β π3 (8 β 5π)π + π2 + 4π½ βοΈ + (β1 + π)2 (β2 + π π2 + 4π½) βοΈ (οΈ )οΈ + π4 β 1 + π π2 + 4π½ βοΈ {οΈ + π2 6 β 4π π2 + 4π½ βοΈ (οΈ )οΈ}οΈ]οΈ + π β 4 + 3π π2 + 4π½ , [οΈ ]οΈ 2 2 3/2 (1 + π) π (2 + π) β π’π΄ (3, 4) = (2 + π)3/2 (π(2 + π) Β± 2 + πΞ)2 β [οΈ ]οΈ (1 + π)2 π(2 + π)(π 2 + ππ Β± Ξ) β + (2 + π)3/2 (π(2 + π) Β± 2 + πΞ)2 β [οΈ ]οΈ (1 + π)2 π(2π 2 + π Β± πΞ(2 + π)) β + , (2 + π)3/2 (π(2 + π) Β± 2 + πΞ)2 β (1 + π)2 2 + π π’π΅ (3, 4) = β [2ππ + πππ 4π(2 + π)5/2 β β 2π Β± 2 + πΞπ], (18) where Ξ =
βοΈ
4π2 + π2 (2 + π).
40
30
Payoffs
ten as
20
10
0 0.0
0.2
0.4
0.6
0.8
Fig. 1. The payoffs of the firms at the classical and quantum Nash equilibria against the substitution parameter π. The values of the parameter π and the marginal cost π are chosen to be 3.5 and 0.1, respectively. The superscripts πΆ and π of π’ represent the classical and quantum cases, respectively. The subscripts π΄ stands for firm π΄.
We present a quantization scheme for the Bertrand duopoly with differentiated products. To analyze the effect of quantum entanglement on the game dynamics, we plot the payoffs of the firms at the classical and quantum Nash equilibria against the substitution
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CHIN. PHYS. LETT. Vol. 27, No. 8 (2010) 080302
parameter π in Fig. 1. The values of parameters π and π are chosen to be 3.5 and 0.1, respectively. The solid line (π’π π΄ (1)) represents quantum mechanical payoffs and the dotted line (π’πΆ π΄ ) represents the classical payoffs of the firms. It is clear from the figure that quantum payoffs of the firms are higher than the classical payoffs for the entire range of substitution parameter π. The maximum entanglement in the initial state of the game makes the firms better off. In Fig. 2, we plot the payoffs of the firms (Eq. (18) against the substitution parameter π at the other three points which are not the Nash equilibria.
0.5
Payoffs
0.0 -0.5 -1.0 -1.5 -2.0 0.0
0.2
0.4
0.6
0.8
Fig. 2. The payoffs of the firms at the second and third points as a function of the substitution parameter π. The values of the parameter π and the marginal cost π are chosen to be 3.5 and 0.1, respectively. The superscripts πΆ and π of π’ represent the classical and quantum cases, respectively. The subscripts π΄ and π΅ correspond to firms π΄ and π΅, respectively. The numbers in the parentheses represent the corresponding Nash equilibrium points.
In conclusion, we have used the Marinatto and Weber quantization scheme to find the quantum version of Bertrand duopoly with differentiated products. We have studied the entanglement behavior on the payoffs of the firms for a maximally entangled initial state,
and found that for large values of substitution parameter π, both firms can achieve significantly higher payoffs as compared to the classical payoffs. Furthermore, for maximally entangled state the quantum payoffs are higher than the classical payoffs for the entire range of substitution parameter and is the best situation for both firms. Thus, the dilemma-like situation in the classical Bertrand duopoly game is resolved.
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