Quantum repeated games with continuous-variable strategies

Physics Letters A 383 (2019) 2874–2877

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Physics Letters A www.elsevier.com/locate/pla

Quantum repeated games with continuous-variable strategies ✩ Zhe Yang a,b,∗ , Xian Zhang a a b

School of Economics, Shanghai University of Finance and Economics, Shanghai, 200433, China Key Laboratory of Mathematical Economics (SUFE), Ministry of Education, Shanghai, 200433, China

a r t i c l e

i n f o

Article history: Received 6 March 2019 Received in revised form 25 May 2019 Accepted 22 June 2019 Available online 2 July 2019 Communicated by M.G.A. Paris

a b s t r a c t By the Li-Du-Massar method, we investigate the quantization of a two-stage repeated duopoly game. By solving the quantum repeated game with continuous-variable strategies, we analyze the effect of two stages’ entanglement levels, and give some numerical evidences to describe our model. © 2019 Elsevier B.V. All rights reserved.

Keywords: Quantum repeated games Entanglement level Continuous-variable strategies

1. Introduction By combining quantum theory and game theory, the quantum game was first introduced by Meyer [1]. As an interdisciplinary field, the quantum game has been studied by many scholars. Eisert et al. [2] gave the quantum prisoners’ dilemma game, and Benjamin and Hayden [3] considered the multiplayer quantum game. Later, Li et al. [4] gave the quantization scheme of continuousvariable games. In [4], Li et al. analyzed the quantum duopoly, which was extended to quantum games with asymmetric information by Du et al. [5]. Inspired by the work of [4], the quantum duopoly was also extended to quantum Stackelberg duopoly [6–8], quantum Bertrand duopoly [9], multiplayer quantum games with continuous-variable strategies [10] and so on. For more contents, we can refer to Frackiewicz’s remarks on quantum duopoly schemes [11]. On the other hand, Iqbal and Toor [12] studied quantum repeated games. They [12] investigated the quantization of two-stage game of prisoners’ dilemma. Furthermore, Frackiewicz [13] revised the model of [12], and used the method of Marinatto and Weber [14] to present a new scheme of quantum repeated 2 games. Some comments on relations between [12] and [13] were given by Frackiewicz.

✩

This research is supported by National Natural Science Foundation of China (No. 11501349) and Graduate Innovation sponsored by Shanghai University of Finance and Economics (No. CXJJ-2019-348). Corresponding author at: School of Economics, Shanghai University of Finance and Economics, Shanghai, 200433, China. E-mail address: [email protected] (Z. Yang).

*

https://doi.org/10.1016/j.physleta.2019.06.030 0375-9601/© 2019 Elsevier B.V. All rights reserved.

It is worth nothing that the current studies on quantum repeated games focus on games with finitely many pure strategies. Inspired by the work of [4] on continuous-variable quantum games, we shall investigate the quantization of repeated games with continuous-variable strategies. In Section 2, we first give a model of two-stage repeated duopoly. In Section 3, we use the LiDu-Massar technique to give the quantum repeated duopoly. By solving the quantum repeated game and using some numerical evidences, we analyze the effect of entanglement levels of different stages. 2. Two-stage repeated duopoly We assume that there exist two firms who monopolize a market of a commodity and play a two-stage repeated game. In the first stage, firms 1 and 2 simultaneously decide the quantities q1 and q2 . The commodity price of the first stage is given by p = a − q1 − q2 . In the second stage, two firms still simultaneously decide the quantities q1 and q2 . The price of the second stage is derived by p  = (1 − s) p + s(a − q1 − q2 ), where s ∈ [0, 1] is the weight. That is, the price is sticky. Assume that the cost function of each firm is C (q) = cq, where a > c > 0. Then, the profits of two-stage repeated duopoly can be expressed as

u 1 (q1 , q2 , q1 , q2 ) = (a − c − q1 − q2 )q1 + [(1 − s)(a − q1 − q2 )

+s(a − q1 − q2 ) − c ]q1 = (a − c − q1 − q2 )q1 + [a − c − (1 − s)q1 −(1 − s)q2 − sq1 − sq2 ]q1 ,

Z. Yang, X. Zhang / Physics Letters A 383 (2019) 2874–2877

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u 2 (q1 , q2 , q1 , q2 ) = (a − c − q1 − q2 )q2 + [a − c − (1 − s)q1

−(1 − s)q2 − sq1 − sq2 ]q2 . Let k = a − c. By ∂∂ uq 1 = 0, ∂∂ uq1 = 0, ∂∂ uq 2 = 0, ∂∂ uq2 = 0, we have 1 2 1 2

k − 2q1 − q2 − (1 − s)q1 = 0 k − (1 − s)q1 − (1 − s)q2 − 2sq1 − sq2 = 0 k − q1 − 2q2 − (1 − s)q2 = 0 k − (1 − s)q1 − (1 − s)q2 − sq1 − 2sq2 = 0. Then we can find the unique Nash equilibrium of the repeated game

q∗1 = q∗2 =

4s − 1 9s − 2(1 − s)2

q1∗ = q2∗ =

k,

2s + 1 9s − 2(1 − s)2

k.

At the equilibrium, the profits are

u 1 (q∗1 , q∗2 , q1∗ , q2∗ ) = u 2 (q∗1 , q∗2 , q1∗ , q2∗ )

=

s(5 − 2s)(4s − 1) + s(2s + 1)2 9s − 2(1 − s)2

∗

Fig. 1. The profit u Q at the quantum equilibrium (x∗1 , x∗2 , y ∗1 , y ∗2 ) with respect to when k = 100, s = 0.5, γ2 = 5.

2

k .

We next give the following statements to explain above model. 1. To ensure the existence of Nash equilibrium, we assume that

s =

q2 = ψ 2f | X 2 | ψ 2f  = y 1 sinh γ2 + y 2 cosh γ2 .

√ 4

q1 = ψ 1f | X 1 | ψ 1f  = x1 cosh γ1 + x2 sinh γ1 , q1 = ψ 2f | X 1 | ψ 2f  = y 1 cosh γ2 + y 2 sinh γ2 ,

that is,

153

Then, the final measurements of stages 1 and 2 are given by

q2 = ψ 1f | X 2 | ψ 1f  = x1 sinh γ1 + x2 cosh γ1 ,

9s − 2(1 − s)2 = 0,

13 −

γ1

.

Moreover, the quantum profits are

2. In our model, it is possible to obtain that q∗1 , q∗2 , q1∗ , q2∗ < 0. We can think that the firm sell the commodity when the output q  0, and buy the commodity from the market when the output q < 0. Our model is a decision problem with two stages. To achieve the maximization of own profit, the firm may buy (sell) the commodity at the first stage and sell (buy) the commodity at the second stage according to the market situation of every stage. Thus, we obtain some negative outputs in our model.

u 1 (x1 , x2 , y 1 , y 2 ) = u 1 (q1 , q2 , q1 , q2 ) Q

= (k − e γ1 x1 − e γ1 x2 )(x1 cosh γ1 + x2 sinh γ1 ) +[k − (1 − s)e γ1 x1 − (1 − s)e γ1 x2 − se γ2 y 1 −se γ2 y 2 ]( y 1 cosh γ2 + y 2 sinh γ2 ), u 2 (x1 , x2 , y 1 , y 2 ) = u 2 (q1 , q2 , q1 , q2 ) Q

= (k − e γ1 x1 − e γ1 x2 )(x1 sinh γ1 + x2 cosh γ1 ) +[k − (1 − s)e γ1 x1 − (1 − s)e γ1 x2 − se γ2 y 1

3. Quantum form of the repeated duopoly In this section, we use the Li-Du-Massar method [4] to give the quantum form of above repeated duopoly. † Let a j (a j ) be the creation (annihilation) operator of firm j. De†

√

fine the position operator of firm j by X j = (a j + a j )/ 2, the

√ † momentum operator of firm j by P j = i (a j − a j )/ 2, and the uni† †

tary operator by J (r ) = exp{−γ (a1 a2 − a1 a2 )}. We assume that γi is the entanglement level of ith stage. For every stage, let |01 and |02 be the initial state, and the set of strategies for firm j be defined by

S j = { D j (x j ) := exp(−ix j P j ) | x j  0}. The final state of stages 1 and 2 are determined by

|ψ 1f 

†

= J (γ1 ) D 1 (x1 ) ⊗ D 2 (x2 ) J (γ1 )|01 |02 ,

|ψ 2f 

= J (γ2 )† D 1 ( y 1 ) ⊗ D 2 ( y 2 ) J (γ2 )|01 |02 .

−se γ2 y 2 ]( y 1 sinh γ2 + y 2 cosh γ2 ). ∂uQ

∂uQ

∂uQ

∂uQ

By ∂ x1 = 0, ∂ y1 = 0, ∂ x2 = 0, ∂ y2 = 0, we have 1 1 2 2

−e γ1 (x1 cosh γ1 + x2 sinh γ1 ) + cosh γ1 (k − e γ1 x1 − e γ1 x2 ) −(1 − s)e γ1 ( y 1 cosh γ2 + y 2 sinh γ2 ) = 0, −se γ2 ( y 1 cosh γ2 + y 2 sinh γ2 ) + cosh γ2 [k − (1 − s)e γ1 x1 −(1 − s)e γ1 x2 − se γ2 y 1 − se γ2 y 2 ] = 0, −e γ1 (x1 sinh γ1 + x2 cosh γ2 ) + cosh γ1 (k − e γ1 x1 − e γ1 x2 ) −(1 − s)e γ1 ( y 1 sinh γ2 + y 2 cosh γ2 ) = 0, −se γ2 ( y 1 sinh γ2 + y 2 cosh γ2 ) + cosh γ2 [k − (1 − s)e γ1 x1 −(1 − s)e γ1 x2 − se γ2 y 1 − se γ2 y 2 ] = 0. Obviously, we obtain x1 = x2 , y 1 = y 2 . Let x1 = x2 = x and y 1 = y 2 = y. Then, we get the equations:

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Z. Yang, X. Zhang / Physics Letters A 383 (2019) 2874–2877

∗

Fig. 2. The profit u Q at the quantum equilibrium (x∗1 , x∗2 , y ∗1 , y ∗2 ) with respect to when k = 100, s = 0.5, γ1 = 5.

γ2

∗

Fig. 4. The profit u Q at the quantum equilibrium (x∗1 , x∗2 , y ∗1 , y ∗2 ) with respect to γ = γ1 = γ2 when k = 100, s = 0.5.

Fig. 5. The nonexistence condition region  of quantum equilibrium.

∗

Fig. 3. The profit u Q at the quantum equilibrium (x∗1 , x∗2 , y ∗1 , y ∗2 ) with respect to and γ2 when k = 100, s = 0.5.

γ1

(1 + 2e γ1 )x + (1 − s)e γ1 +γ2 y = k cosh γ1 (1 − s)(e γ1 +γ2 + e γ1 −γ2 )x + s(1 + 2e 2γ2 ) y = k cosh γ2 . Thus, we solve the quantum equilibrium

x∗ = x∗1 = x∗2

=

s(1 + 2e 2γ2 ) cosh γ1 − (1 − s)e γ1 +γ2 cosh γ2 s(1 + 2e 2γ1 )(1 + 2e 2γ2 ) − (1 − s)2 (e 2(γ1 +γ2 ) + e 2γ1 )

k,

y ∗ = y ∗1 = y ∗2

=

(1 + 2e 2γ1 ) cosh γ2 − (1 − s)(e γ1 +γ2 + e γ1 −γ2 ) cosh γ1 k. s(1 + 2e 2γ1 )(1 + 2e 2γ2 ) − (1 − s)2 (e 2(γ1 +γ2 ) + e 2γ1 )

To ensure the existence of quantum equilibrium, we assume that

s(1 + 2e 2γ1 )(1 + 2e 2γ2 ) − (1 − s)2 (e 2(γ1 +γ2 ) + e 2γ1 ) = 0.

It is obvious that x∗1 = x∗2 = q∗1 = q∗2 and y ∗1 = y ∗2 = q1∗ =

∗ q2∗ when γ1 = γ2 = 0, and we denote u Q Q Q ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ u 1 (x1 , x2 , y 1 , y 2 ) = u 2 (x1 , x2 , y 1 , y 2 ).

∗

∗

= u 1Q = u 2Q =

We next use numerical evidences to show whether two firms cooperate at every stage when two entanglement levels γ1 and γ2 converge to infinite. ∗ The Fig. 1 shows the profit u Q at the quantum equilibrium (x∗1 , x∗2 , y ∗1 , y ∗2 ) as a function of the entanglement level γ1 of the first stage when k = 100, s = 0.5, γ2 = 5. ∗ The Fig. 2 shows the profit u Q at the quantum equilibrium ∗ ∗ ∗ ∗ (x1 , x2 , y 1 , y 2 ) as a function of the entanglement level γ2 of the second stage when k = 100, s = 0.5, γ1 = 5. ∗ The Fig. 3 shows the profit u Q at the quantum equilibrium ∗ ∗ ∗ ∗ (x1 , x2 , y 1 , y 2 ) as a function of the entanglement levels γ1 and γ2 of two stages when k = 100, s = 0.5. ∗ The Fig. 4 shows the profit u Q at the quantum equilibrium ∗ ∗ ∗ ∗ (x1 , x2 , y 1 , y 2 ) as a function of the entanglement level γ = γ1 = γ2 when k = 100, s = 0.5. From Figs. 1–4, we deduce that the quantum profits increase and achieve the maximum as the entanglements converge to infinite.

Z. Yang, X. Zhang / Physics Letters A 383 (2019) 2874–2877

Finally, Fig. 5 shows the nonexistence condition region  of quantum equilibrium, that is,

 = {(γ1 , γ2 , s) | γ1 , γ2  0, s ∈ [0, 1], s(1 + 2e 2γ1 )(1 + 2e 2γ2 ) − (1 − s)2 (e 2(γ1 +γ2 ) + e 2γ1 ) = 0}. References [1] D.A. Meyer, Quantum strategies, Phys. Rev. Lett. 82 (1999) 1052. [2] J. Eisert, M. Wilkens, M. Lewenstein, Quantum games and quantum strategies, Phys. Rev. Lett. 83 (1999) 3077. [3] S.C. Benjamin, P.M. Hayden, Multiplayer quantum games, Phys. Rev. A 64 (2001) 030301. [4] H. Li, J. Du, S. Massar, Continuous-variable quantum games, Phys. Lett. A 306 (2002) 73–78. [5] J. Du, H. Li, C. Ju, Quantum games of asymmetric information, Phys. Rev. E 68 (2003) 016124.

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