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Physics Letters A 339 (2005) 10–17

www.elsevier.com/locate/pla

Multiplayer quantum games with continuous-variable strateg

Jie Zhou∗, Lei Ma, Yun Li

Department of Physics, East China Normal University, Shanghai 200062, PR China

Received 15 December 2004; received in revised form 1 March 2005; accepted 4 March 2005

Available online 14 March 2005

Communicated by P.R. Holland

Abstract

Based on the extended case of the Cournot’s duopoly, we investigate a case of three firms in which the different entaparameters can vary arbitrarily. An analytical formula is presented and some interesting features are demonstrateFurthermore, we find that the quantum entanglement can make an arbitrary number of players cooperate to the large 2005 Elsevier B.V. All rights reserved.

Keywords: Quantum game; Entanglement; Multiplayer; Continuous variable

1. Introduction

The theory of quantum information has become an increasingly important focus of study. Many other fielas calculations, code communication and communicate capability, etc., have combined with it and resulseries of new findings being discovered. Many interesting and surprising properties have been uncoveredconcept of quantum information was applied to classical game theory. First, Meyer showed the power of qstrategies by which the player can always beat his classical opponent[1]. Then, Eisert et al. showed how the powof entanglement can be used to eliminate the dilemma which exists in classical game[2]. After that, noisy quantumgames[3,4], multiplayer quantum games[5,6], continuous-variable quantum games[7,8] and other interestingaspects[9–15]were studied.

The properties of two player quantum games have been discussed extensively, but the case of multiplatum games has received less attention. Multiplayer games contains many properties which do not exisplayer games. In practice, many situations should be represented by multiplayer game. So, from both a thand a practice standpoint, the study of multiplayer games is important. In this Letter, we investigate a muquantum game which is extended from the case of Cournot’s duopoly[16].

* Corresponding author.E-mail addresses: gasta20002000@yahoo.com.cn(J. Zhou),lma@phy.ecnu.edu.cn(L. Ma).

0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2005.03.006

J. Zhou et al. / Physics Letters A 339 (2005) 10–17 11

nuantumetion ofe highestto EPRay be

xtendinconve-

ed

ing theo, in the

s

Cournot’s duopoly model of a two player case has been discussed by Du et al.[7,8]. They showed that whethe players’ qubits are entangled in an EPR state the best profits for the two firms results. By utilizing the qstructure where every pair of players are entangling with each other[6], we arrive at an analytic formula of thprofits for the case of three firms. By illustrating the variety of the sum of these players’ profits with the variathe three different parameters which present the extent of entanglement, we find the conditions to attain thof the sum are not unique. Furthermore, when all of the players, presented by qubit, entangled pairwisestate, they will obtain the highest profits while maintaining the symmetry of the game. This conclusion mapplicable to arbitrary number of participators.

2. The extension of the Cournot’s duopoly

The Cournot’s duopoly is a model of two firms monopolizing the market of a certain commodity. We ethis model by adding the number of new firms. Suppose there aren firms monopolizing the market of a certacommodity. Every firm may decide its own quantity of product, and all of their products are the same. Fornience, we denote the quantity of thej th firm asqj and the total quantity asQ, soQ = q1 + · · · + qn. Suppose theprice isP(Q), then we have

(1)P(Q) ={

a − Q for Q � a,

0 for Q > a.

Taking the cost of each product asc with c < a. For convenience, letk = a − c. Then the profits can be representas

(2)uj (q1, . . . , qn) = qj (k − Q),

wherej means thej th firm. So the Nash equilibrium of the game is

(3)q∗1 = · · · = q∗

n = k

n + 1.

The profits at the equilibrium are

(4)u1(q∗

1, . . . , q∗n

) = · · · = un

(q∗

1, . . . , q∗n

) = k2

(n + 1)2.

However, this equilibrium is not the optimal solution. If we restrict the quantities at

(5)q ′1 = · · · = q ′

n = k

2n.

The profits of the firms will be

(6)u1

(1

2n, . . . ,

1

2n

)= · · · = un

(1

2n, . . . ,

1

2n

)= k2

4n.

This is a better solution of the game. Actually, it is the highest profit they can ever attain while maintainsymmetry of the game. However, they will never escape the Nash equilibrium because of selfishness. Sextended Cournot’s duopoly, a dilemma-like situation also exists.

3. Quantum form of the model with three firms

We use three single-mode electromagnetic fields. The quantum structure is shown inFig. 1. The game startfrom the vacuum state|0〉 ⊗ |0〉 ⊗ |0〉 . First, the state passes through a unitary operatorJ which is known to all

1 2 3

12 J. Zhou et al. / Physics Letters A 339 (2005) 10–17

e

in-

y iss

romag-tains twoter

to

Fig. 1. The quantum structure of the model.

firms. Consequently, it is sent to the firms for employing their strategies markedD1, D2, D3. These strategies arlocal unitary operators. Then, before taking measurement, it will be acted on byJ+. The entangling operatorJ isgiven by

(7)J (γ1, γ2, γ3) = exp{−γ1

(a+

1 a+2 − a1a2

) − γ2(a+

2 a+3 − a2a3

) − γ3(a+

3 a+1 − a3a1

)}.

Let the observables of the final measurement beXj = (a+j + aj )/

√2 which is the ‘position’ operators of form

j where a+j (aj ) is the creation (annihilation) operator of firmj ’s field. We assume that the light beam is

finitely squeezed, thusXj could be measured precisely. Supposexj to be the result, soqj = xj represents thequantity of firm j , hence the profit isuj (D1, D2, . . . , Dn) = uj (x1, x2, . . . , xn), Pj = i(a+

j − aj )/√

2 which is

“momentum” operator. WhenJ (γ1, γ2, γ3) = J (γ1, γ2, γ3)+ = I (the identity operators) and the set of strateg

Sj = {Dj (xj ) = exp(−ixj Pj ) | xj ∈ [0,∞]}, it will go back to classical game forqj = xj . So the classical form ithe subset of the quantum structure.

J is a kind of three-mode operator which can be realized via the following method. First, let every electnetic field generate two homogeneous beams, then combine the 6 beams into 3 pairs while every pair condifferent ones. Actually, each pair is a two-mode squeezed vacuum state.γj is known as the squeezing parameof the pair and can be reasonably regarded as a measure of entanglement, here we suppose that all ofγj are non-negative real numbers. Whenγj → ∞, which is the infinite squeezing limit, the relevant pair will approximatethe EPR state[17–19].

Noticing thatPj could be represented byi√2(a+

j − aj ), if we know the expressions ofJ+a+j J andJ+aj J , it

will be subsequently to get the forms ofJ+Pj J . For convenience we puta+j andaj into a row vector markΛ,

(8)ΛT = (a+

1 , a+2 , a+

3 , a1, a2, a3),

and suppose

(9)Λ′ = J (γ1, γ2, γ3)+ΛJ (γ1, γ2, γ3).

So

(10)

P ′1

P ′2

P ′3

= J (γ1, γ2, γ3)

+ P1

P2

P3

J (γ1, γ2, γ3).

It is easy to get the following equation:

(11)J (γ1, γ2, γ3)+ = 1

2ΛT

(Γ 00 −Γ

)Λ,

here

(12)Γ =( 0 γ1 γ3

γ1 0 γ2

).

γ3 γ2 0

J. Zhou et al. / Physics Letters A 339 (2005) 10–17 13

From Ref.[20], we obtained that

(13)

P ′1

P ′2

P ′3

= eΓ

(P1P2P3

).

The way of spectral decompose could be used to represent the expression ofeΓ . BecauseΓ is a normal matrix of3⊗ 3, we assume that

(14)Γ = η1|α1〉〈α1| + η2|α2〉〈α2| + η3|α3〉〈α3|,whereηj is eigenvalue ofΓ and|αj 〉 is eigenvector ofΓ . Therefore,

(15)

η1 = 2√

p cos(

ϕ3

),

η2 = 2√

p cos(

ϕ3 + 2π

3

),

η2 = 2√

p cos(

ϕ3 + 4π

3

),

here

p = 1

3

(γ 2

1 + γ 22 + γ 2

3

), q = −γ1γ2γ3, cosϕ = q

−p√

p,

(16)|αj 〉 = 1√(γ1γ2 + ηjγ3)2 + (γ1γ3 + ηjγ2)2 + (η2

j − γ 21 )2

γ1γ2 + ηjγ3

γ1γ3 + ηjγ2

η2j − γ 2

1

.

Utilizing the orthogonal normalization[21],

(17)eΓ = eη1|α1〉〈α1| + eη2|α2〉〈α2| + eη3|α3〉〈α3|,which can be evolved in

eΓ =3∑

j=1

eηj

(γ1γ2 + ηjγ3)2 + (γ1γ3 + ηjγ2)2 + (η2j − γ 2

1 )2

(18)×

(γ1γ2 + ηjγ3)2 (γ1γ2 + ηjγ3)(γ1γ3 + ηjγ2) (γ1γ2 + ηjγ3)

(η2

j − γ 21

)(γ1γ2 + ηjγ3)(γ1γ3 + ηjγ2) (γ1γ3 + ηjγ2)

2 (γ1γ3 + ηjγ2)(η2

j − γ 21

)(γ1γ2 + ηjγ3)

(η2

j − γ 21

)(γ1γ3 + ηjγ2)

(η2

j − γ 21

) (η2

j − γ 21

)2

.

For convenience, we mark

eΓ =(

u b c

b v d

c d w

),

b, c, d,u, v,w being the respective formations.So,

J (γ )+D1(x1)J (γ ) = exp{−ix1(uP1 + bP2 + cP3)

},

J (γ )+D2(x2)J (γ ) = exp{−ix1(bP1 + vP2 + dP3)

},

(19)J (γ )+D3(x3)J (γ ) = exp{−ix1(cP1 + dP2 + wP3)

}.

14 J. Zhou et al. / Physics Letters A 339 (2005) 10–17

fter

y otheraten

ity

Under

After taking the measurement, the quantities of the firms are

(20)

q1 = ux1 + bx2 + cx3,

q2 = bx1 + vx2 + dx3,

q3 = cx1 + dx2 + wx3,

and

(21)Q = q1 + q2 + q3 = (u + b + c)x1 + (v + b + d)x2 + (w + c + d)x3.

Referring to Eq.(1), the quantum profits of the firms are

(22)uj (x1, x2, x3) = uj (q1, q2, q3) = qj (k − Q).

Under the condition of Nash equilibrium the solution ofQ is

(23)Q = k

(1+ 1

uu+b+c

+ vv+b+d

+ ww+c+d

)−1

.

We assign

(24)S = u

u + b + c+ v

v + b + d+ w

w + c + d.

So,

(25)

q1 = uu+b+c

(k

S+1

),

q2 = vv+b+d

(k

S+1

),

q3 = ww+c+d

(k

S+1

),

and

(26)

u1 = uu+b+c

(k

S+1

)2,

u2 = vv+b+d

(k

S+1

)2,

u3 = ww+c+d

(k

S+1

)2.

Now let us focus on the sum of the profits of the firms.

(27)U = u1 + u2 + u3 = S

(k

S + 1

)2

.

We may easily find whenS = 1, U (represents the sum of all the firm’s profits) will be the maximum value. Achecking the condition of it, we find whenγ1, γ2, γ3 → ∞, S = 1 andU = 1

4k2. This is similar with the situationof γ1 = γ2 = γ3 = γ . The feature means that the sum under this condition is highest and there has not ansuperior condition. It can be viewed in theFig. 2, in which we fixedγ3 at 0.05 (a satisfied degree to demonstrthe variety ofU ), thatU increases monotonously with the increasing ofγ1 andγ2. Because of symmetry, we cainfer thatU also has the same relation withγ3. In Fig. 3, we plotU with different settings ofγ3. It illustrativelyshows the relationship ofγ3 andU we have mentioned above.Fig. 4 is the section of the diagonal ofγ1 andγ2of Fig. 3. But γ1, γ2, γ3 → ∞ is not the only condition ofS = 1. We find that just two parameters trend to infinis enough. However, the profits of the firms will be asymmetric. For example, whenγ1 = γ2 = γ , γ3 = 0, i.e., firm1 entangles with firm 2 and firm 2 entangles with firm 3, but firm 1 does not entangle with firm 3 directly.

J. Zhou et al. / Physics Letters A 339 (2005) 10–17 15

ntangle.

Fig. 2. The sum of profits versusγ1 andγ2 plot with γ3 = 0.05. Fig. 3. The plots of different setting ofγ3 are combined in here.

Fig. 4. The section with the diagonal ofγ1 andγ2 of Fig. 3.

this condition, the relevant relations are

(28)u

u + b + c= v

v + b + d= 2+ ch(

√2r)

2 ch(√

2γ ) + √2 sh(

√2γ )

,

(29)w

w + c + d= ch(

√2γ )

ch(√

2γ ) + √2 sh(

√2γ )

,

so, whenγ → ∞,

(30)S = u

u + b + c+ v

v + b + d+ w

w + c + d= 1.

This feature indicates that the sum of profits may also attain the highest value even if two firms do not edirectly but have a firm to link them. The property may be useful in defining the multi-partite entanglement

16 J. Zhou et al. / Physics Letters A 339 (2005) 10–17

So,pletely

Fig. 5. The quantum structure of the model.

4. Extension for arbitrary firms when the entanglement parameters vary consistently

We suppose there aren firms participating the game and entangle in the same degree, i.e.,γ1 = γ2 = · · ·= γn = γ . J (γ ) is constructed in the same way shown inFig. 5. Therefore,

(31)J (γ ) = exp

{−γ

[n∑

i,j=1,i =j

(a+j a+

i − aj ai

)]}.

Utilizing the way of Baker–Campbell–Hausdorff relation and induction, we have

(32)J (γ )+Dj (xj )J (γ ) = exp

{−ixj

[Pj

1

n

(e(n−1)γ + (n − 1)e−γ

) +n∑

i=1,i =j

Pi

1

n

(e(n−1)γ − e−γ

)]}.

Then, after taking measurement, we get the quantities of the firms as

(33)qj = xj

1

n

(e(n−1)γ + (n − 1)e−γ

) +n∑

i=1,i =j

xi

1

n

(e(n−1)γ − e−γ

),

(34)Q =n∑

j=1

xj e(n−1)γ .

The solution of Nash equilibrium(∂uj

∂xj= 0) is

(35)q∗j = 1

n

k((e(n−1)γ + (n − 1)e−γ ))

2e(n−1)γ + (n − 1)e−γ,

and the profit is

(36)u∗j = k2

n

e(n−1)γ ((e(n−1)γ + (n − 1)e−γ ))

(2e(n−1)γ + (n − 1)e−γ )2,

here we mark the sum of the profits asU (= ∑n1 uj ). Particularly, whenn = 3

(37)q∗1 = q∗

2 = q∗3 = 1

3

k(e2γ + 2e−γ )

2e2γ + 2e−γ,

(38)u∗1 = u∗

2 = u∗3 = k2

3

e2γ (e2γ + 2e−γ )

(2e2γ + 2e−γ )2.

It shows how the profits at the Nash equilibrium vary withγ . When the game is not entangling, i.e.,γ = 0, thequantum game goes back to the original classical form. However, at the extent of infinite squeezing limitγ → ∞,

these pairs all fall into|EPR〉 andu1 = u2 = · · · = un → k2

4n. It is the best situation they can ever achieve.

we concluded that, whatever the number of firms is, the dilemma-like situation of the game may be comremoved owing to the entanglement.

J. Zhou et al. / Physics Letters A 339 (2005) 10–17 17

is repre-subset ofonouslyofbro-here allglement

9300 of

. 251–272,

5. Conclusion

From the extended model of the Cournot’s duopoly, we construct a quantum structure where each firmsented by a qubit and then we let these firms entangle pairwise. In this structure the classical form is thethe quantum game. In the three firms case we find the sum of the profits of the firms will increases monotwith the increasing of theγj (measure of entanglement) when theγj vary arbitrarily. Furthermore, the conditionattaining the highest sum of profits (U ) is not unique, though the symmetry of the profits of the firms may beken. This may be helpful to define the degree of multi-partite entanglement. By considering the condition wof theγj are varying consistently, we find even when the game has an arbitrary number of firms the entancan completely remove the dilemma-like situation.

Acknowledgements

This work was supported by the National Fundamental Research Program Grant No. 2001CB30PR China. We also thank Dr. Jiangfeng Du for beneficial discussion.

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