n-player quantum minority game
TRANSCRIPT
(2001)sical ones
Physics Letters A 327 (2004) 98–102
www.elsevier.com/locate/pla
N-player quantum minority game
Qing Chena,c,∗, Yi Wangb, Jin-Tao Liuc, Ke-Lin Wangc
a Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, PR Chinab Department of Physics, University of Science and Technology of China, Hefei 230026, PR China
c Department of Modern Physics, University of Science and Technology of China, Hefei 230026, PR China
Received 25 March 2004; received in revised form 5 May 2004; accepted 6 May 2004
Available online 13 May 2004
Communicated by P.R. Holland
Abstract
We investigate theN-player quantum minority game under the scheme proposed by Benjamin et al. [Phys. Rev. A 64030301] when they share a nonmaximally entangled state. We find all pure quantum strategies simply reduce to claswhenN is an odd number. WhenN is an even number, fundamentally nonclassical equilibria become available and the playerscan get higher expected payoffs at the same time. 2004 Elsevier B.V. All rights reserved.
PACS: 03.67.-a; 02.50.Le
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Classical game theory is a mature field of applmathematics which has found numerous applicatiin economy, psychology, ecology, and biology[1–3].It concerns the study of multiperson decision prolems, where two or more individuals make rational dcisions limited in a certain set ofstrategy space andevery one’s decision influence others’ payoffs. Rcently, game theory based on quantum strategiesattracted intense study because it has been foundnew strategies are available to the players if the ccepts of quantum superposition[4,5] and quantum entanglement[6–11]are introduced.
Minority game has attracted much interest recen[12]. It is the mathematical formulation of “EI Faro
* Corresponding author.E-mail address: [email protected] (Q. Chen).
0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserveddoi:10.1016/j.physleta.2004.05.012
t
Bar” problem [13]. It is believed that the gamcaptures an essential feature of systems where placompete for limited resources, such as, tradinga financial market and choosing which eveningvisit an overcrowded bar. In minority game, theareN players and each player privately choosesalternatives, say “0” and “1”. The choices are thcompared and the players who have made the minodecision will win. If all players have made the samchoice, or if there is an even split, there is no winnIn the case most paper concerned[12], the game isplayed repeatedly. Every player only has a limitedof differentS strategies and each player does not knanything about the others, they base their decisionthe knowledge of theM last winning alternatives anthe performance of their strategies in the past. Inevolution game, cooperation appears even thoughplayer do not care about others.
.
Q. Chen et al. / Physics Letters A 327 (2004) 98–102 99
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More recently, a quantum minority game (QMGhas been proposed by Benjamin et al.[7]. Differentfrom the model above, it is a static game in whichplayers play only once and every player has the spublic quantum strategies sets, thus game theorybe applied. In their pioneering work, they investiga3-player and 4-player QMG where the entanglemis maximal, and they found no new Nash equilibriuappears in 3-player QMG, interestingly, new Naequilibria which give the optimal expected payoffevery player appear in 4-player QMG.
In this Letter, the results of theN -player QMG un-der the elegant scheme proposed in Refs.[6,7] are re-ported. We find the game reduce to the classicalwhen N is odd. Interestingly, whenN is even, newNash equilibria independent of the game’s entanment are discovered. Expected payoffs higher thanof the classical game are obtained and the payincrease monotonously when entanglement increaBesides, asN (even) increases, the expected payoincrease monotonously, but the difference betweenexpected payoff of quantum game and that of clacal game decreases. In particular, whenN approachesinfinity, both expected payoffs approach 1/2, which isthe upper limit of expected payoff in minority game
In classical minority game (CMG), Nash equilirium appears when all players choose randomlytween the 0 and 1 actions. There exist 2N final possi-ble configurations which appear with the same probility, and every player gets the same expected pay
$∗c_odd =
{1
2− �
(N2
)2√
π �(
N+12
)}
,
(1)$∗c_even =
{1
2− �
(N+1
2
)√
π �(
N2 + 1
)}
,
where�(x) is the gamma function, $∗c_odd ($∗c_even) is
the expected payoff whenN is odd (even).In our quantum game, as shown inFig. 1, the arbiter
uses the entangling gate
J = √p I⊗N + i
√1− p σ⊗N
x
to generate aN -particle GHZ state from theN -particleproduct state|00. . .0〉, wherep ∈ [0,1] is a measureof the entanglement of the initial state. Then tplayers act with local unitary operators(s1, . . . , sN )
on their own qubits which are sent by the arbit
.
Fig. 1. The schematic diagram ofN -player quantum minoritygame[7].
Finally, the disentangling gateJ+ is carried out andthe system is measured in the computational ba{|0〉, |1〉}, and the expected payoff of every playerdecided. When comparing the quantum and classgames, it is of fundamental importance to chooseappropriate strategy space. Our quantum game redto the classical one if each one’s strategies are limin {I , σx}. Any superset of this classical space canchosen in principle. In this Letter, we take the mgeneral one-qubit operation for playeri,
si (αi, βi , γi) = cosαi (sinβi iσx + cosβi iσy)
+ sinαi (cosγi I + sinγi iσz)
with three parameter[7], where αi ∈ [0,π/2] andβi, γi ∈ [−π,π).
Now we calculate the expected payoffs of the plers. The final state prior to measurement is a suposition of eigenstates|x1, . . . , xN 〉, where x1, . . . ,
xN = 0, or 1. We note that the states|x1, . . . , xN 〉and |x1, . . . , xN 〉 give the same winners, wherex =NOT(x). It is convenient to calculate
P(x1, . . . , xN) = px1...xN + px1...xN
in the following discussions, where
px1...xN = ∣∣〈x1 . . . xN |J+s1 . . . sN J |0 . . .0〉∣∣2is the probability that the measurement outcome(x1, . . . , xN). As we will see, the expression ofpx1...xN
is a very complex one, and that ofP(x1, . . . , xN) ismuch concise and clear. We obtain
100 Q. Chen et al. / Physics Letters A 327 (2004) 98–102
ucethe
f
is
ear
gy
to
,
holds
tion,
〈x1 . . . xN |J+s1 . . . sN J |0 . . .0〉= (−1)N−j p
∏{xk=0}
sinαk eiγk∏
{xl=1}cosαl e
−iβl
− i(−1)j√
p(1− p)
×∏
{xk=0}cosαk e−iβk
∏{xl=1}
sinαl eiγl
+ i√
p(1 − p)
×∏
{xk=0}cosαk eiβk
∏{xl=1}
sinαl e−iγl
(2)
+ (1− p)∏
{xk=0}sinαk e−iγk
∏{xl=1}
cosαl eiβl ,
〈x1 . . . xN |J+s1 . . . sN J |0 . . .0〉= (−1)jp
∏{xk=0}
cosαk e−iβk∏
{xl=1}sinαl e
iγl
− i(−1)N−j√
p(1− p)
×∏
{xk=0}sinαk eiγk
∏{xl=1}
cosαl e−iβl
+ i√
p(1 − p)
×∏
{xk=0}sinαk e−iγk
∏{xl=1}
cosαl eiβl
(3)
+ (1− p)∏
{xk=0}cosαk eiβk
∏{xl=1}
sinαl e−iγl ,
wherej is the number of zeros in(x1, . . . , xN). Aftera bit of calculations, we get
(4)P(x1, . . . , xN) = Λ − (−1)j + (−1)N−j
2∆,
where
Λ =∏
{xk=0}sin2 αk
∏{xl=1}
cos2 αl
+∏
{xk=0}cos2 αk
∏{xl=1}
sin2 αl,
(5)
∆ = 2−(N−2)√
p(1− p)
×N∏
i=1
sin2αi sin
[N∑
i=1
(βi − γi)
].
WhenN is an odd number, the last term ofEq. (4)vanishes. Thus all pure quantum strategies redto classical ones. Nash equilibria appear when
strategiessi (π/4, β ′, γ ′) are adopted, whereβ ′, γ ′ arearbitrary numbers in[−π,π). The expected payofof every player is the same and equals to $∗
c_odd.Moreover, it is worth to note that the reductionindependent of the game’s entanglement.
WhenN is an even number,Eq. (4)becomes
(6)P(x1, . . . , xN) = Λ − (−1)j∆.
In this case, we will prove that Nash equilibria appif
α∗1 = · · · = α∗
N = π
4,
(7)N∑
i=1
(β∗
i − γ ∗i
) = (−1)N/2 π
2
are satisfied.1
To prove this result, let’s begin with the strateset which accords withEq. (7). When an arbitrarilychosen playerq unilaterally changes her strategysq(αq ,βq, γq), P(x1, . . . , xN) turns to
(8)P(x1, . . . , xN) = Λ0 − (−1)j∆0,
where
Λ0 = 2−(N−1),
(9)
∆0 = 2−(N−2)√
p(1− p)sin 2αq
× sin
[(−1)N/2π
2− (
β∗q − γ ∗
q
) + (βq − γq)
].
Notice thatΛ0 and∆0 are independent of(x1, . . . ,
xN), andP(x1, . . . , xN) is related with the parity ofj .Thus the expected payoff of playerq can be obtained
(10)$q_even(q) =N2 −1∑j=1
P(x1, . . . , xN)Cj−1N−1,
whereCmn is the binomial coefficient.
1 It is obvious that several Nash equilibria occur whenN is aneven number. Players may use criteria, such as every playerthe same strategy,
s∗(π4 , (−1)
N2 π
4N, (−1)
N2 +1 π
4N
),
to establish a focal point, but this becomes a psychological quessee also in[7].
Q. Chen et al. / Physics Letters A 327 (2004) 98–102 101
,
Fig. 2. The expected payoff $∗q_even at Nash equilibria as a function of the entanglement parameterp in quantum form. From the bottom to topthe number of players areN = 4, N = 6, N = 8, respectively.
y
aysn
tos atenteoffsled,to
ieso
ee,
t isdand
isirly.nd
t is
osens
Inteps,her
hether
timehe
CombiningEqs. (8) and (9), Eq. (10)reduce to
$q_even(q) ={
1
2− �
(N+1
2
)√
π�(
N2 + 1
)}
(11)+ (−1)N/2∆�(N − 1)
�(
N2 − 1
)�
(N2 + 1
) .
On the other hand, whenEq. (7)are satisfied, everplayer has the same expected payoff,
$∗q_even =
{1
2− �
(N+1
2
)√
π �(
N2 + 1
)}
(12)+ ∆max�(N − 1)
�(
N2 − 1
)�
(N2 + 1
) ,
where∆max = 2−(N−2)√
p(1 − p). ComparingEqs.(11) and (12), one easily see that whatever playerq
changes her strategy, her excepted payoff is alwless than or equal to $∗
q_even. So we get the conclusiothat Nash equilibria appear ifEq. (7)are satisfied.
Obviously, the first term ofEq. (12) is equal to$∗c_even, and the second term can be attributed
entanglement. It is interesting to see how the payoffNash equilibria vary with respect to the entanglemp, as depicted inFig. 2. It shows that the stronger thgame’s entanglement is, the higher expected payall players can obtain. When the game is not entangi.e., p = 0 or p = 1, the quantum game reduce
the original classical one. Moreover, all strategsatisfyingEq. (7)are Nash equilibria of the game nmatter what degree the entanglement is.
We notice that there is an upper limit to thexpected payoff of each player in minority gamno matter what strategy is used. The upper limireached when[(N −1)/2] players choose 0 (or 1), anthe expected payoff of every player is the sameequals to[(N − 1)/2]/N , where the symbol[·] meansto take the integer part of a real number. This limitour aim: resources are used most efficiently and faFig. 3 shows the expected payoff of CMG, QMG athe upper limit vs. the number of playersN . We cansee that QMG is always better than the CMG, but ibelow the upper limit with only one exception,N = 4.One can analyze as follows. In CMG, players chorandomly between 0 and 1, thus all configuratio(x1, . . . , xN) happen with the same probability.QMG, from Eq. (6), we can see that the final staprior to measurement can be divided into two grouone with odd number of 0s (and 1s) and the otwith even number of 0s. In each group,|x1, . . . , xN 〉and |x1, . . . , xN 〉 “form” a pair, and P(x1, . . . , xN)
is the same for all the pairs at Nash equilibria. Tplayers can enhance one group and weaken the oto enlarge everyone’s expected payoff at the sameby exploiting their strategies. Especially, when tentanglement becomes maximal,p = 1/2, one group
102 Q. Chen et al. / Physics Letters A 327 (2004) 98–102
MG
Fig. 3. The expected payoff of CMG (square),QMG (triangle) and the upper limit of minoritygame (downward triangle) as functions ofN . InQMG, the players share a maximally entangled state (p = 1/2). The difference (circle) between the expected payoff of QMG and that of Cis also shown.h theyoff
Gect
achin
oreralallical
getting
s-ure).
s,
.rd,
82
83
87
01)
01.
.
disappears and the players’ expected payoffs reacmaximum. But even in this case, the expected pais still lower than the upper limit, except whenN = 4,because the remaining groupinevitably contain thosestate pairs with minority players less than[(N −1)/2].
Another interesting thing shown inFig. 3 is thatthe difference of expected payoff in QMG and CMis a monotonously decreasing function with respto the number of playersN . In particular, whenNapproaches infinity, both expected payoffs appro1/2, which is the upper limit of expected payoffminority game.
To conclude, in this paper theN -player quantumminority game was investigated by seeking mavailable Nash equilibria. Taking the most geneone-qubit operation for every player, we find thatpure quantum strategies simply reduce to the classones whenN is odd. However, whenN is even, theplayers are able to reach new Nash equilibria andhigher expected payoffs at the same time by exploientanglement.
Acknowledgement
Q. Chen thanks L. Fu and J. Li for helpful discusions. This work was supported by the National NatScience Foundation of China (Grant No. 10231050
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