Collective quantum games with Werner-like states

Accepted Manuscript Collective quantum games with Werner-like states Ramón Alonso-Sanz

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S0378-4371(18)30886-0 https://doi.org/10.1016/j.physa.2018.07.022 PHYSA 19847

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Physica A

Received date : 12 April 2018 Revised date : 4 July 2018 Please cite this article as: R. Alonso-Sanz, Collective quantum games with Werner-like states, Physica A (2018), https://doi.org/10.1016/j.physa.2018.07.022 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

*Highlights (for review)

Highlights  



The implementation of the imitation of the best evolving rule proves to be a very useful tool to analyze the collective behaviour of two-person games via simulation. Werner-like states enable to scrutinize the way in which the features of the studied games vary from the quantum entangled scenario up to that of independent players with uniform random strategies In the Prisoner's Dilemma, Hawk and Dove, Samaritan's Dilemma game-types, the new Nash equilibriums achieved with highly correlated games maximize the sum of the payoffs of both players, i.e., they provide its (unique) so called social welfare solution. The Battle of the Sexes game-type turns out much more challenging at this respect, because it has two social welfare solutions.

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Collective quantum games with Werner-like states. Ram´ on Alonso-Sanz Technical University of Madrid, ETSIAAB (Estadistica, GSC). C.Universitaria. Madrid 28040, Spain. [email protected]

Abstract This article studies the collective behaviour of quantum games with Werner-like states, where every player interacts with four partners and four mates. Four two-person game-types are scrutinized by allowing the players to adopt the strategy of his best paid mate. Particular attention is paid in the study to the effect of variable degree of entanglement and fidelity to the quantum model on Nash equilibrium strategy pairs. Keywords : collective; quantum; Werner; games

1

Introduction.

Quantum games have allowed to connect quantum mechanics, which determines the behaviour of systems at microscopic scales, with game theory, which has proven to be a useful and versatile tool dealing with many economical, social and evolutive problems. Since the inception of the subject circa 1999 [1, 2], the extension of game theory into the quantum realm, i.e., quantum game theory, has attracted much attention [3, 4, 5] . No exempt of criticism as summarized in [6], so that in the exteme cases, the field may be either neglected [7] or fully vindicated [8, 9] . To investigate how the implementation of quantum mechanic tools affects the field of game theory, a variety of games have been taken into consideration. This paper considers the four twoperson (A and B), 2×2 non-zero sum game-types defined by the payoff matrices given in Table 1. Namely, the Prisoner’s Dilemma (PD), the Hawk-Dove (HD), the Samaritan’s Dilemma (SD), and the Battle of the Sexes (BOS) games described below. .

C

A D

PD C R R

B

D T S

S T

.

P P

T>R>P>S

.

D

A H

HD D R R

B

.

SD

.

.

BOS

B W

H T S

S T

.

A

A

P

A

P

T>R>S>P

2 3 1 -1

L 3 -1 0 0

F

A B

F r R

B

B 0 0

0 0

.

R r

R>r

Table 1: Four game-types. From left to right : Prisoner’s Dilemma (PD), Hawk-Dove (HD), Samaritan’s Dilemma (SD), Battle of the Sexes (BOS). In the Prisoner’s Dilemma (PD) game, both players may choose either to cooperate (C) or to defect (D). Mutual cooperators each scoring the reward R, mutual defectors score the punishment P ; D scores the temptation T against C, who scores S (sucker’s payoff) in such an encounter. In the PD it is : T > R > P > S . According to [10], in this study the PD payoff values will be T = 5, R = 3, P = 1 and S = 0. The PD with these payoffs will be referred to as PD(5,3,1,0). 1

In the Hawk-Dove (HD) game, the structure of the payoffs matrices is similar to that in the PD, but in the HD it is P < S instead of P > S as is the PD. In this study, will be keep T = 5, R = 3 for the HD, but P = 0 and S = 1. The HD with these payoffs will be referred to as HD(5,3,0,1). In the Samaritan’s Dilemma (SD) game, the charity (or Samaritan) player A may choose Aid/No Aid people in need, whereas the beneficiary player B may choose simply rely on the handout (Loaf) rather than try to improve their situation (Work). Many people may have experienced this dilemma when confronted with people in need: Although there is a desire to help them, there is the recognition that a handout may be harmful to the long-run interests of the recipient [11]. Following the references [12, 13, 14], we adopt here the payoff matrices given in the SD panel in Table 1, and the SD with these payoffs will be referred to as SD(3,2,1,-1). In the so called battle of the sexes (BOS) game, the rewards R > r > 0 quantify the preferences in a conventional couple fitting the traditional stereotypes : The male player A prefers to attend a F ootball match, whereas the female player B prefers to attend a Ballet performance. Both players hope to coordinate their choices, but the conflict is also present because their preferred activities differ [15, 16] . In this study the BOS with R = 5, r = 1 , referred to as BOS(5,1), will be analyzed. The PD and HD games are symmetric i.e., the payoff matrices of both players coincide after transposition, whereas the SD and the BOS games are not symmetric. In symmetric games the role of both players are somehow interchangeable, whereas in asymmetric games every player has to be studied separately.

2

The classical and quantum games.

In the somehow canonical approach to game theory, both players choose their strategies independently of each other. In an alternative approach, an external (probabilistic) mechanism sends a signal to each player, so that, in principle, the players do not have any active role. Both approaches, as well as a quantum mechanism for combining them, are featured in this section.

2.1

The classical context.

In conventional classical games both players decide independently their probabilistic strategies x = (x, 1 − x)0 and y = (y, 1 − y)0 , that give rise to the joint probability distribution Π = xy0 . As a result, in a game with PA and PB payoff matrices, the expected payoffs (p) of both players are ( indicates element-by-element matrix multiplication, 10 =(1,1)) : pA(x, y) = 10 PA Π1 = x0 PA y ,

pB(x, y) = 10 PB Π1 = x0 PB y

(1)

The strategy pair (x, y), referred here to as (x, y), is in Nash equilibrium (NE), if x is the best response to y and y is the best response to x . In the PD game, mutual defection, i.e., x∗ = y ∗ = 0, is the only pair in NE. The HD game has three strategy pairs in NE, two of them with pure strategies : (x∗ = 1,y ∗ = 0) and (x∗ = 0,y ∗ = 1), and a third one with mixed strategies, which in the particular case of the HD(5,3,0,1) turns out to be: x∗ = y ∗ = 1/3, leading to pA,B = 5/3 . Please, note that (x = y = 0) is not in NE in the HD game. The SD game has only one NE, that in the particular case of the SD(3,2,1,-1) becomes: (x∗ = 1/2, y ∗ = 1/5), leading to (pA = −0.2 , pB = 1.5) . The BOS has two strategy pairs in NE with pure strategies : (x∗ = y ∗ = 1) and (x∗ = y ∗ = 0), and a third r Rr R , y∗ = ), leading to pA,B = < r. one with mixed strategies (x∗ = R+ r R+ r R +r

2

Social welfare (SW) functions may be envisaged as summarizing some particular conception of the common good [17]. In its simplest form, SW solutions maximize the sum of the payoffs of both players. The (1,1) strategy is the only SW solution in the PD(5,3,1,0), HD(5,3,0,1) and SD(3,2,1,-1) games; whereas in the BOS(5,1) game both (1,1) and (0,0) are SW solutions. In a different  game scenario, that of correlated games, an external probability distribution π11 π12 Π= assigns probability to every combination of player choices [16], given rise to the π21 π22 expected payoffs pA(Π) = 10 PA Π1, pB(Π) = 10 PB Π1 . The quantum approach described in the next subsection, combines both the independent players and the Π-correlated game models all at once.

2.2

Quantum games: The basic EWL model

Within the EWL quantization scheme [1], the purely classical strategies C and D, which will be ˆ in order to explicitly manifest that we are playing a quantum hereafter respectively named Cˆ and D, game, act in a two level Hilbert space spanned by the two bases vectors |0i and |1i. The state of the game is a vector in the tensor product space spanned by the basis vectors |00i, |01i, |10i, and |11i . ˆ The EWL protocol starts with an initial entangled  state  |Ψi i = J|00i, where the symmetric unitary   γ ˆ ⊗2 , being D ˆ = 0 1 , and 0 ≤ γ ≤ π/2 the entanglement factor, operator equals Jˆ = exp i D −1 0 2 which tunes the entanglement degree. The initial state then becomes |Ψi i = cos γ2 |00i + i sin γ2 |11i. The players perform independently their quantum strategies described by local unitary operators ˆ in the SU(2) space. Namely with strategies of the form, U ! eiα cos θ2 sin θ2 eiβ θ ∈ [0, π] ˆ (θ, α, β) = U , (2) θ −iβ θ −iα α ∈ [0, π/2], β ∈ [0, π/2] − sin 2 e e cos 2 The case of strategies with three parameters (3P) will be treated only in section 5 , whereas the rest of this section and sections 3 and 4 consider only the two-parameter (2P) subset of SU(2) with β = 0 . The 2P model may be criticized as being just a subset of the general SU(2) space of unitary strategies [18, 19]. Thus, the 2P-restriction in the original formulation of the EWL model [1] is a subject of continuing discussions. Anyway, the 2P-EWL model has proved to be a good (and widely used) test-bed to show how the quantum approach in game theory may solve dilemmas, by allowing for Nash equilibrium strategies out of the scope of the classical approach [20, 21] . Last but not least, according to the study on purely classical correlated games [22, 23], the 2P-EWL (somehow semi-quantum), emerges as an excellent intermediate step from classical correlation to the full quantum approach (3P), and then it still deserves attention on its own. ˆA ⊗ U ˆB )J|00i. ˆ After the application of these strategies, the state of the game changes to |Ψf i = (U † Prior to measurement, the Jˆ gate is applied and then the state of the game becomes |Ψf i = ! ρ

ρ

11 12 ˆA ⊗ U ˆB )J|00i, ˆ Jˆ† (U with density matrix ρf = |Ψf ihΨf |. Finally, Πf = . 22 ρ21 ρ ˆ= U ˆ (0, π/2)= i 0 stands out, because In the 2P model, the (purely quantum) strategy Q 0 −i ˆ Q} ˆ pair becomes in NE for high values of γ in the PD, HD and SD games yielding π Q,Q = the {Q, 11 C,C ˆ interacts with the pure D ˆ π11 = 1 regardless of γ (see Sec. 3). The Π matrices emerging when Q

3

ˆ and D ˆ encounters strategy are given in Eqs. (3) below, and the graphs of the payoffs arising in the Q in the QPD(5,3,1,0) game are shown in the right frame Fig. 1 . ΠD,D = f

2.3

 0 0

 0 , 1

ΠQ,D = f



 0 cos2 γ , sin2 γ 0

ΠD,Q = f



0 cos2 γ

 sin2 γ , 0

Q,Q ΠC,C = f = Πf

 1 0

0 0



(3)

Quantum games: Werner-like states

According to [10], we will consider here Werner-like states [24] by means of, ρ = (1 − δ)I/4 + δρf ,

0≤δ≤1

 1 1 Where δ ponders the fidelity to ρf . If δ = 0 it is ρ = I/4, so that Π = 4 1 game it is pA (γ, 0) = pB (γ, 0) = K, with K = (T + P + R + S)/4 . From Eq. (4), it is,   1−δ 1 1 + δΠf , 0 ≤ δ ≤ 1 Π= 1 1 4

(4)  1 , so that in the PD 1

(5)

Therefore, in the PD game the payoffs achieved with ρf are to be multiplied by δ and increased D,D Q,Q by (1 − δ)K . It is, pD,D A,B = (1 − δ)K + δP , and pA,B = (1 − δ)K + δR , leading to pA,B (γ, 1) = P and pQ,Q A,B (γ, 1) = R . In the QPD(5,3,1,0) context of Fig. 1, with K = (5 + 3 + 1 + 0)/4 = 2.25, D,D D,D pA,B (δ = 0.5) = (2.25+0) = 1.625, pQ,Q A,B (δ = 0.5) = (2.25+3)/2 = 2.265 (left frame), and pA,B (δ = Q,D 1.0) = P = 1.0, pQ,Q = (1−δ)K+δ(S cos2 γ+T sin2 γ) . A,B (δ = 1.0) = R = 3.0 (right frame). It is, pA D,D Q,D 2 2 ? As a result, q the  intersection of pA and pA is given with P = S cos γ + T sin γ, i.e., at γ =

arcsin

P −S T−S

D,Q which does not depend on δ . Similarly, pA = (1 − δ)K + δ(S sin2 γ + T cos2 γ) ,

so thatq the intersection of pD,Q and pQ,Q is given with S sin2 γ + T cos2 γ = R, i.e., at γ • = A A  T−R arcsin which neither depends on δ . In the QPD(5,3,1,0) context of Fig. 1 it is : γ ? = p T−S p arcsin 1/5 = 0.464, closely over π/8 = 0.393, and γ • = arcsin 2/5 = 0.685, closely under π/4 = 0.785 .

3

Spatial games.

In the spatial games we deal with in this section, the same number of players of type A and of type B are involved. In the spatial version we deal with in this section, each player occupies a site (i, j) in a two-dimensional N × N lattice, alternating in the site occupation in a chessboard form. Consequently every player is surrounded by four partners (A-B, B-A), and four mates (A-A, B-B) . The game is played in the cellular automata (CA) manner, i.e., with uniform, local and synchronous interactions [25] . In this way, every player plays with his four adjacent partners, so (T ) that the payoff pi,j of a given individual at time-step T is the sum over these four interactions. The evolution is ruled by the (deterministic) imitation of the best paid mate neighbour, so that in the next generation, every generic player (i, j) will adopt the quantum parameters of his mate player with the highest payoff. The just described setup is reminiscent of that proposed in [26] . 4

p

(θ,α)-QPD(5,3,1,0)

p

δ = 0.5

5.0

5.0

4.5

4.5

4.0 3.5 3.0 2.5 2.0

pQD pDQ A B

4.0

pQQ pQQ A B

3.0

pDD pDD A B

2.0

δ = 1.0 pQD pDQ A B

3.5

pQQ pQQ A B

2.5

1.5

1.5

pDQ pQD A B

1.0

pDD pDD A B

1.0

0.5 0.0

(θ,α)-QPD(5,3,1,0)

0.5

γ⋆ π/8

γ• π/4

3π/8

π/2

γ

0.0

γ⋆ π/8

pDQ pQD A B

γ• π/4

3π/8

π/2

γ

Figure 1: Payoffs of the interactions of the D and Q strategies in the QPD(5,3,1,0) game. Left : δ = 0.5, Right : δ = 1.0 . Please note that the transition rule is both memoryless and myopic, i.e., neither the past nor the future does matter, only the present time-step T determines T + 1 . Initial quantum parameter values will be assigned at random. Thus, initially : θ ' π/2 and α = π/4 . Five simulations are implemented in every scenario. The mean payoffs (p) and the underlying mean value of the quantum parameters (θ, α) are shown at T = 200 in simulations with variable entanglement factor γ . As a rule, the results regarding player A are shown in red color, and those regarding player B are shown in blue color. The computations have been performed by a double precision Fortran code run on a mainframe. Next subsection 3.1 deals only with the symmetric PD and HD games by fixing one of the parameters γ and δ, whereas subsection 3.2 scrutinizes also the SD and the BOS games with both γ and δ variable.

3.1

Either γ or δ fixed.

In the QPD(5,3,1,0)-CA simulations of Fig. 2, mutual defection arises below the lower γ • = 0.464 threshold and mutual Q beyond the higher γ F = 0.685 threshold. In the (γ ? , γ • ) transition interval, where both (Q, D) and (D, Q) are in NE, the dynamics is highly conditioned by spatial effects as explained below when commenting Fig. 2 . Remarkably, not only the values of the γ ? and γ • trhesholds are unaffected by δ, neither the evolution of the θ and α mean quantum parameters are affected by δ, as it is shown in the lower frames of Fig. 2 . In short, implementing ρ instead of ρf only affects the payoffs. It is noticeable that for γ < γ • , as both θ’s are selected to be π, both α’s are irrelevant, as cos(π/2) = 0 annihilates the influence of α in Eq. (2), whereas for γ > γ ? , both θ’s are set to zero and both α’s to π/2, i.e., the (Q, Q) pair is stated. Figure 2 shows also the mean-field payoffs (p∗ ) achieved in a single hypothetical two-person game with players adopting the average parameter values appearing in the collective dynamic simulation. Namely, with strategies of the form :  iα   iα  e A cos θA /2 e B cos θB /2 sin θA /2 sin θB /2  , U ∗ (θB , αB ) =   UA∗ (θA , αA ) =  B −iα −iα − sin θA /2 e A cos θA /2 − sin θB /2 e B cos θB /2 5

p

(θ,α)-QPD(5,3,1,0)-CA N=200 T=200

p

δ = 0.5

p p A B

p p A B

p p A B

2.5

2.5

2.0

p p A B

p∗p∗

1.5

1.5

A B

1.0

1.0

γ⋆

0.5

γ•

π/8

π/4

3π/8

π/2

p p A B

p∗p∗ A B

γ⋆

0.5

γ

γ•

π/8

π/4

3π/8

π/2

γ

θ, α

θ, α π

δ = 1.0

p p A B

3.0

3.0

2.0

(θ,α)-QPD(5,3,1,0)-CA N=200 T=200

3.5

3.5

θA θB

π

3π/4

θA θB

3π/4

θA θB αA αB

π/2

θA θB αA αB

π/2

αA αB

αA αB π/4

π/4

αA αB

γ⋆ π/8

γ• π/4

θA θB 3π/8

π/2

γ

αA αB

γ⋆ π/8

γ• π/4

θA θB 3π/8

π/2

γ

Figure 2: The QPD(5,3,1,0) with variable γ in five spatial simulations at T=200. Left : δ = 0.5. Right : δ = 1.0 . Upper : Mean payoffs. Lower : Mean quantum parameters.

Mean-field payoff approaches are colored brown for player A and green for player B, somehow approaching the red and blue colors featuring the respective actual mean payoffs. In Fig. 2, the mean-field payoffs coincide with the actual mean payoffs with γ below γ ? and over γ • . But in the (γ ? ,γ • ) interval the mean-field payoff approaches underestimate the actual mean payoffs. The lack of coincidence of both the mean-field and actual mean payoffs reflects the emergence of quantum parameter patterns that impede an approach based on the mean values of said parameters. An example of this is given in the next Fig. 3 . Figure 3 takes care of one simulation in the left frame scenario of Fig. 2 at γ in the center of the (γ ? , γ • ) transition interval. Its far left frame shows the dynamics up to T = 30 of mean payoffs and parameters, the standard deviations (σ) of these magnitudes, and the mean-field payoff approaches. It demonstrates that the dynamics induced by the imitation of the best paid mate implemented in this article actuates in a straightforward manner, so that the permanent regime is achieved very soon. This in fact applies not only in the context considered in this figure, but in a general manner, regardless the game and conditions under scrutiny. Thus, iterating up to T = 200 may be excessive, less iterations would suffice in order to reach stable configurations. Thus, the simulations from now on will be are run up to T = 100. In the patterns at T = 200 shown in Fig. 3 , increasing grey levels indicate increasing values of parameter and payoffs, both in the whole 200×200 patterns (upper) and in their zooms of the 20 × 20 central part shown below. The whole patterns exhibit a kind of patchwork aspect, where irregular borders separate (Q, D) and (D, Q) clusters. This is particularly enhanced in the zoom of the θ parameter, where two D(θ = π)-Q(θ = 0) regions are separated by a 6

black spot (border) formed by defectors (θ = π). As the players in the borders are defectors mostly surrounded by defectors, they get a low payoff, which is reflected in clear border cells in the payoff pattern (far right). The spatial structure of the patterns in the γ transition interval, far from the initial random configuration and of the fixed point reached with low or high γ, explains why the mean-field estimations of the payoffs (p∗ ) differ from the actual ones (p) in the simulations with γ in the (γ ? , γ • ) transition interval.

p,p∗,σ (θ,α) (θ,α)-QPD(5,3,1,0)-CA γ = (γ ⋆+γ •)/2 = 0.574 π 3.0 δ = 0.5

2.5

θAθB

θ200

α200

p200

pA pB

2.0

1.5

1.0 0.8 0.6 0.4 0.2

∗ p∗ A pB

π 2

σθ σθ A B αA αB

σαAσαB σpAσpB 15

π 4

30

T

Figure 3: A simulation in the QPD(5,3,1,0) scenario of the left frame of Fig. 2 . Far left frame : Dynamics up to T = 30. Right : Patterns at T = 200. Increasing grey levels indicate increasing values in the patterns. In the QPD(5,3,1,0)-CA simulations of Fig. 4, the entanglement factor is fixed to its maximum, i.e., γ = π/2, whereas δ is the free parameter. In the left frame of Fig. 4, both players update their ˆ Q} ˆ pair, and consequently pQ,Q = (1 − δ)K + δR , that in strategies. The updating leads to the {Q, A,B Q,Q PD(5,3,1,0) game, varies from pQ,Q (δ = 0.0) = K = 2.25 up to p (δ = 1.0) = 3.0 . In the right A,B A,B frame of Fig. 4, only player A updates its strategies, whereas player B remains passive, so that in a mean-field analysis, he is to be assigned the Middle-level strategy M = U (π/2, π/4) . In turn, player ˆ as αA appears closely under π/2, and θA not far from zero. It A resorts to a strategy not far from Q     2 1 1 1 12 0 1 − sin γ 1 − sin2 γ Q,M Q,M 2 is Πf (γ) = , so that Πf (γ = π/2) = . Consequently, 2 1 sin2 γ 2 2 1 12 2 sin γ

1 1 1 1 pQ,M (π/2, δ) = (1 − δ)K + δ (R + P ) + T , pQ,M (π/2, δ) = (1 − δ)K + δ (R + P ) + S . A B 2 2 2 2 9 5 9 5 Q,M In the QPD(5,3,1,0) context of Fig. 4 it is : pQ,M (π/2, δ) = + δ , p (π/2, δ) = − δ , with A B 4 4 4 4 Q,M Q,M pQ,M (π/2, 1) = 14/4 = 3.50 , pB (π/2, 1) = 4/4 = 1.00 . The latter A,B (π/2, 0) = K = 2.25 , pA values are not far from the actual mean payoffs in the simulations in the right panel of Fig. 4, where pA (δ = 1) = 3.2 , pB (δ = 1) = 1.01 . 7

p

(θ,α)-QPD(5,3,1,0)-CA N=200 T=200

p

γ = π/2

3.5

(θ,α)-QPD(5,3,1,0)-CA N=200 T=200

3.5

p p A B

3.0

p A p A

3.0

2.5

2.5

2.0

2.0

1.5

1.5

1.0

p∗ B p B

1.0

0.5

0.25

0.50

θ, α

0.75

1.00

δ

0.5

0.25

0.50

θ, α

αA αB

π/2

θA θB 0.50

0.75

1.00

1.00

δ

αA αB

π/4

0.25

0.75

θB

π/2

π/4

γ = π/2

∗

θA

δ

0.25

0.50

0.75

1.00

δ

Figure 4: The QPD(5,3,1,0) with γ = π/2 and variable δ in five spatial simulations at T=200. Left : The two players update strategies. Right : Only player A updates strategies. In the QHD(5,3,0,1)-CA simulations of Fig. 5, mutual Q arises, as it!does in the QPD, at the r

Q,Q intersection of pD,Q and pA , thus at γ • = arcsin A

T−R=5−3 1 = T−S =5−1 2

= π/4. But at variance

with what happens in the PD, mutual defection is never in NE, neither for low entanglement. As a result, in the QHD, below γ • the mean payoff and parameter pattern graphs reflect what happens in the transition interval of the QPD: They exhibit a noisy aspect, and high spatial effects induce mean-field approaches that underestimate the actual mean payoffs. The mean payoffs trend to the payoff achieved with mutual Q as γ increases from K = 2.25 at γ = 0.0 . The standard deviations (σ) of the actual payoffs decrease in a fairly smooth way as γ increase in the two scenarios of Fig. 5, albeit they plummet to zero at γ = γ • . As pointed out before, the graphs of the mean quantum parameters are unaffected by δ, therefore only one quantum parameters frame is shown below in Fig. 5 .

3.2

Both γ and δ variable.

This subsection deals with a spatial simulation run up to T = 100 with both γ and δ variable in the four games featured in section 1 . Both γ and δ have been sampled in one hundred equidistant points. Figure 6 shows the mean payoffs of the player A in a spatial simulation of the QPD(5,3,1,0) (left frame) and the QHD(5,3,0,1) (right frame) games. In the QPD (left) it stands out how the (γ ? , γ • ) transition interval is unaffected by δ and the mean-field approaches underestimate the actual mean payoffs in said transition interval. In the QHD (right) the transition to mutual Q at γ • is fairly smooth, as explained when commenting Fig. 5. The mean-field approaches underestimate the actual payoffs before γ • = π/4 in the QHD game as do in the transition interval in the QPD. With no entanglement and full δ in the QHD(5,3,0,1) it is pA (0.0, 1.0) = 2.28 , close to the average of the payoffs achieved in the three NE in the HD classical game : (5+1+5/3)/3=2.56 . The mentioned greater values of the actual mean payoffs compared to the mean-field approaches seems to indicate that spatialization boosts the payoffs of both players in the PD and HD. This is so much as in the way reported in the seminal paper [27] in the classical PD scenario. This 8

p, p∗,σ

2P-QPD(5,3,0,1)-CA N=200 T=100

3.0

3.0

p p A B

p p A B

2.5

p, p∗,σ

δ = 0.5

2P-QPD(5,3,0,1)-CA N=200 T=100

δ = 1.0 p p A B

p p A B

2.5

2.0

2.0

p∗ p∗ A B

1.5 1.0

1.5

p∗ p∗ A B

1.0

σpA σpB

0.5

σpA σpB

0.5

γ• π/8

π/4

3π/8

π/2

γ•

γ

π/8

π/4

3π/8

π/2

γ

θ, α 3π/4

θA θB θA θB

π/2

π/4

αAαB αA αB

γ• π/8

π/4

3π/8

π/2

γ

Figure 5: The QHD(5,3,0,1) with variable γ in five spatial simulations at T=100. Above: Mean payoffs and their standard deviations at δ = 0.5 (left) and δ = 1.0 (right). Below : Mean quantum parameters. HD A

3

3

2

2

pA

pA

PD A

1

1

1 0.5 0

0

3π/8

1

π/2

•

0.5 δ

3

2

2

0

0

γ π/4

π/8

3π/8

π/2

γ

∗

pA

3

∗

pA

δ

• γ∗ γ π/4 π/8 γ

1

1

1 0.5 δ

0

0

∗ γ• γ π/4 π/8 γ

3π/8

1

π/2

•

0.5 δ

0

0

γ π/4

π/8

3π/8

π/2

γ

Figure 6: Mean payoffs of the player A in a spatial simulation at T=100 of the QPD(5,3,1,0) (left) and the QHD(5,3,0,1) (right). Upper : Actual mean payoffs. Lower : Mean-field payoffs.

9

article considers only pure Cooperators and Defectors arranged in a 2D lattice that evolve following the imitation of the best neighbour rule. As a result of the local interaction in the lattice, it is found that Cooperation is not fully discarded, but survives in a no negligible proportion of the cells. Provided that the temptation is not high. Otherwise, Defection will fully occupy the lattice. Figure 7 shows the mean payoffs of a spatial simulation of the QSD(3,2,1,-1) game. For δ = 0.0, the players get the average of the elements of their payoff matrices, i.e., pA = 0.25 , pB = 1.5 ˆ Q} ˆ is the only pair in NE for γ > γ • = π/4 . This is so because, regardless γ . In the QSD game, {Q, ˆ ˆ ˆ ˆ ˆQ ˆ ˆD ˆ Q D i) pQ = 2 and pQ = 3 cos2 γ + sin2 γ = 3 − 2 sin2 γ , so that pQ > pQ for γ > γ • = π/4 , B B B B ˆQ ˆ ˆQ ˆ ˆ ˆ ˆ ˆ 2 Q D Q Q D Q and ii) pA = 3 and pA = −1 sin γ − cos2 γ = −1 , so that pA > pA ∀γ . Incidentally, the QHD(5,3,0,1) and the QSD(3,2,1,-1) games share the same critical γ • = π/4 . Consequently, the Samaritan player A overrates the beneficiary player B if γ > π/4 , with pA (γ > π/2, 1.0) = 3.0 , pB (γ > π/2, 1.0) = 2.0 . Opposite to this, if γ > π/4 the beneficiary player B overrates the Samaritan player A, which gets negative payoffs with high δ. In particular, with no entanglement and full δ it is pA (0.0, 1.0) = −0.028 , pB (0.0, 1.0) = 1.549 , close to the payoffs achieved in the NE in the classical SD(3,2,1,-1) game given in the subsection 2.1 , i.e., pA = −0.2 , pB = 1.50 . Unexpectedly after studying the QHD game, no particular spatial effects arise in the QSD(3,2,1,-1) game before γ • = π/4 , so that the mean-field payoffs approach very well the actual mean payoffs, and consequently they have been not included in Fig. 7. As an example, in the particular case of no entanglement and full δ just mentioned, it is p∗A (0.0, 1.0) = −0.013 , p∗B (0.0, 1.0) = 1.564 . SD B

3

3

2.5

2.5

2

2

1.5

1.5

pB

pA

SD A

1 0.5

1 0.5

0

0

−0.5

−0.5

1

1 π/2 •

γ

0.5

3π/8

π/2 γ• π/4

0.5

π/4 π/8

δ

0

0

3π/8

π/8 δ

γ

0

0

γ

Figure 7: The QSD(3,2,1,-1) in a spatial simulation with variable γ and δ. Mean payoffs at T=100. Left : Player A, Right : Player B . Figure 8 shows the mean payoffs of a spatial simulation of the QBOS(5,1) game, where, as due, it is pA (γ, 0.0) = pB (δ, 0) = (5 + 1)/4 = 1.5 . With no entanglement and δ = 1 it is pA (0.0, 1.0) = 2.27 and pB (0.0, 1.0) = 2.64 , very close to the average of the payoffs achieved in the three NE in the classical BOS(5,1) game : (5+1+5/6)/3=2.28 . It is remarkable that the original formulation of the EWL model is somehow biased toward the player B (female) in the BOS game. This is so 1 as with middle-level election of the quantum parameters (θ = π/2, α = π/4) it is π11 = cos2 γ, 4 1 2 π22 = (1 + sin γ) . Thus, in the QBOS game with middle-level election of the parameters it 4     r 1 R 1 2  is : p  = cos γ + (1 + sin γ)2 . Finally, as (1 + sin γ) > cos γ in γ ∈ [0, π/2], A r 4 R 4 B 10

it is pB > pA . As a reflect of this bias, in Fig.8 it turns out that pB > pA in simulations with very high entanglement. So for example with maximum entanglement, it is pA (π/2, 1.0) = 2.47 , pB (π/2, 1.0) = 3.01 . But opposite to this, with low entanglement it turns out that pB < pA , with maximal advantage of player B nearly before γ = π/8 with δ = 1.0, where pB = 4.25 , pA = 1.25 . Both payoffs equalize at γ = 3π/8 in the δ = 1.0 scenario [28]. Spatial effects are particularly important in the BOS game. Thus, with maximum entanglement it is p∗A (π/2, 1.0) = 0.97 < 2.45 , p∗B (π/2, 1) = 4.83 > 2.47 . The structures that emerge in the spatial simulations of the BOS game show a maze-like aspect described in [29], that are not comparable to the patchkworks found in the QPD spatial simulations as that shown in Fig. 3. BOS B

5

5

3

3

pB

pA

BOS A

1

1

1 0.5 0

0

π/8

3π/8

1

π/2

0.5 δ

γ

5

5

3

3

p∗B

p∗A

δ

π/4

1

0

0

π/4

π/8

3π/8

π/2

γ

1

1 0.5 δ

0

0

π/4

π/8

3π/8

1

π/2

0.5 δ

γ

0

pi/8

pi/4

3pi/8

pi/2

γ

Figure 8: The QBOS(5,1) in a spatial simulation with variable γ and δ. Mean payoffs at T=100. Left : Player A, Right : Player B . Upper : Actual Mean payoffs. Lower : Mean-field approaches.

4

Games on random networks.

In the games on networks we deal with in this section, all the premises stated for the spatial games studied in the preceding section 3 are preserved, except that the players are connected at random, without any spatial structure. The payoffs of player A in a simulation of the QPD and the QHD games on a random network in Fig. 9 are to be compared to those in a spatial simulation shown in Fig. 6. Regarding the QPD(5,3,1,0) game it is to be remarked that the mutual defection regime persists beyond the γ • = 0.685 landmark featuring the spatial simulation, and that the behaviour of the payoffs in the transition interval is rather erratic : some simulations render low payoff, other ones render high payoffs, in particular those just after the mutual defection regime that rocket up to around 4.0 . In the spatial simulations of the QPD, the payoffs become stabilized in the transition interval (e.g., in Fig. 2) because both (Q, D) and (D, Q) coexist (as explained when commenting Fig. 3 ). In contrast 11

to this, in the network simulations of the QPD, in the transition interval either (Q, D) or (D, Q) prevails. If (Q, D) prevails, player A get a high payoff (and player B get a low payoff), and the contrary happens if (D, Q) prevails. The just described considerations about the CA-NW contrast in the behaviour of the payoffs in the transition interval of the QPD, apply for the QHD before the emergence of the (Q, Q) pair in NE. Thus, in the network QHD simulation in the right panel of Fig. 9, (Q, D) prevails, so that at δ = 1 the payoff of player A grows as pA = S +(T−S) sin2 γ = 1+4 sin2 γ, from pA = 1 at γ = 0 up to pA = 3 at γ = π/2 , whereas the (not shown in Fig. 9) payoff of player B decays at δ = 1 as pB = T − (T − S) sin2 γ = 5 − 2 sin2 γ, from pA = 5 at γ = 0 up to pA = 3 at γ = π/2 . Anyhow, the behaviour of the network simulations in the QHD before the emergence of mutual Q are dependent of the initial random assignments of the (θ, α) parameters, so that (D, Q) instead of (Q, D) may emerge before γ • . Let us conclude the comments regarding the QPD and QHD stressing that simulations on networks are influenced by the the initial conditions, whereas the spacialization (immediate previous section) induces the stabilization in the behaviour of the asymptotic strategies and consequently in the payoffs, so that the the asymptotic results are free of the initial conditions stated in the simulation. This is so much as in the seminal paper [27] dealing with the spatial simulation of the classical PD game. HD A

4

4

3.5

3.5

3

3

2.5

2.5

2

pA

pA

PD A

1.5

2 1.5

1

1

0.5

0.5

0

0

1

1 π/2 0.5

δ

γ∗

γ•

3π/8

π/8 0

0

π/2 γ•

0.5

π/4

δ

γ

3π/8

π/4 π/8 0

0

γ

Figure 9: The QPD(5,3,1,0) and QHD(5,3,0,1) games with variable γ and δ in a random network. Mean payoffs of the player A at T=100. Left : QPD, Right : QHD . The general form of both surface payoffs for the QSD(3,2,1,-1) in Fig. 10 is very similar to those in Fig. 7, albeit two relevant differences are to be remarked. First, the critical value of the entanglement γ • that indicates the emergence of (Q, Q) in NE, emerges in Fig. 10 close after the π/4, but not exactly at this middle level of gamma as happens in the spatial simulations of Fig. 7. Second, before γ • both players get lower payoffs in the simulations in networks, particularly the Samaritan player A that gets payoffs around -0.4 in simulations with full δ and γ < γ • . The general form of the surface payoffs of both QBOS(5,1)-players in Fig. 11 notably differs from to that in Fig. 8. In the network simulations in Fig. 11 the discontinuities in the values of the payoffs that affect in Fig. 8 only to player A with low entanglement, appear in Fig. 11 affecting to both players and almost in the whole range of variation of γ . As a rule, with low entanglement player A get higher payoffs than player B which in turn tends to receive higher payoffs when the entanglement increases (this much as in the spatial simulations of Fig. 8). The general payoff features observed in the simulation shown in Fig. 11 are preserved with different initial random assignments of the (θ, α) parameters, though the details are altered in a detectable way when varying said initial conditions,

12

SD A

SD B

3

3

2.5

2.5

pB

2 1.5

pA

2 1.5 1

1

0.5

0.5

0

0

−0.5

−0.5

1

1 γ• 0.5

π/2

γ•

3π/8

0.5

π/4 π/8

δ

0

π/8

δ

γ

0

π/2 3π/8

π/4 0

γ

0

Figure 10: The QSD(3,2,1,-1) with variable γ and δ in a random network. Mean payoffs at T=100. Left : Player A, Right : Player B . as it is shown in [30] at δ = 1.0 with five different random assigments of the (θ, α) parameters. BOS A

BOS B

4

3

3

p

pB

5

4

A

5

2

2

1

1

0

0

1

1 π/2

δ

π/2

3π/8

0.5 π/8 0

0

3π/8

0.5

π/4

δ

γ

π/4 π/8 0

0

γ

Figure 11: The QBOS(5,1) with variable γ and δ in a random network. Mean payoffs at T=100. Left : Player A, Right : Player B . Quantum games on networks is currently a fairly active are of reseach. Likely due not on only to its purely scientific interest but also due to the extraordinary importance that networks have nowadays. To the best of our knowledge, the articles produced in this area are mainly devoted to the PD with full entanglement (e.g., [31, 32, 33]). References considering full entanglement may also be found in the literature (e.g., [34, 35]), but in this case typically involving only the C, D and Q strategies (much in the realm of [27]), not the whole set of available strategies. In our opinion, other games than the PD, variable entanglement and quantum general strategies also deserve to be considered in the study of quantum games on networks. A kind of general approach that we plan to implement in a future work.

13

5

Three quantum parameter strategies.

In this section, both players follow general SU(2) strategies, i.e., three-parameter (3P) strategies with the β parameter active in the U structure given in Eq. (2) . Figures 12 and 13 deal with spatial simulations of the QPD(5,3,1,0) game in their left frames, and the QHD(5,3,0,1) game in the right frames when the players are allowed to follow 3P strategies. Figure 12 concerns only to player A in a simulation with both γ and δ variable, whereas Fig. 13 concerns to both players in five simulations with fixed δ = 1.0 . PD A

HD A

p

A

3.5

3.5

3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

1

1 π/2 γ#

0.5

3π/8

π/2 3π/8

0.5

π/4

π/4

π/8 δ

0

0

π/8 γ

0

0

Figure 12: The spatial QPD(5,3,1,0) and QHD(5,3,0,1) games with three quantum parameter strategies. Mean payoffs at T=100 with variable γ and δ. Left : QPD, Right : QHD . At variance with what happens in the two parameter scenario (Fig.2), in the 3P-QPD the mean payoff increases fairly monotonically. In [36] pure strategies in Nash equilibrium are described r P −S 1
in the 3P-QPD(δ=1) scenario below the threshold γ # = arcsin = =0.612 , T+P −R−S 3 providing the same payoffs for both players : pA,B = P + (R − P ) sin2 γ. This equation applies in fact also after γ # in the simulation, with no relevant discontinuity at γ # , so that the payoff of both players well fit pA,B = 1 + 2 sin2 γ, from pA,B (0.0, 1.0) = P = 1 up to pA,B (1.0, 1.0) = R = 3 . Thus, it is conjecturable that the dynamics in this scenario resorts somehow to the mixed strategies in Nash equilibrium described in [21] . The left frames of Fig.13 deal with five spatial simulations of the 3P-QPD(5,3,1,0)(δ = 1.0) game with variable γ at T = 100. Its upper-left frame shows the actual mean and mean-field payoffs, and its lower-left frame the mean values of the quantum parameters. In this scenario, both θA and θB are set to π/2 (which makes α irrelevant), except in the proximity of γ # where some turbulence is observed. With low entanglement, the β parameter of both players is not far from its middle value, and  set to exactly π/4 with high entanglement. In short, the 1 0 1+i Q = U (π, α, π/4) = √ strategy dominates the scene in the 3P-QPD. The pair 0 2 1−i  2  sin γ 0 Q,Q (Q, Q) generates the joint probability distribution Π = , and consequently in 0 cos2 γ the 3P-QPD(δ=1) scenarios of Figs.12 and 13 , the afore mentioned pA,B = P + (R − P ) sin2 γ = 1 + 2 sin2 γ equalitarian payoff emerges in the simulation. In contrast with what happens in the 3P-QPD, the right frames of Figs. 12 and 13 show that the (Q, Q) pair only emerges with high entanglement in the 3P-QHD. Thus, the lower-right frame 14

p 3.5

(θ,α,β)-QPD(5,3,1,0)-CA N=200 T=100

p

δ = 1.0

p p A B

3.0

δ = 1.0 p∗ p∗ A B

3.5

p∗ p∗ A B

2.5

p p A B

3.0

2.0

2.5

1.5

2.0 1.5

1.0 0.5

3P-QHD(5,3,0,1)-CA N=200 T=100

4.0

γ

#

π/8

π/4

3π/8

π/2

1.0

γ

σ pAσ pB

0.5

θ, α, β θA θB

π

π/8

π/4

3π/8

π/2

γ

θ, α, β π

θA θB

3π/4 3π/4

π/2 π/2

βAβB αB αA

π/4

γ π/8

#

π/4

3π/8

π/2

βAβB αB αA

π/4

γ

π/8

π/4

3π/8

π/2

γ

Figure 13: The δ = 1.0 QPD and QHD with three quantum parameter strategies and variable γ . Five spatial simulations at T=100 . Left : QPD(5,3,1,0). Right : QHD(5,3,0,1) . Upper : Mean payoffs. Lower : Mean quantum parameters. of Fig. 13 indicates that, although the α and β parameters oscillate close the middle value π/2 (as in the 3P-QPD), the θ parameter values only approach π with very high γ . The upper-right frame of Fig. 13 in turn indicates that spatial effects (absent in the 3P-QPD) induce mean-field payoffs that vary erratically, mainly underestimating the actual mean payoffs in the 3P-QHD(δ=1), while the standard deviations (σ) of the actual payoffs decrease in a fairly smooth way down to zero as γ increases. Also noticeable is that in agreement with what happens in the 2P-QHD simulation in Fig.6, the actual mean payoffs of both players in the 3P-QHD depart at γ = 0.0 from p(0.0, 1.0) ' 2.25 , close to 2.56, i.e., to the average of the payoffs achieved in the three NE in the HD classical game, instead of from p(0.0, 1.0) = 1.0 as happens in the 3P-QPD. Figure 14 deals with a spatial simulation of the QSD(3,2,1,-1) game with three quantum parameter strategies, instead of with two parameters as in Fig. 10 . At variance with what happens in the latter, the actual mean payoffs of both players monotonically increase their values as the entanglement increases Fig.14, with no emergence of the (Q, Q) pair in NE. In the δ = 1.0 scenario, the payoff of player A from increases from approximately zero up approximately 1.5, and that of player B from approximately 1.5 up to approximately 2.25 . Thus, player B overrates player A all along the γ variation, albeit in a lower degree as γ grows. It is shown in [12] that in the δ = 1.0 QSD(3,2,1,-1) scenario, heavy spatial effects arise, given rise to a rather erratic behaviour of the mean-field estimations. The surface payoffs of both players in the spatial 3P-QBOS(5,1) simulation shown in Fig.15 appear to be much smoother than those in the 2P-QBOS(5,1) simulations in Fig.8 . Also relevant is that both players achieve similar payoffs in Fig.15, at variance with the trends favouring to one of the players depending on the entanglement level that emerge in Fig.8 . The strong spatial effects 15

SD A

SD B

2.5

2.5

2

2

1.5

1.5

B

1

p

pA

1 0.5

0.5

0

0

−0.5

−0.5

1

1 π/2

π/2

3π/8

0.5

3π/8

0.5

π/4

π/4

π/8 δ

0

π/8

0

δ

γ

0

0

γ

Figure 14: The spatial QSD(3,1,1,-1) game with three quantum parameter strategies . Mean payoffs at T=100 with variable γ and δ. Left : Player A, Right : Player B . that arise in the simulations in the δ = 1.0 scenario of Fig.15 are described in [37] . BOS B

4

4

2

2

pB

pA

BOS A

0 1 0.5 δ

0 0

π/4

π/8

3π/8

0 1

π/2

0.5 δ

γ

0 0

π/4

π/8

3π/8

π/2

γ

Figure 15: The spatial QBOS(5,1) game with three quantum parameter strategies. Mean payoffs at T=100 with variable γ and δ. Left : Player A, Right : Player B .

6

Conclusions.

This article studies iterated two-person quantum games with Werner-like states via simulation. Thus, in every iteration, every player of a large collective of players interact with their nearest neighbours in two steps: First playing with their partners and then adopting the strategy of their best paid mate neighbour for the next round. The implementation of such imitation of the best evolving rule proves to be a very useful tool to analyze the collective behaviour of two-person games via simulation. It is described how high entanglement enables the emergence of new Nash equilibriums. In three of the game-types here studied (Prisoner’s Dilemma, Hawk and Dove, Samaritan’s Dilemma) the new Nash equilibriums achieved with highly correlated games maximize the sum of the payoffs of both players, i.e., they provide its (unique) so called social welfare solution. The case of the fourth 16

game-type studied here, the Battle of the Sexes, appears to be the most challenging one at this respect because it has two social welfare solutions. The implementation of Werner-like states made in this article enables to scrutinize the way in which the features of the studied games vary from the quantum entagled scenario up to that of independent players with uniform random strategies.

Acknowledgments This work has been funded by the Spanish Grant MTM2015-63914-P. Part of the computations of this work were performed in FISWULF, an HPC machine of the International Campus of Excellence of Moncloa, funded by the UCM and Feder Funds.

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[13] Ozdemir,S.K.,Shimamura,J.,Morikoshi,F.,Imoto,N.(2004). Dynamics of a discoordination game with classical and quantum correlations. Physics Letters A, 333, 218-231. [14] Rasmussen,E.(2001). Games and Information, An Introduction to Game Theory. Blackwell,Oxford. [15] Binmore,K.(2007). Game Theory: A very short introduction. Oxford UP. [16] Owen,G.(1995). Game Theory. (Academic Press). [17] Binmore,K.(1998). Just Playing: Game Theory and the Social Contract II. MIT Press, (ISBN 0-262-02444-6). [18] Benjamin,S.C.,Hayden,P.M.(2001). Comment on “Quantum Games and Quantum Strategies”. Phys. Rev.Lett., 87,069801. [19] Benjamin,S.C.,Hayden,P.M.(2001). Multiplayer quantum games. Phys. Rev. A, 64,030301. [20] Eisert,J.,Wilkens,M.,Lewenstein.M.(2001). Comment on ”Quantum games and quantum strategies”-Reply. Phys. Rev. Lett. 87, 069802. [21] Eisert,J.,Wilkens,M.(2000). Quantum Games. Journal of Modern Optics, 47,14-15,2543-2556. [22] Alonso-Sanz,R.(2017). Spatial correlated games. Royal Society Open Science, 4,11,171361. [23] Iqbal,A.,Chappell,J.M.,Abbott,D.(2016). On the equivalence between non-factorizable mixedstrategy classical games and quantum games. Royal Society Open Science, 3,150477. [24] Werner,R.F.(1989). Quantum states with Einstein-Podolsky-Rossen correlation admitting a hidden variable model. Physical Review A, 40,8,4277-4281. [25] Schiff,J.L.(2008). Cellular automata : A discrete view of the world. Wiley. [26] Ellison,G.(1993). Learning, local interaction, and coordination. Physical Review Letters, 685,1047-1071. [27] Nowak,M.A.,May,R.A. (1992). Evolutionary games and spatial chaos. Nature, 359,826-829. [28] Alonso-Sanz,R.(2014). Variable entangling in a quantum battle of the sexes cellular automaton. ACRI-2014, LNCS,8751,125-135. [29] Alonso-Sanz,R.(2012). A quantum battle of the sexes cellular automaton. Proc. R. Soc. A, 468,3370-3383. [30] Alonso-Sanz,R.(2018). On collective quantum games. Quantum, (submitted). [31] Li,Q.,Chen,M.,Perc,M.,Iqbal,A.,Abbott,D.(2013). Effects of adaptive degrees of trust on coevolution of quantum strategies on scale-free networks. Scientific reports, 3,2949. [32] Li,Q.,Iqbal,A.,Perc,M.,Chen,M.,Abbott,D.(2013). Coevolution of quantum and classical strategies on evolving random networks. PloS one, 8(7),e68423. [33] Li,Q.,Iqbal,A.,Chen,M.,Abbott,D.(2012). Evolution of quantum strategies on a small-world network. The European Physical Journal B, 85(11), 376. 18

[34] Li,A.,Yong,X.(2015). Emergence of super cooperation of prisoners dilemma games on scalefree networks. PloS one, 10(2),e0116429. [35] Li,A.,Yong,X.(2014). Entanglement guarantees emergence of cooperation in quantum prisoner’s dilemma games on networks. Scientific reports, 4,6286. [36] Du,J.F.,Li,H.,Xu,X.D.,Zhou,X.,Han,R.(2003). Phase-transition-like behaviour of quantum games. J. Phys. A : Math. and Gen., 36, 23, 6551-6562. [37] Alonso-Sanz,R.(2013). On a three-parameter quantum battle of the sexes cellular automaton. Quantum Information Processing, 12,5,1835-1850.

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