Duality, Phase Structures and Dilemmas in Symmetric Quantum Games

Izumi Tsutsui; Tsubasa Ichikawa

REVIEW

Duality, Phase Structures and Dilemmas in Symmetric Quantum Games

Tsubasa Ichikawa, Izumi Tsutsui

Quantum Physics · Submitted 22 Feb 2006 · 21 references

Pith's verdict: A useful, mostly sound classification of edge QNE for symmetric 2x2 quantum games; the dilemma-resolution claims overreach because the non-edge analysis assumes T-symmetric equilibria without proof.

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CitationIzumi Tsutsui, Tsubasa Ichikawa. "Duality, Phase Structures and Dilemmas in Symmetric Quantum Games." arXiv:quant-ph/0602178 (2006). https://pith.science/paper/YAM7WCLX

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Editor's note

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Read the paper. The core contribution is genuinely useful: for 2x2 symmetric games in the pure-state scheme, the authors map the correlation family to a rectangle on the (G'_+, G'_−) plane and show that the edge QNE phases fall into eight types determined by the classical payoff matrix. The duality map between S- and T-symmetric games is clean and does real work. The edge-QNE analysis — Hessian conditions, rectangle geometry, BoS/PD/SH labels — checks out modulo the sign typo. The self-citation to [7] is legitimate: the scheme comes from there, but the general classification is new.

Where it gets shaky is the non-edge analysis. In Section III.B they write the T-symmetric ansatz and derive conditions, but they never prove that every non-edge QNE is T-symmetric. For a T-symmetric game, the twisted partner of a QNE is also a QNE, but there is no reason an equilibrium pair has to be invariant under the twist. That matters because the PD conclusion — 'for the standard PD there is no non-edge QNE' — is drawn from conditions obtained under that ansatz. An asymmetric non-edge QNE could in principle give both players at least the edge-QNE payoff and spoil the claimed full resolution. The same gap hangs over the SH discussion, where they only check standard values and even concede non-edge QNE 'could alter the nature of the dilemmas.' The BoS statement that no non-edge QNE exists without the ansatz is asserted, not shown.

Also, Eq. (3.16) gives the BoS domain as H_+ < 0, H_− < 0, while Table II and the BoS section require H_+, H_− > 0. Likely a sign typo, but it is exactly where a reader will check the classification.

None of this kills the paper. The edge-QNE phase classification and the duality are solid, and the non-edge gap is addressable: prove or disprove the T-symmetric restriction (or at least bound asymmetric equilibria) for the standard PD/SH payoffs, fix the sign, and the dilemma statements become trustworthy.

Who is this for? People working on quantum game theory and 2x2 quantization schemes; the phase diagram is a handy reference even if the broader significance is moderate. I would send it to a serious referee, with the non-edge question flagged as the main thing to demand.

Referee report

Significance

If the results are correct, the paper provides a useful and fairly complete classification of equilibrium phase structures for symmetric 2x2 quantum games in this particular scheme, and it sharpens the ongoing discussion of scheme dependence in quantum game theory. The derivations in Sections II and III are explicit, the duality map is elegant, and the eight-type classification is derived rather than imposed, with no fitted parameters; the paper also gives falsifiable statements connecting classical payoff matrices to the possibility of dilemma resolution. The main limitation is that the non-edge equilibrium analysis relies on an unproven symmetry ansatz, so the headline claims about full resolution of the PD dilemma and generic resolution of the SH dilemma are conditional unless that gap is closed.

Major objections (3)

[Section III.B, Eq. (3.26)] The non-edge QNE analysis is performed under the T-symmetric ansatz |α*,β*> = |bar β*, bar α*>, but the paper gives no argument that every non-edge QNE of a T-symmetric game must be invariant under the twist. This is load-bearing, because the PD claim in Section IV.B that there is no non-edge QNE for standard payoffs is inferred from conditions (3.33) and (3.34), which were derived from this ansatz, and the SH discussion in Section IV.C explicitly concedes that non-edge QNE 'could alter the nature of the dilemmas.' If an asymmetric non-edge QNE exists with payoffs at least as good as the edge QNE, the conclusion that the classical PD dilemma disappears after quantization would not follow. The authors should either prove that every QNE can be taken to be T-symmetric, or analyze general non-edge equilibria, or explicitly state the dilemma-resolution conclusions as conditional on the ansatz.
[Section III.A, Eq. (3.16)] The stated BoS condition H+ < 0, H- < 0 is inconsistent with Table II, where both |0,0> and |1,1> are QNE exactly when H+ > 0 and H- > 0, and it is also inconsistent with Figure 1 and the BoS line-segment argument in Section IV.A. As printed, (3.16) describes the region where neither of the diagonal edge strategies is a QNE, so the inequalities should have the opposite signs.
[Section IV.A, after Eq. (4.4)] The paper asserts that, without making use of the ansatz (3.26), one can confirm by the Hessian condition (3.9) that there is no non-edge QNE for the BoS except the one at γ2=0,π, but no derivation is shown for this assertion. Since the BoS no-go result is a central negative claim of the paper, this missing argument needs to be supplied, either as a proof in the text or as a precise reference to a derivation.

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The paper

The paper claims

For two-player, two-qubit symmetric games the paper classifies all equilibrium structures into eight types fixed by the classical payoff matrix, and shows quantization removes the Prisoner's Dilemma and Stag Hunt dilemmas but not the…

The reading

This paper aims to show that symmetric quantum games built from two-player, two-qubit pure states have a small and predictable menu of equilibrium phase structures, and that this menu determines when quantum correlations can resolve classical dilemmas. The authors work in a quantization scheme where each player's strategy is a pure qubit state, the joint strategy is any pure state of two qubits produced by the correlation operator $J(\gamma)=e^{i\gamma_1 S/2}e^{i\gamma_2 T/2}$, and the payoff operators are fixed uniquely by the classical payoff matrix. They prove that the two natural classes of symmetric games, S-symmetric and T-symmetric, are dual to one another under a unilateral flip of one player's strategy labels, so classifying one class classifies the other. The classification yields exactly eight phase structures for the edge equilibria, determined by the signs and differences of the classical payoffs. The payoff of the result is concrete: for the Prisoner's Dilemma and Stag Hunt, suitable correlations move the game into a dilemma-free phase, while for the Battle of the Sexes every correlation leaves the dilemma in place, with maximal entanglement only equalizing the two equilibria.

What carries the argument

The load-bearing object is the duality map $C_A|i,j\rangle=|\bar{i},j\rangle$ together with the $(G'_+,G'_-)$ plane. The map exchanges the S-symmetric and T-symmetric game classes through $C_A S C_A=T$ and $C_A T C_A=S$, so results for one class transfer to the other, with the correlation parameters swapping as $(\gamma_1,\gamma_2)\mapsto(\gamma_2,\gamma_1)$. On the plane, the four edge Hessian conditions become the two straight lines $H_\pm(\gamma)=\tau(A)\pm[G'_+(\gamma)+G'_-(\gamma)]=0$, where $G'_+=(A_{00}-A_{11})\cos\gamma_2$ and $G'_-=(A_{01}-A_{10})\cos\gamma_1$. The correlation family of a game forms a rectangle with horizontal length $2|A_{00}-A_{11}|$ and vertical length $2|A_{01}-A_{10}|$, and the eight phase structures are the eight possible positions of that rectangle relative to the two stability lines for $\tau(A)>0$ and $\tau(A)<0$. For non-edge equilibria, the paper adopts the T-symmetric ansatz $|\alpha^*,\beta^*\rangle=|\bar{\beta}^*,\bar{\alpha}^*\rangle$, which reduces the stationarity conditions to two algebraic equations whose solutions extend the classical mixed Nash equilibrium.

Extended reading notes

Core claim

The paper establishes that, for two-player, two-qubit symmetric games quantized in this pure-state scheme, the quantum Nash equilibria on the four classical strategy edges have exactly eight possible phase structures, fixed by the four classical payoff entries. Concretely, the correlation family of any symmetric game traces a rectangle in the $(G'_+,G'_-)$ plane, and the Hessian stability lines $H_\pm(\gamma)=0$ cut that rectangle into domains labelled by which edge equilibrium survives; the sign of $\tau(A)=A_{00}-A_{01}-A_{10}+A_{11}$ and the rectangle's position relative to those stability lines leave eight distinct patterns. The same classification controls dilemma resolution: in the Battle of the Sexes the rectangle lies entirely inside the two-equilibrium dilemma domain, so no correlation removes the dilemma (maximal entanglement only makes the two equilibria payoff-identical); in the Prisoner's Dilemma and Stag Hunt the rectangle reaches dilemma-free domains, and the standard Prisoner's Dilemma admits no non-edge equilibrium to spoil the resolution, while in the Stag Hunt the non-edge equilibrium occupies only narrow regions and generically does not spoil it. Because the duality map converts S-symmetric games into T-symmetric ones with the same effective payoffs, the eight patterns cover both symmetry classes at once.

Load-bearing premise

The analysis of non-edge equilibria assumes without proof that every non-edge quantum Nash equilibrium of a symmetric game is T-symmetric, so the dilemma-resolution conclusions could be incomplete if asymmetric non-edge equilibria exist.

Editorial extensions

If this is right

For any symmetric game, the classical payoff matrix fixes the full menu of edge-equilibrium phases; no amount of entanglement can produce a phase outside that menu.
The Battle of the Sexes dilemma is robust under this quantization: every correlation keeps the two diagonally split equilibria, and maximal entanglement merely makes their payoffs equal.
For the Prisoner's Dilemma with the standard payoffs, there is no non-edge quantum equilibrium, so reaching a dilemma-free domain of the rectangle is a complete resolution rather than an edge-only artefact.
For the Stag Hunt, the dilemma-free domains are reached by edge equilibria, and the non-edge equilibrium present near the classical limit remains confined to narrow regions for generic payoff values.
Because of the duality map, every phase-structure statement about T-symmetric games applies equally to S-symmetric games after flipping one player's strategy labels, and conversely.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The eight-phase classification provides a natural benchmark for comparing quantization schemes: any scheme that claims a different equilibrium menu for the same classical game must differ in which parts of the two-qubit state space it allows, or in the ordering of strategy choice and correlation.
A concrete testable consequence is that in an experimental two-qubit realization of the Battle of the Sexes, the payoff difference between the two equilibria should shrink to zero at maximal entanglement but should never switch the game into a single-equilibrium phase.
The same rectangle-and-stability-line method should extend to two-player games on $d$-dimensional systems, with the four edge lines replaced by the Hessian hyperplanes of $d^2$ edge states; the number of phase structures would then grow with $d$ while the underlying classification logic stays intact.

Original abstract

Symmetric quantum games for 2-player, 2-qubit strategies are analyzed in detail by using a scheme in which all pure states in the 2-qubit Hilbert space are utilized for strategies. We consider two different types of symmetric games exemplified by the familiar games, the Battle of the Sexes (BoS) and the Prisoners' Dilemma (PD). These two types of symmetric games are shown to be related by a duality map, which ensures that they share common phase structures with respect to the equilibria of the strategies. We find eight distinct phase structures possible for the symmetric games, which are determined by the classical payoff matrices from which the quantum games are defined. We also discuss the possibility of resolving the dilemmas in the classical BoS, PD and the Stag Hunt (SH) game based on the phase structures obtained in the quantum games. It is observed that quantization cannot resolve the dilemma fully for the BoS, while it generically can for the PD and SH if appropriate correlations for the strategies of the players are provided.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.84 · human participants 0.02 · controlled comparison 0.08 · randomized assignment 0.01 · states quantitative result 0.94 · compares with prior results 0.21 · mentions code or data availability 0.02.

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

Symmetric quantum games for 2-player, 2-qubit strategies are analyzed in detail by using a scheme in which all pure states in the 2-qubit Hilbert space are utilized for strategies. We consider two different types of symmetric games exemplified by the familiar games, the Battle of the Sexes (BoS) and the Prisoners' Dilemma (PD). These two types of symmetric games are shown to be related by a duality map, which ensures that they share common phase structures with respect to the equilibria of the strategies. We find eight distinct phase structures possible for the symmetric games, which are determined by the classical payoff matrices from which the quantum games are defined. We also discuss the possibility of resolving the dilemmas in the classical BoS, PD and the Stag Hunt (SH) game based on the phase structures obtained in the quantum games. It is observed that quantization cannot resolve the dilemma fully for the BoS, while it generically can for the PD and SH if appropriate correlations for the strategies of the players are provided.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Symmetric quantum games for 2-player, 2-qubit strategies are analyzed in detail by using a scheme in which all pure states in the 2-qubit Hilbert space are utilized for strategies.	method 0.57	0.91	0.04	asserted 1.00
s1	We consider two different types of symmetric games exemplified by the familiar games, the Battle of the Sexes (BoS) and the Prisoners' Dilemma (PD).	contribution 0.59	0.76	0.03	asserted 0.98
s2	These two types of symmetric games are shown to be related by a duality map, which ensures that they share common phase structures with respect to the equilibria of the strategies.	result 0.78	0.87	0.19	asserted 1.00
s3	We find eight distinct phase structures possible for the symmetric games, which are determined by the classical payoff matrices from which the quantum games are defined.	result 1.00	0.95	0.18	asserted 1.00
s4	We also discuss the possibility of resolving the dilemmas in the classical BoS, PD and the Stag Hunt (SH) game based on the phase structures obtained in the quantum games.	contribution 0.78	0.79	0.16	hedged possibility 0.99
s5	It is observed that quantization cannot resolve the dilemma fully for the BoS, while it generically can for the PD and SH if appropriate correlations for the strategies of the players are provided.	result 0.95	0.86	0.52	asserted 0.93

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

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full text
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1: Phase structures of QNE in terms of edge strategies:

**Figure 2.** FIG. 2: Four patterns of rectangles which are possible in re

**Figure 3.** FIG. 3: Possible patterns of allowed regions by (3.33) in the

**Figure 4.** FIG. 4: Phase structure of edge QNE in the BoS game. The

**Figure 6.** FIG. 6: Phase structures of edge QNE (left) and non-edge

Showing 5 of the paper's 6 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:quant-ph/0602178 by the authors.

References

Works cited (21)

[7]

shows that there exists a new scheme in which the dilemma can be resolved even with a full strategic space. Similarly, the resolution of the dilemma in the BoS has been argued using diﬀerent reasonings depending on the schemes [3, 8, 9] (for a generalized scheme, with no analy- sis on dilemmas, see [10]), casting a doubt on the genuine nature of the resol...

View on Pith
[1]

J. Eisert, M. Wilkens and M. Lewenstein, Quantum games and quantum strategies , Phys. Rev. Lett. 83 (1999) 3077-3080

View on Pith 1999
[2]

Similarly, for A00 > A 11 the mid- point ( G′ +, G ′ −) = ( Lh/ 2, 0) corresponds to J(0, π/ 2) un- der which the edge state |00⟩ becomes ( |00⟩+ i|11⟩)/ √ 2. The four corners of the rectangle, on the other hand, cor- respond to J(mπ, nπ ) for m, n = 0, 1, which are S, T , C, and I operations, and hence the resultant joint strategies are all unentangled. ...

View on Pith
[3]

J. Shimamura, S. K. ¨Ozdemir, F. Morikoshi and N. Imoto, Quantum and classical correlations between play- ers in game theory , Int. Journ. Quant. Inf. 2 (2004) 79-89

View on Pith 2004
[4]

J. Eisert and M. Wilkens, Quantum Games, J. Mod. Opt. 47 (2000) 2543-2556

View on Pith 2000
[5]

L. Marinatto and T. Weber, A quantum approach to static games of complete information , Phys. Lett. A272 (2000) 291-303

View on Pith 2000
[6]

S. C. Benjamin and P. M. Hayden, Comment on “Quan- tum Games and Quantum Strategies” , Phys. Rev. Lett. 87 (2001) 069801

View on Pith 2001
[8]

J. Du, X. Xu, H. Li, X. Zhou and R. Han, Playing Pris- oner’s Dilemma with Quantum Rules , Fluct. and Noise Lett. 2 (2002) 189-203

View on Pith 2002
[9]

A. Nawaz and A. H. Toor, Dilemma and Quantum Battle of Sexes , J. Phys. A: Math. Gen. 37 (2004) 4437-4443

View on Pith 2004
[10]

T. Cheon and I. Tsutsui, Classical and Quantum Con- tents of Solvable Game Theory on Hilbert Space , Phys. Lett. A348 (2006) 147-152

View on Pith 2006
[11]

in a diﬀerent scheme. This suggests that the phase structures may exhibit a scheme-independent, intrinsic features of quantum games under consideration. The aim of the present paper is to support this idea by providing a convenient tool to analyze quantum games in general terms. We consider 2-player, 2-qubit strategy games, which are the simplest nontrivi...

View on Pith
[12]

L. Marinatto and T. Weber, Reply to “Comment on: A Quantum Approach to Static Games of Complete Infor- mation ”, Phys. Lett. A277 (2000) 183-184

View on Pith 2000
[13]

A. Nawaz and A. H. Toor, Generalized Quantization Scheme for Two-Person Non-Zero-Sum Games , J. Phys. A: Math. Gen. 37 (2004) 11457-11463

View on Pith 2004
[14]

J. Du, X. Xu, H. Li, X. Zhou and R. Han, Experimental realization of quantum games on a quantum computer , Phys. Rev. Lett. 88 (2002) 137902

View on Pith 2002
[15]

A. P. Flitney and D. Abbott, Advantage of a quantum player over a classical one in 2x2 quantum games , Poc .R. Soc. (London) A459 (2003) 2463-2474

View on Pith 2003
[16]

E. Rasmusen, An Introduction to Game Theory , Cam- bridge Univ. Press, Cambridge, 1989

View on Pith 1989
[17]

J. F. Nash, Equilibrium points in N -person games, Proc. Nat. Acad. Sci. U.S.A. 36 (1950) 48-49

View on Pith 1950
[18]

C. F. Lee and N. Johnson, Quantum Game Theory , Phys. Rev. A67 (2003) 022311

View on Pith 2003
[19]

S. C. Benjamin, Comment on “A quantum approach to static games of complete information ” , Phys. Lett. A277 (2000) 180-182

View on Pith 2000
[20]

Apart from the irrelevant freedoms concerning the over - all phase and the normalization, the dimensionality of the joint strategy space is dim H − 2 = 6 in real variables. Since the individual strategies |α ⟩A and |β ⟩B are speciﬁed by 2 + 2 = 4 parameters ( e.g., see (3.3)), the correlation factor must have another 2 parameters to cover the full joint s...

View on Pith
[21]

In the present paper, we consider quantum joint strate- gies given by pure states only. The space of pure states is not convex, and hence the Nash theorem [14], which en- sures the existence of NE for a classical game with mixed strategies, is no longer available. The existence of QNE in quantum game is, therefore, non-trivial [15]

View on Pith

How the literature uses it

Papers whose main result relates to this one's

How each label was checked · JSON

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States a conflicting result 1 paper

For a particular choice of initial entangled quantum state it is shown that the classical dilemma of the battle of sexes can be resolved and a unique solution of the game can be obtained.

Dilemma and Quantum Battle of Sexes · 0.85

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [7]

shows that there exists a new scheme in which the dilemma can be resolved even with a full strategic space. Similarly, the resolution of the dilemma in the BoS has been argued using diﬀerent reasonings depending on the schemes [3, 8, 9] (for a generalized scheme, with no analy- sis on dilemmas, see [10]), casting a doubt on the genuine nature of the resol...

View on Pith

[2] [1]

J. Eisert, M. Wilkens and M. Lewenstein, Quantum games and quantum strategies , Phys. Rev. Lett. 83 (1999) 3077-3080

View on Pith 1999

[3] [2]

Similarly, for A00 > A 11 the mid- point ( G′ +, G ′ −) = ( Lh/ 2, 0) corresponds to J(0, π/ 2) un- der which the edge state |00⟩ becomes ( |00⟩+ i|11⟩)/ √ 2. The four corners of the rectangle, on the other hand, cor- respond to J(mπ, nπ ) for m, n = 0, 1, which are S, T , C, and I operations, and hence the resultant joint strategies are all unentangled. ...

View on Pith

[4] [3]

J. Shimamura, S. K. ¨Ozdemir, F. Morikoshi and N. Imoto, Quantum and classical correlations between play- ers in game theory , Int. Journ. Quant. Inf. 2 (2004) 79-89

View on Pith 2004

[5] [4]

J. Eisert and M. Wilkens, Quantum Games, J. Mod. Opt. 47 (2000) 2543-2556

View on Pith 2000

[6] [5]

L. Marinatto and T. Weber, A quantum approach to static games of complete information , Phys. Lett. A272 (2000) 291-303

View on Pith 2000

[7] [6]

S. C. Benjamin and P. M. Hayden, Comment on “Quan- tum Games and Quantum Strategies” , Phys. Rev. Lett. 87 (2001) 069801

View on Pith 2001

[8] [8]

J. Du, X. Xu, H. Li, X. Zhou and R. Han, Playing Pris- oner’s Dilemma with Quantum Rules , Fluct. and Noise Lett. 2 (2002) 189-203

View on Pith 2002

[9] [9]

A. Nawaz and A. H. Toor, Dilemma and Quantum Battle of Sexes , J. Phys. A: Math. Gen. 37 (2004) 4437-4443

View on Pith 2004

[10] [10]

T. Cheon and I. Tsutsui, Classical and Quantum Con- tents of Solvable Game Theory on Hilbert Space , Phys. Lett. A348 (2006) 147-152

View on Pith 2006

[11] [11]

in a diﬀerent scheme. This suggests that the phase structures may exhibit a scheme-independent, intrinsic features of quantum games under consideration. The aim of the present paper is to support this idea by providing a convenient tool to analyze quantum games in general terms. We consider 2-player, 2-qubit strategy games, which are the simplest nontrivi...

View on Pith

[12] [12]

L. Marinatto and T. Weber, Reply to “Comment on: A Quantum Approach to Static Games of Complete Infor- mation ”, Phys. Lett. A277 (2000) 183-184

View on Pith 2000

[13] [13]

A. Nawaz and A. H. Toor, Generalized Quantization Scheme for Two-Person Non-Zero-Sum Games , J. Phys. A: Math. Gen. 37 (2004) 11457-11463

View on Pith 2004

[14] [14]

J. Du, X. Xu, H. Li, X. Zhou and R. Han, Experimental realization of quantum games on a quantum computer , Phys. Rev. Lett. 88 (2002) 137902

View on Pith 2002

[15] [15]

A. P. Flitney and D. Abbott, Advantage of a quantum player over a classical one in 2x2 quantum games , Poc .R. Soc. (London) A459 (2003) 2463-2474

View on Pith 2003

[16] [16]

E. Rasmusen, An Introduction to Game Theory , Cam- bridge Univ. Press, Cambridge, 1989

View on Pith 1989

[17] [17]

J. F. Nash, Equilibrium points in N -person games, Proc. Nat. Acad. Sci. U.S.A. 36 (1950) 48-49

View on Pith 1950

[18] [18]

C. F. Lee and N. Johnson, Quantum Game Theory , Phys. Rev. A67 (2003) 022311

View on Pith 2003

[19] [19]

S. C. Benjamin, Comment on “A quantum approach to static games of complete information ” , Phys. Lett. A277 (2000) 180-182

View on Pith 2000

[20] [20]

Apart from the irrelevant freedoms concerning the over - all phase and the normalization, the dimensionality of the joint strategy space is dim H − 2 = 6 in real variables. Since the individual strategies |α ⟩A and |β ⟩B are speciﬁed by 2 + 2 = 4 parameters ( e.g., see (3.3)), the correlation factor must have another 2 parameters to cover the full joint s...

View on Pith

[21] [21]

In the present paper, we consider quantum joint strate- gies given by pure states only. The space of pure states is not convex, and hence the Nash theorem [14], which en- sures the existence of NE for a classical game with mixed strategies, is no longer available. The existence of QNE in quantum game is, therefore, non-trivial [15]

View on Pith