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arxiv: 2606.08227 · v1 · pith:UNQO3TMTnew · submitted 2026-06-06 · 🪐 quant-ph · cs.GT· econ.TH· math-ph· math.MP

Entanglement in the Quantum Volunteer's Dilemma

Noah Dane Hebdon , Dax Enshan Koh This is my paper

Pith reviewed 2026-06-27 19:36 UTC · model grok-4.3

classification 🪐 quant-ph cs.GTecon.THmath-phmath.MP
keywords quantum volunteer's dilemmaentanglement thresholdNash equilibriaquantum gamesEisert-Wilkens-Lewensteinsymmetric equilibriatunable entanglementresource-limited quantum devices
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The pith

Symmetric Nash equilibria in the quantum volunteer's dilemma exist above an analytically derived entanglement threshold rather than requiring maximal entanglement.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies a quantum volunteer's dilemma in which n players share an entangled state and each controls one qubit. It derives conditions on the tunable entanglement strength γ under which symmetric Nash equilibria appear for two families of strategy profiles. These equilibria yield higher payoffs than the classical game, but only when γ exceeds a size-dependent threshold that the authors compute in closed form. The finding shows that perfect entanglement is unnecessary for the quantum advantage to survive.

Core claim

Within the Eisert-Wilkens-Lewenstein framework, the generalized quantum volunteer's dilemma with entanglement parameter γ admits symmetric Nash equilibria whenever γ lies above an explicit threshold. The threshold is obtained analytically for the strategy profile valid when 2 ≤ n ≤ 9 and for the profile valid when n is even; equilibrium behavior therefore persists for any entanglement level above this value and disappears below it.

What carries the argument

The tunable entanglement parameter γ that parametrizes the shared two-qubit or n-qubit entangled state in the Eisert-Wilkens-Lewenstein quantum game.

Load-bearing premise

The analysis assumes the Eisert-Wilkens-Lewenstein framework in which each player manipulates one qubit of a shared entangled state controlled by γ.

What would settle it

Measuring whether symmetric Nash equilibria cease to exist exactly when γ falls below the derived threshold for a chosen n and strategy profile would confirm or refute the claimed dependence.

Figures

Figures reproduced from arXiv: 2606.08227 by Dax Enshan Koh, Noah Dane Hebdon.

Figure 1
Figure 1. Figure 1: FIG. 1. Lower bound on the entanglement parameter [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Lower bound on the entanglement parameter [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
read the original abstract

A well-known model in game theory, the Volunteer's Dilemma describes a group of $n$ players who decide whether to volunteer for a collective benefit at a personal cost, or to abstain and risk forfeiting the benefit altogether. A quantum version of this dilemma, developed within the Eisert-Wilkens-Lewenstein framework, allows each player to manipulate one qubit of a shared entangled state, leading to symmetric Nash equilibria with higher expected payoffs than in the classical game. Existing analyses, however, assume maximal entanglement. Within the same framework, we introduce a generalized Quantum Volunteer's Dilemma with a tunable entanglement parameter $\gamma$ and study the extent to which equilibrium behavior depends on the level of entanglement. We derive explicit conditions relating $\gamma$, the number of players, and the players' strategies under which symmetric Nash equilibria exist, focusing on two canonical strategy profiles: one for $2\leq n\leq 9$, and one for even $n$. We find that maximal entanglement is not required to sustain symmetric equilibria. Instead, equilibrium behavior persists above a threshold value, which we compute analytically in both cases. We also demonstrate that the threshold value directly depends on system size. This characterization is directly relevant for implementations on resource-constrained quantum devices, where entanglement is inherently limited.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript generalizes the Quantum Volunteer's Dilemma in the Eisert-Wilkens-Lewenstein framework by introducing a tunable entanglement parameter γ. It analytically derives explicit threshold conditions on γ (depending on player number n) under which symmetric Nash equilibria exist for two specified canonical strategy profiles—one valid for 2 ≤ n ≤ 9 and one for even n—showing that maximal entanglement is not required.

Significance. If the derivations are correct, the result is significant for near-term quantum implementations: it supplies concrete, n-dependent lower bounds on γ below which the symmetric equilibria disappear, directly addressing resource constraints on entanglement generation. The analytic (rather than numerical) character of the thresholds is a clear strength.

minor comments (2)
  1. The abstract states that thresholds are computed analytically but does not indicate whether the payoff parameters of the underlying Volunteer's Dilemma (e.g., benefit and cost values) enter the final expressions; a brief statement clarifying this would improve readability.
  2. Notation for the two strategy profiles should be introduced with explicit labels (e.g., “Profile A for 2≤n≤9”) the first time they appear, rather than only in the results section, to aid cross-referencing.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work and for noting its relevance to resource-constrained quantum devices. The recommendation of minor revision is noted. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper extends the established EWL quantum game framework by introducing a tunable entanglement parameter γ and performs an analytical derivation of threshold conditions for symmetric Nash equilibria in two canonical strategy profiles. These thresholds are obtained directly from the payoff expressions and Nash equilibrium conditions within the model, without reducing to fitted parameters, self-definitional loops, or load-bearing self-citations. The EWL framework and its parameterization are external to the authors, and the central claim (equilibria persist above a computable γ threshold) follows from explicit algebraic conditions on γ and n rather than by construction or renaming. No steps match the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim depends on the EWL framework and the parameterization of entanglement by γ, which is a standard extension in quantum game theory. No free parameters are fitted to data; γ is tunable. No invented entities are introduced.

free parameters (1)
  • entanglement parameter γ
    Tunable parameter introduced to generalize the model from maximal entanglement; thresholds are computed as functions of γ rather than fitted.
axioms (1)
  • domain assumption The game is modeled within the Eisert-Wilkens-Lewenstein framework using a shared entangled state with tunable parameter γ.
    Stated in the abstract as the basis for the quantum version of the dilemma.

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Reference graph

Works this paper leans on

109 extracted references · 2 canonical work pages

  1. [1]

    Similarly, the second case implies thatx= 1 n

    1. Similarly, the second case implies thatx= 1 n. The third implies thatx= 0 n, and finally, the fourth case implies thatx= 00. . .0 1 ↑a

  2. [2]

    With this in mind, we now substitute this result into Eq

    0. With this in mind, we now substitute this result into Eq. (10), which yields $γ a(θ, ϕ) (θi,ϕi)=Q,∀i̸=a = 2 X x∈{0,1}n x̸=0n xa=0 pθ,ϕ(x) + nX k=1 2− 1 k X x∈{0,1}n xa=1 wt(x)=k pθ,ϕ(x) (θi,ϕi)=Q,∀i̸=a = 2pθ,ϕ(1. . .101. . .1) + 2− 1 1 pθ,ϕ(0. . .010. . .0) + 2− 1 n pθ,ϕ(1n) (θi,ϕi)=Q,∀i̸=a = 2 cos2 π n sin2 θa 2 + cos2(γ) sin2 θa 2 sin2 π n + sin2(γ) ...

  3. [3]

    Horodecki, P

    R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009)

    2009

  4. [4]

    Ekert and R

    A. Ekert and R. Jozsa, Quantum algorithms: entan- glement–enhanced information processing, Philosophi- cal Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences356, 1769 (1998)

    1998

  5. [5]

    Jozsa and N

    R. Jozsa and N. Linden, On the role of entanglement in quantum-computational speed-up, Proceedings of the Royal Society A: Mathematical, Physical and Engineer- ing Sciences459, 2011 (2003)

    2011

  6. [6]

    Bruß and C

    D. Bruß and C. Macchiavello, Multipartite entangle- ment in quantum algorithms, Phys. Rev. A83, 052313 (2011)

    2011

  7. [7]

    J. D. Biamonte, M. E. S. Morales, and D. E. Koh, En- tanglement scaling in quantum advantage benchmarks, Phys. Rev. A101, 012349 (2020)

    2020

  8. [8]

    Díez-Valle, D

    P. Díez-Valle, D. Porras, and J. J. García-Ripoll, Quan- tum variational optimization: The role of entangle- ment and problem hardness, Phys. Rev. A104, 062426 (2021)

    2021

  9. [9]

    Wu, Exploring variational quantum eigensolver ansatzes for the long-range XY model, arXiv preprint arXiv:2109.00288 (2021)

    J.-B.You, D.E.Koh, J.F.Kong, W.-J.Ding, C.E.Png, and L. Wu, Exploring variational quantum eigensolver ansatzes for the long-range XY model, arXiv preprint arXiv:2109.00288 (2021)

    arXiv 2021

  10. [10]

    Katabarwa, S

    A. Katabarwa, S. Sim, D. E. Koh, and P.-L. Dallaire- Demers, Connecting geometry and performance of two- qubit parameterized quantum circuits, Quantum6, 782 (2022)

    2022

  11. [11]

    Horodecki, P

    M. Horodecki, P. Horodecki, and R. Horodecki, Mixed- state entanglement and quantum communication, in Quantum Information: An Introduction to Basic Theo- retical Concepts and Experiments(Springer Berlin Hei- delberg, Berlin, Heidelberg, 2001) pp. 151–195

    2001

  12. [12]

    Ursin, F

    R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Bar- bieri, H. Weinfurter, and A. Zeilinger, Entanglement- based quantum communication over 144km, Nature Physics3, 481 (2007)

    2007

  13. [13]

    Zou, Quantum entanglement and its application in quantum communication, Journal of Physics: Confer- ence Series1827, 012120 (2021)

    N. Zou, Quantum entanglement and its application in quantum communication, Journal of Physics: Confer- ence Series1827, 012120 (2021)

    2021

  14. [14]

    J. Xing, T. Feng, Z. Fan, H. Ma, K. Bharti, D. E. Koh, and Y. Xiao, Fundamental limitations on com- munication over a quantum network, arXiv preprint arXiv:2306.04983 (2023)

    arXiv 2023

  15. [15]

    J. Xing, Y. Li, D. Qu, L. Xiao, Z. Fan, H. Ma, P. Xue, K. Bharti, D. E. Koh, and Y. Xiao, Teleportation with embezzling catalysts, Communications Physics7, 357 (2024)

    2024

  16. [16]

    A.K.Ekert,QuantumcryptographybasedonBell’sthe- orem, Phys. Rev. Lett.67, 661 (1991)

    1991

  17. [17]

    Jennewein, C

    T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, Quantum cryptography with entangled photons, Phys. Rev. Lett.84, 4729 (2000)

    2000

  18. [18]

    Yin, Y.-H

    J. Yin, Y.-H. Li, S.-K. Liao, M. Yang, Y. Cao, L. Zhang, J.-G. Ren, W.-Q. Cai, W.-Y. Liu, S.-L. Li, R. Shu, Y.- M. Huang, L. Deng, L. Li, Q. Zhang, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, X.-B. Wang, F. Xu, J.-Y. Wang, C.- Z. Peng, A. K. Ekert, and J.-W. Pan, Entanglement- based secure quantum cryptography over 1,120 kilome- tres, Nature582, 501 (2020)

    2020

  19. [19]

    Giovannetti, S

    V. Giovannetti, S. Lloyd, and L. Maccone, Advances in quantum metrology, Nature Photonics5, 222 (2011)

    2011

  20. [20]

    Huang, C

    Z. Huang, C. Macchiavello, and L. Maccone, Useful- nessofentanglement-assistedquantummetrology,Phys. Rev. A94, 012101 (2016)

    2016

  21. [21]

    Augusiak, J

    R. Augusiak, J. Kołodyński, A. Streltsov, M. N. Bera, A. Acín, and M. Lewenstein, Asymptotic role of entan- glementinquantummetrology,Phys.Rev.A94,012339 (2016)

    2016

  22. [22]

    Eisert, M

    J. Eisert, M. Wilkens, and M. Lewenstein, Quantum games and quantum strategies, Physical Review Letters 83, 3077 (1999)

    1999

  23. [23]

    Eisert and M

    J. Eisert and M. Wilkens, Quantum games, Journal of Modern Optics47, 2543 (2000)

    2000

  24. [24]

    A. P. Flitney and D. Abbott, An introduction to quan- tumgametheory,FluctuationandNoiseLetters2,R175 (2002)

    2002

  25. [25]

    F. S. Khan, N. Solmeyer, R. Balu, and T. S. Hum- ble, Quantum games: a review of the history, current state, and interpretation, Quantum Information Pro- cessing17, 309 (2018)

    2018

  26. [26]

    D. A. Meyer, Quantum strategies, Physical Review Let- ters82, 1052 (1999)

    1999

  27. [27]

    S.C.BenjaminandP.M.Hayden,Multiplayerquantum games, Phys. Rev. A64, 030301(R) (2001)

    2001

  28. [28]

    Tsakiroglou, I

    M. Tsakiroglou, I. Liliopoulos, G. D. Varsamis, K. Milchanowski, and I. G. Karafyllidis, Graph-encoded strategic interactions in multiplayer quantum games, Quantum Information Processing25, 109 (2026)

    2026

  29. [29]

    Essalmi, F

    K. Essalmi, F. Garrido, and F. Nashashibi, Multi- player, multi-strategy quantum game model for interaction-aware decision-making in autonomous driv- ing, arXiv preprint arXiv:2602.03571 (2026)

    arXiv 2026

  30. [30]

    H. Li, J. Du, and S. Massar, Continuous-variable quan- tum games, Physics Letters A306, 73 (2002). 17

    2002

  31. [31]

    R. Kay, N. F. Johnson, and S. C. Benjamin, Evolution- ary quantum game, Journal of Physics A: Mathematical and General34, L547 (2001)

    2001

  32. [32]

    Sanz-Martín, G

    L. Sanz-Martín, G. Rivas, N. Clavijo-Buriticá, M. Her- rera, and J. Parra-Domínguez, Mapping the frontier: a review of quantum and evolutionary game theory for complex decision-making, Quantum Information Pro- cessing24, 291 (2025)

    2025

  33. [33]

    J. Du, X. Xu, H. Li, X. Zhou, and R. Han, Nash equi- librium in the quantum battle of sexes game, arXiv preprint quant-ph/0010050 (2000)

    Pith/arXiv arXiv 2000

  34. [34]

    Piotrowski and J

    E. Piotrowski and J. Sładkowski, Quantum market games, Physica A: Statistical Mechanics and its Appli- cations312, 208 (2002)

    2002

  35. [35]

    Q. Chen, Y. Wang, J.-T. Liu, and K.-L. Wang, N-player quantum minority game, Physics Letters A327, 98 (2004)

    2004

  36. [36]

    Nawaz and A

    A. Nawaz and A. Toor, Dilemma and quantum battle of sexes, Journal of Physics A: Mathematical and General 37, 4437 (2004)

    2004

  37. [37]

    Frąckiewicz, The ultimate solution to the quantum battle of the sexes game, Journal of Physics A: Mathe- matical and Theoretical42, 365305 (2009)

    P. Frąckiewicz, The ultimate solution to the quantum battle of the sexes game, Journal of Physics A: Mathe- matical and Theoretical42, 365305 (2009)

    2009

  38. [38]

    Consuelo-Leal, A

    A. Consuelo-Leal, A. G. Araujo-Ferreira, E. Lucas- Oliveira, T. J. Bonagamba, and R. Auccaise, Pareto- optimal solution for the quantum battle of the sexes, Quantum Information Processing19, 41 (2020)

    2020

  39. [39]

    Szopa, Efficiency of classical and quantum games equilibria, Entropy23, 506 (2021)

    M. Szopa, Efficiency of classical and quantum games equilibria, Entropy23, 506 (2021)

    2021

  40. [40]

    Kastampolidou and T

    K. Kastampolidou and T. Andronikos, Quantum tap- silou—aquantumgameinspiredbythetraditionalgreek coin tossing game tapsilou, Games14, 72 (2023)

    2023

  41. [41]

    Frąckiewicz and M

    P. Frąckiewicz and M. Szopa, Permissible extensions of classical to quantum games combining three strategies, Quantum Information Processing23, 75 (2024)

    2024

  42. [42]

    Frąckiewicz, A

    P. Frąckiewicz, A. Gorczyca-Goraj, and M. Szopa, Per- missible four-strategy quantum extensions of classi- cal games, Quantum Information Processing24, 201 (2025)

    2025

  43. [43]

    D. E. Koh, K. Kumar, and S. T. Goh, Quantum vol- unteer’s dilemma, Physical Review Research7, 013104 (2025)

    2025

  44. [44]

    Andronikos, C

    T. Andronikos, C. Bitsakos, K. Nikas, G. I. Goumas, and N. Koziris, A GHZ-based protocol for the dining information brokers problem, Future Internet17, 408 (2025)

    2025

  45. [45]

    Essalmi, F

    K. Essalmi, F. Garrido, and F. Nashashibi, Quantum game models for interaction-aware decision-making in automated driving, in2025 IEEE International Confer- ence on Advanced Robotics (ICAR)(2025) pp. 155–162

    2025

  46. [46]

    R.Smoliński, P.Frąckiewicz, K.Grzanka,andM.Szopa, Quantum negotiation games: Toward ethical equilibria, Entropy28, 51 (2026)

    2026

  47. [47]

    Vlachos and I

    P. Vlachos and I. G. Karafyllidis, Quantum game simu- lator, using the circuit model of quantum computation, Computer Physics Communications180, 1990 (2009)

    1990

  48. [48]

    Kotara, T

    P. Kotara, T. Zawadzki, and K. Rycerz, Software aided analysis of EWL based quantum games, inPar- allel Processing and Applied Mathematics, edited by R. Wyrzykowski, J. Dongarra, E. Deelman, and K. Kar- czewski (Springer International Publishing, Cham,

  49. [49]

    Grzanka, A

    K. Grzanka, A. Gorczyca-Goraj, M. Szopa,et al., QEGS:AMathematicapackagefortheanalysisofquan- tum extended games, arXiv preprint arXiv:2505.00714 (2025)

    arXiv 2025

  50. [50]

    N. F. Johnson, Playing a quantum game with a cor- rupted source, Phys. Rev. A63, 020302(R) (2001)

    2001

  51. [51]

    L. Chen, H. Ang, D. Kiang, L. Kwek, and C. Lo, Quan- tum prisoner dilemma under decoherence, Physics Let- ters A316, 317 (2003)

    2003

  52. [52]

    A. P. Flitney and D. Abbott, Quantum games with decoherence, Journal of Physics A: Mathematical and General38, 449 (2004)

    2004

  53. [53]

    Shuai, F

    C. Shuai, F. Mao-Fa, and Z. Xiao-Juan, The effect of quantum noise on multiplayer quantum game, Chinese Physics16, 915 (2007)

    2007

  54. [54]

    Huang and D

    Z. Huang and D. Qiu, Quantum games under decoher- ence, International Journal of Theoretical Physics55, 965 (2016)

    2016

  55. [55]

    Khan and S

    S. Khan and S. Alam, The dynamics of Nash equilib- rium under non-Markovian classical noise in quantum prisoners’ dilemma, Reports on Mathematical Physics 81, 399 (2018)

    2018

  56. [56]

    Kairon, K

    P. Kairon, K. Thapliyal, R. Srikanth, and A. Pathak, Noisy three-player dilemma game: robustness of the quantum advantage, Quantum information processing 19, 327 (2020)

    2020

  57. [57]

    A. R. Legón and E. Medina, Joint probabilities ap- proach to quantum games with noise, Entropy25, 1222 (2023)

    2023

  58. [58]

    A. T. M. Makram-Allah, M. Y. Abd-Rabbou, and N. Metwally, The possibility of fairly mitigating penal- ties in the quantum prisoner’s dilemma via classical noise channels, The European Physical Journal Plus 141, 456 (2026)

    2026

  59. [59]

    J. Lu, L. Zhou, and L.-M. Kuang, Linear optics imple- mentation for quantum game with two players, Physics Letters A330, 48 (2004)

    2004

  60. [60]

    I. M. Buluta, S. Fujiwara, and S. Hasegawa, Quantum games in ion traps, Physics Letters A358, 100 (2006)

    2006

  61. [61]

    Mitra, K

    A. Mitra, K. Sivapriya, and A. Kumar, Experimental implementation of a three qubit quantum game with corrupt source using nuclear magnetic resonance quan- tum information processor, Journal of Magnetic Reso- nance187, 306 (2007)

    2007

  62. [62]

    Prevedel, A

    R. Prevedel, A. Stefanov, P. Walther, and A. Zeilinger, Experimental realization of a quantum game on a one- way quantum computer, New Journal of Physics9, 205 (2007)

    2007

  63. [63]

    Schmid, A

    C. Schmid, A. P. Flitney, W. Wieczorek, N. Kiesel, H. Weinfurter, and L. C. Hollenberg, Experimental im- plementation of a four-player quantum game, New Jour- nal of Physics12, 063031 (2010)

    2010

  64. [64]

    Xu and Z

    K. Xu and Z. Wu, Experimental implementation of quantum prisoner dilemma on IBM quantum comput- ers, in2022 15th International Conference on Advanced Computer Theory and Engineering (ICACTE)(IEEE,

  65. [65]

    G. D. Agreda, C. A. D. Paredes, M. B. Samboni, J. A. Andrade, and S. C. Ordoñez, Bridging theory and prac- tice in quantum game theory: Optimized implementa- tion of the battle of the sexes with error mitigation on NISQ hardware, in2025 IEEE CHILEAN Conference on Electrical, Electronics Engineering, Information and Communication Technologies (CHILECON)(20...

    2025

  66. [66]

    G. D. D. Agreda, J. A. A. Hoyos, C. A. D. Paredes, S. C. Ordoñez, N. D. Hebdon, S. T. Goh, and D. E. Koh, Experimental implementation of the quantum vol- unteer’sdilemmaonNISQhardware: Noiseanalysisand digital-twin validation, arXiv preprint arXiv:2605.30676 (2026)

    Pith/arXiv arXiv 2026

  67. [67]

    J.Du, X.Xu, H.Li, X.Zhou,andR.Han,Entanglement playing a dominating role in quantum games, Physics Letters A289, 9 (2001)

    2001

  68. [68]

    J.Du, H.Li, X.Xu, X.Zhou,andR.Han,Entanglement enhanced multiplayer quantum games, Physics Letters A302, 229 (2002)

    2002

  69. [69]

    S. K. Özdemir, J. Shimamura, and N. Imoto, A nec- essary and sufficient condition to play games in quan- tum mechanical settings, New Journal of Physics9, 43 (2007)

    2007

  70. [70]

    Nawaz and A

    A. Nawaz and A. Toor, Quantum games and quantum discord, arXiv preprint arXiv:1012.1428 (2010)

    Pith/arXiv arXiv 2010

  71. [71]

    Ahmad, The strategic form of quantum prisoners’ dilemma, Chinese Physics Letters30, 050302 (2013)

    N. Ahmad, The strategic form of quantum prisoners’ dilemma, Chinese Physics Letters30, 050302 (2013)

    2013

  72. [72]

    Li and X

    A. Li and X. Yong, Entanglement guarantees emergence of cooperation in quantum prisoner’s dilemma games on networks, Scientific Reports4, 6286 (2014)

    2014

  73. [73]

    Wei and S

    Z. Wei and S. Zhang, Quantum game players can have advantage without discord, Information and Computa- tion256, 174 (2017)

    2017

  74. [74]

    A. C. Santos, Entanglement and coherence in quantum prisoner’s dilemma, Quantum Information Processing 19, 13 (2019)

    2019

  75. [75]

    N. M. A. A. Mohamed, H. Taisheng, and P. Jinhui, Quantum game theory on entangled players, inInterna- tional Conference on Intelligent Information Technolo- gies for Industry(Springer, 2023) pp. 291–301

    2023

  76. [76]

    Bugu, Entanglement as a strategic resource in adver- sarial quantum games, arXiv preprint arXiv:2510.22444 (2025)

    S. Bugu, Entanglement as a strategic resource in adver- sarial quantum games, arXiv preprint arXiv:2510.22444 (2025)

    arXiv 2025

  77. [77]

    G. D. Varsamis, I. Liliopoulos, H. Mohammadbagher- poor, A. K. Kostopoulos, K. Milchanowski, E. T. Kara- matskos, P. Dimitrakis, R. P. Padbury, and I. G. Karafyllidis, Analysis of the effect of entanglement oper- ators and the scalability of players’ payoff computation in n-player quantum games, Advanced Quantum Tech- nologies8, e00375 (2025)

    2025

  78. [78]

    Diekmann, Volunteer’s dilemma, Journal of Conflict Resolution29, 605 (1985)

    A. Diekmann, Volunteer’s dilemma, Journal of Conflict Resolution29, 605 (1985)

    1985

  79. [79]

    J. E. Smith, S. Gavrilets, M. B. Mulder, P. L. Hooper, C. E. Mouden, D. Nettle, C. Hauert, K. Hill, S. Perry, A. E. Pusey, M. van Vugt, and E. A. Smith, Leader- ship in mammalian societies: Emergence, distribution, power, and payoff, Trends in Ecology & Evolution31, 54 (2016)

    2016

  80. [80]

    Weesie and A

    J. Weesie and A. Franzen, Cost sharing in a volunteer’s dilemma, Journal of Conflict Resolution42, 600 (1998)

    1998

Showing first 80 references.