{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,17]],"date-time":"2025-10-17T20:07:28Z","timestamp":1760731648916,"version":"build-2065373602"},"reference-count":137,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2024,8,26]],"date-time":"2024-08-26T00:00:00Z","timestamp":1724630400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Games"],"abstract":"<jats:p>We start from the fundamental premise that any physical interaction can be interpreted as a game. To demonstrate this, we draw upon the free energy principle and the theory of quantum reference frames. In this way, we place the game-theoretic Nash Equilibrium in a new light in so far as the incompleteness and undecidability of the concept, as well as the nature of strategies in general, can be seen as the consequences of certain no-go theorems. We show that games of the generic imitation type follow a circularity of idealization that includes the good regulator theorem, generalized synchrony, and undecidability of the Turing test. We discuss Bayesian games in the light of Bell non-locality and establish the basics of quantum games, which we relate to local operations and classical communication protocols. In this light, we also review the rationality of gaming strategies from the players\u2019 point of view.<\/jats:p>","DOI":"10.3390\/g15050030","type":"journal-article","created":{"date-parts":[[2024,8,26]],"date-time":"2024-08-26T10:01:51Z","timestamp":1724666511000},"page":"30","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Nash Equilibria and Undecidability in Generic Physical Interactions\u2014A Free Energy Perspective"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4812-0744","authenticated-orcid":false,"given":"Chris","family":"Fields","sequence":"first","affiliation":[{"name":"Allen Discovery Center, Tufts University, Medford, MA 02155, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8335-221X","authenticated-orcid":false,"given":"James F.","family":"Glazebrook","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Eastern Illinois University, Charleston, IL 61920, USA"},{"name":"Adjunct Faculty, Department of Mathematics, University of Illinois at Urbana\u2013Champaign, Urbana, IL 61801, USA"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,26]]},"reference":[{"key":"ref_1","unstructured":"von Neumann, J., and Morgenstern, O. (1944). Theory of Games and Economic Behavior, Princeton University Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1073\/pnas.36.1.48","article-title":"Equilibrium points in n-person games","volume":"36","author":"Nash","year":"1950","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"286","DOI":"10.2307\/1969529","article-title":"Non-cooperative Games","volume":"54","author":"Nash","year":"1951","journal-title":"Ann. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"3999","DOI":"10.1073\/pnas.0308738101","article-title":"The Nash equilibrium: A perspective","volume":"101","author":"Holt","year":"2004","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1017\/S0266267100003722","article-title":"On the foundations of Nash equilibrium","volume":"12","author":"Jacobsen","year":"1996","journal-title":"Econ. Phil."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"526","DOI":"10.1090\/noti1375","article-title":"What Is... Nash Equilibrium?","volume":"63","author":"Sethi","year":"2016","journal-title":"Not. Amer. Math. Soc."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Maynard Smith, J. (1982). Evolution and the Theory of Games, Cambridge University Press.","DOI":"10.1017\/CBO9780511806292"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1830","DOI":"10.1257\/000282803322655581","article-title":"Uncoupled dynamics do not lead to Nash equilibrium","volume":"93","author":"Hart","year":"2003","journal-title":"Am. Econ. Rev."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"e2305349120","DOI":"10.1073\/pnas.2305349120","article-title":"An impossibility theorem in game dynamics","volume":"120","author":"Milionis","year":"2023","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_10","first-page":"4902","article-title":"The prevalence of chaotic dynamics in games with many players","volume":"8","author":"Sanders","year":"2018","journal-title":"Nat. Sci. Rep."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"eaat1328","DOI":"10.1126\/sciadv.aat1328","article-title":"Best reply structure and equilibrium convergence in generic games","volume":"5","author":"Pangallo","year":"2019","journal-title":"Sci. Adv."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1016\/j.tcs.2012.10.053","article-title":"On the complexity of deciding degeneracy in a bimatrix game with sparse payoff matrix","volume":"472","author":"Du","year":"2013","journal-title":"Theor. Comp. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1137\/070699652","article-title":"The complexity of computing a Nash equilibrium","volume":"39","author":"Daskalakis","year":"2009","journal-title":"SIAM J. Comp."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Copeland, B.J., Posy, C.J., and Shagrir, O. (2013). Why philosophers should care about computational complexity. Computability: Turing, G\u00f6del, Church, and Beyond, MIT Press.","DOI":"10.7551\/mitpress\/8009.001.0001"},{"key":"ref_15","first-page":"173","article-title":"\u00dcber formal unentscheidbare s\u00e4tze der Principia Mathematica und verwandter systeme, I","volume":"38","year":"1931","journal-title":"Monatsh. Math. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1023\/A:1004220919753","article-title":"The incompleteness of theories of games","volume":"27","author":"Tsuji","year":"1998","journal-title":"J. Philos. Logic"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1023\/A:1004959623042","article-title":"The undecidability of the spatialized prisoner\u2019s dilemma","volume":"42","author":"Grim","year":"1997","journal-title":"Theory Decis."},{"key":"ref_18","unstructured":"Shannon, C.E., and McCarthy, J. (1956). Gedankenexperiments on sequential machines. Autonoma Studies, Princeton University Press."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1093\/mind\/LIX.236.433","article-title":"Computing machines and intelligence","volume":"59","author":"Turing","year":"1950","journal-title":"Mind"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1023\/B:MIND.0000021747.28850.16","article-title":"Undecidability in the imitation game","volume":"14","author":"Sato","year":"2004","journal-title":"Minds Mach."},{"key":"ref_21","unstructured":"Milnor, J. (1951). Games against Nature, RAND Corp."},{"key":"ref_22","unstructured":"Feynman, R.P. (1972). Statistical Mechanics, Benjamin."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1038\/nrn2787","article-title":"The free-energy principle: A unified brain theory?","volume":"11","author":"Friston","year":"2010","journal-title":"Nat. Rev. Neurosci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"20130475","DOI":"10.1098\/rsif.2013.0475","article-title":"Life as we know it","volume":"10","author":"Friston","year":"2013","journal-title":"J. R. Soc. Interface"},{"key":"ref_25","unstructured":"Friston, K.J. (2019). A free energy principle for a particular physics. arXiv."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"20220029","DOI":"10.1098\/rsfs.2022.0029","article-title":"On Bayesian mechanics: A physics of and by beliefs","volume":"13","author":"Ramstead","year":"2023","journal-title":"R. Soc. Interface Focus"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/j.plrev.2023.08.016","article-title":"Path integrals, particular kinds, and strange things","volume":"47","author":"Friston","year":"2023","journal-title":"Phys. Life Rev."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"20140182","DOI":"10.1098\/rspa.2014.0182","article-title":"When does a physical system compute?","volume":"470","author":"Horsman","year":"2014","journal-title":"Proc. R. Soc. A"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1016\/j.pbiomolbio.2022.05.006","article-title":"A free energy principle for generic quantum systems","volume":"173","author":"Fields","year":"2022","journal-title":"Prog. Biophys. Mol. Biol."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1109\/TMBMC.2023.3272150","article-title":"Control flow in active inference systems, Part I: Formulations of classical and quantum active inference","volume":"9","author":"Fields","year":"2023","journal-title":"IEEE Trans. Mol. Biol. Multi-Scale Commun."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Fields, C., and Glazebrook, J.F. (2020). Representing measurement as a thermodynamic symmetry breaking. Symmetry, 12.","DOI":"10.3390\/sym12050810"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Addazi, A., Chen, P., Fabrocini, F., Fields, C., Greco, E., Lulli, M., Marcian\u00f2, A., and Pasechnik, R. (2021). Generalized holographic principle, gauge invariance and the emergence of gravity \u00e0 la Wilczek. Front. Astron. Space Sci., 8.","DOI":"10.3389\/fspas.2021.563450"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Fields, C., Glazebrook, J.F., and Marcian\u00f2, A. (2021). Reference frame induced symmetry breaking on holographic screens. Symmetry, 13.","DOI":"10.3390\/sym13030408"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Ali, A., Ellis, J., and Randjbar-Daemi, S. (1993). Dimensional reduction in quantum gravity. Salamfestschrift, World Scientific.","DOI":"10.1142\/9789814535717"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"6377","DOI":"10.1063\/1.531249","article-title":"The world as a hologram","volume":"36","author":"Susskind","year":"1995","journal-title":"J. Math. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"825","DOI":"10.1103\/RevModPhys.74.825","article-title":"The holographic principle","volume":"74","author":"Bousso","year":"2002","journal-title":"Rev. Mod. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"72","DOI":"10.12743\/quanta.v11i1.206","article-title":"The physical meaning of the holographic principle","volume":"11","author":"Fields","year":"2022","journal-title":"Quanta"},{"key":"ref_38","unstructured":"Nielsen, M.A., and Chuang, I.L. (2000). Quantum Computation and Quantum Information, Cambridge University Press."},{"key":"ref_39","unstructured":"Zurek, W. (1989). Information, physics, quantum: The search for links. Complexity, Entropy, and the Physics of Information, CRC Press."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1047","DOI":"10.1103\/PhysRevA.50.1047","article-title":"Eavesdropping on quantum cryptographical systems","volume":"50","author":"Ekert","year":"1994","journal-title":"Phys. Rev. A"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"913","DOI":"10.1080\/09500340110109412","article-title":"Quantum theory of preparation and measurement","volume":"49","author":"Pegg","year":"2010","journal-title":"J. Mod. Opt."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"368","DOI":"10.1103\/PhysRevD.30.368","article-title":"Quantum frames of reference","volume":"30","author":"Aharonov","year":"1984","journal-title":"Phys. Rev. D"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1103\/RevModPhys.79.555","article-title":"Reference frames, superselection rules, and quantum information","volume":"79","author":"Bartlett","year":"2007","journal-title":"Rev. Mod. Phys."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1080\/0952813X.2018.1544282","article-title":"A mosaic of Chu spaces and Channel Theory I: Category-theoretic concepts and tools","volume":"31","author":"Fields","year":"2019","journal-title":"J. Expt. Theor. Artif. Intell."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1080\/0952813X.2020.1836034","article-title":"Information flow in context-dependent hierarchical Bayesian inference","volume":"34","author":"Fields","year":"2022","journal-title":"J. Expt. Theor. Artif. intell."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Barwise, J., and Seligman, J. (1997). Information Flow: The Logic of Distributed Systems, Cambridge University Press.","DOI":"10.1017\/CBO9780511895968"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"2200104","DOI":"10.1002\/prop.202200104","article-title":"Sequential measurements, topological quantum field theories, and topological quantum neural networks","volume":"70","author":"Fields","year":"2022","journal-title":"Fortschr. Phys."},{"key":"ref_48","first-page":"202400049","article-title":"Communication protocols and QECCs from the perspective of TQFT, Part I: Constructing LOCC protocols and QECCs from TQFTs","volume":"72","author":"Fields","year":"2024","journal-title":"Fortschr. Phys."},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Pearl, J. (1988). Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference, Morgan Kaufmann.","DOI":"10.1016\/B978-0-08-051489-5.50008-4"},{"key":"ref_50","unstructured":"Hutter, M. (2006). Universal Artificial Intellegence Sequential Decisions Based on Algorithmic Probability, Springer."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1049\/el:19950331","article-title":"Free-energy minimisation algorithm for decoding and cryptoanalysis","volume":"31","author":"MacKay","year":"1995","journal-title":"Electron. Lett."},{"key":"ref_52","first-page":"nix019","article-title":"An algorithmic information theory of consciousness","volume":"2017","author":"Ruffini","year":"2017","journal-title":"Neurosci. Cons."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1093\/comjnl\/42.4.270","article-title":"Minimum message length and Kolmogorov complexity","volume":"42","author":"Wallace","year":"1999","journal-title":"Comput. J."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"1480","DOI":"10.3758\/s13423-015-0890-8","article-title":"The interface theory of perception","volume":"22","author":"Hoffman","year":"2015","journal-title":"Psychon. Bull. Rev."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1038\/323533a0","article-title":"Learning representations by back-propagating errors","volume":"323","author":"Rumelhart","year":"1986","journal-title":"Nature"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1109\/TEVC.2006.890271","article-title":"Intrinsic motivation systems for autonomous mental development","volume":"11","author":"Oudeyer","year":"2007","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Keil, F., and Wilson, R. (2000). Explanation as orgasm and the drive for causal understanding: The evolution, function and phenomenology of the theory-formation system. Cognition and Explanation, MIT Press.","DOI":"10.7551\/mitpress\/2930.001.0001"},{"key":"ref_58","doi-asserted-by":"crossref","unstructured":"Fields, C., Glazebrook, J.F., and Levin, M. (2024). Principled limitations on self-representation for generic physical systems. Entropy, 26.","DOI":"10.3390\/e26030194"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1007\/s10773-023-05406-9","article-title":"Separability, contextuality, and the quantum Frame Problem","volume":"62","author":"Fields","year":"2023","journal-title":"Int. J. Theor. Phys."},{"key":"ref_60","first-page":"463","article-title":"Some philosophical problems from the standpoint of artificial intelligence","volume":"Volume 4","author":"Michie","year":"1969","journal-title":"Machine Intelligence"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1287\/mnsc.14.3.159","article-title":"Games with incomplete information played by Bayesian players, Part I","volume":"14","author":"Harsanyi","year":"1967","journal-title":"Manag. Sci."},{"key":"ref_62","unstructured":"Quine, W.V.O. (1960). Word and Object, MIT Press."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1162\/NECO_a_00912","article-title":"Active inference: A process theory","volume":"29","author":"Friston","year":"2017","journal-title":"Neural Comput."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.plrev.2017.09.001","article-title":"Answering Schr\u00f6dinger\u2019s question: A free-energy formulation","volume":"24","author":"Ramstead","year":"2018","journal-title":"Phys. Life Rev."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1016\/j.plrev.2018.12.002","article-title":"Variational ecology and the physics of sentient systems","volume":"31","author":"Ramstead","year":"2019","journal-title":"Phys. Life Rev."},{"key":"ref_66","unstructured":"Raup, D.M. (1991). Extinction: Bad Genes or Bad Luck?, Norton."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"390","DOI":"10.1016\/j.concog.2014.12.003","article-title":"Duet for one","volume":"36","author":"Friston","year":"2015","journal-title":"Conscious. Cogn."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/j.cortex.2015.03.025","article-title":"Active inference, communication and hermeneutics","volume":"68","author":"Friston","year":"2015","journal-title":"Cortex"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"6708","DOI":"10.1103\/PhysRevE.54.6708","article-title":"Detecting dynamical interdependence and generalized synchrony through mutual prediction in a neural ensemble","volume":"54","author":"Schiff","year":"1996","journal-title":"Phys. Rev. E"},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"520","DOI":"10.1063\/1.166278","article-title":"Fundamentals of synchronization in chaotic systems, concepts, and applications","volume":"7","author":"Pecora","year":"1997","journal-title":"Chaos"},{"key":"ref_71","doi-asserted-by":"crossref","unstructured":"Friston, K., Breakspear, M., and Deco, G. (2012). Perception and self-organized instability. Front. Comp. Neurosci., 6.","DOI":"10.3389\/fncom.2012.00044"},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1109\/JPROC.2014.2306251","article-title":"Cognitive dynamics: From attractors to active inference","volume":"102","author":"Friston","year":"2014","journal-title":"Proc. IEEE"},{"key":"ref_73","first-page":"6412","article-title":"The emergence of synchrony in networks of mutually inferring neurons","volume":"9","author":"Palacios","year":"2019","journal-title":"Nature Sci. Rep."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"119038","DOI":"10.1016\/j.neuroimage.2022.119038","article-title":"Directed coupling in multi-brain networks underlies generalized synchrony during social exchange","volume":"252","author":"Bilek","year":"2022","journal-title":"NeuroImage"},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"358","DOI":"10.1090\/S0002-9947-1953-0053041-6","article-title":"Classes of recursively enumerable sets and their decision problems","volume":"74","author":"Rice","year":"1953","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1112\/plms\/s2-42.1.230","article-title":"On computable numbers, with an application to the Entscheidungsproblem","volume":"42","author":"Turing","year":"1937","journal-title":"Proc. London Math. Soc."},{"key":"ref_77","unstructured":"Hopcroft, J.E., and Ullman, J.D. (1979). Introduction to Automata Theory, Languages, and Computation, Addison-Wesley."},{"key":"ref_78","doi-asserted-by":"crossref","unstructured":"Dietrich, E., and Fields, C. (2020). Equivalence of the Frame and Halting problems. Algorithms, 13.","DOI":"10.3390\/a13070175"},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1146\/annurev.anthro.32.061002.093158","article-title":"The social brain: Mind, language and society in evolutionary perspective","volume":"32","author":"Dunbar","year":"2003","journal-title":"Annu. Rev. Anthropol."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"3077","DOI":"10.1103\/PhysRevLett.83.3077","article-title":"Quantum games and quantum strategies","volume":"83","author":"Eisert","year":"1999","journal-title":"Phys. Rev. Lett."},{"key":"ref_81","unstructured":"Myerson, R.B. (1991). Game Theory: An Analysis of Conflict, MIT Press."},{"key":"ref_82","doi-asserted-by":"crossref","unstructured":"Chater, N. (2018). The Mind Is Flat, Allen Lane.","DOI":"10.12987\/9780300240610"},{"key":"ref_83","first-page":"255","article-title":"Complex type 4 structure changing dynamics of digital agents: Nash equilibria of a game with arms race innovations","volume":"4","author":"Markose","year":"2017","journal-title":"J. Dyn. Games Am. Inst. Math. Sci."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1016\/j.plrev.2018.12.003","article-title":"Self-referential basis of undecidable dynamics: From the Liar Paradox and the Halting Problem to the Edge of Chaos","volume":"31","author":"Prokopenko","year":"2019","journal-title":"Phys. Life Rev."},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1016\/S0167-2789(99)00115-3","article-title":"Dynamics of internal models in game players","volume":"134","author":"Taiji","year":"1999","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_86","unstructured":"Ikegami, T., and Taiji, M. (1998). Uncertainty, Possible Worlds and Coupled Dynamical Recognizers, University of Tokyo."},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"2543","DOI":"10.1080\/09500340008232180","article-title":"Quantum games","volume":"47","author":"Eisert","year":"2000","journal-title":"J. Modern Opt."},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/0165-1765(85)90026-6","article-title":"Bounded complexity justifies cooperation in the finitely repeated prisoners\u2019 dilemma","volume":"19","author":"Neyman","year":"1985","journal-title":"Econ. Lett."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1080\/00207727008920220","article-title":"Every good regulator of a system must be a model of that system","volume":"1","author":"Conant","year":"1970","journal-title":"Int. J. Syst. Sci."},{"key":"ref_90","unstructured":"Worden, R.P. (2024). The requirement for cognition, in an equation. arXiv."},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1016\/0304-3975(94)90041-8","article-title":"Rice\u2019s theorem for the limit sets of cellular automata","volume":"127","author":"Kari","year":"1994","journal-title":"Theor. Comp. Sci."},{"key":"ref_92","first-page":"1075","article-title":"On the undecidability of the limit behavior of Cellular Automata","volume":"411","author":"Margara","year":"2020","journal-title":"Theor. Comp. Sci."},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"1052","DOI":"10.1103\/PhysRevLett.82.1052","article-title":"Quantum strategies","volume":"82","author":"Meyer","year":"1999","journal-title":"Phys. Rev. Lett."},{"key":"ref_94","doi-asserted-by":"crossref","unstructured":"Minai, A., Braha, D., and Bar-Yam, Y. (2010). Quantum Nash equiibria and quantum computing. Unifying Themes in Complex Systems, Springer.","DOI":"10.1007\/978-3-642-17635-7"},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"1225","DOI":"10.1103\/RevModPhys.77.1225","article-title":"Quantum cloning","volume":"77","author":"Scarani","year":"2005","journal-title":"Rev. Mod. Pys."},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"012416","DOI":"10.1103\/PhysRevA.105.012416","article-title":"Entanglement in the quantum Game of Life","volume":"105","author":"Ney","year":"2022","journal-title":"Phys. Rev. A"},{"key":"ref_97","doi-asserted-by":"crossref","unstructured":"Kaur, H., and Kumar, A. (2017, January 22\u201323). Analysing the role of entanglement in the three-qubit Vaidman\u2019s game. Proceedings of the 2017 International Conference on Intelligent Communication and Computational Techniques (ICCT), Jaipur, India.","DOI":"10.1109\/INTELCCT.2017.8324027"},{"key":"ref_98","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1007\/s00220-014-1953-9","article-title":"Everything you always wanted to know about LOCC (but were afraid to ask)","volume":"328","author":"Chitambar","year":"2014","journal-title":"Comms. Math. Phys."},{"key":"ref_99","doi-asserted-by":"crossref","first-page":"1862","DOI":"10.1103\/PhysRevD.26.1862","article-title":"Environment-induced superselection rules","volume":"26","author":"Zurek","year":"1982","journal-title":"Phys. Rev. D"},{"key":"ref_100","first-page":"233","article-title":"The emergence of classical properties Through interaction with the environment","volume":"59","author":"Joos","year":"1985","journal-title":"Zeitschr. Phys. B"},{"key":"ref_101","doi-asserted-by":"crossref","unstructured":"Zurek, W.H. (2003). Decoherence, einselection, and the quantum origins of the classical. Rev. Mod. Phys., 715\u2013775.","DOI":"10.1103\/RevModPhys.75.715"},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"062310","DOI":"10.1103\/PhysRevA.73.062310","article-title":"Quantum Darwinism: Entanglement, branches, and the emergent classicality of redundantly stored quantum information","volume":"73","author":"Zurek","year":"2006","journal-title":"Phys. Rev. A"},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1038\/nphys1202","article-title":"Quantum Darwinism","volume":"5","author":"Zurek","year":"2009","journal-title":"Nature Phys."},{"key":"ref_104","doi-asserted-by":"crossref","first-page":"460","DOI":"10.1103\/PhysRevLett.47.460","article-title":"Experimental tests of realistic local theories via Bell\u2019s theorem","volume":"47","author":"Aspect","year":"1981","journal-title":"Phys. Rev. Lett."},{"key":"ref_105","first-page":"374","article-title":"How the Bell tests changed quantum physics","volume":"3","author":"Georgescu","year":"2021","journal-title":"Nat. Phys."},{"key":"ref_106","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1103\/PhysicsPhysiqueFizika.1.195","article-title":"On the Einstein-Podolsky-Rosen paradox","volume":"1","author":"Bell","year":"1964","journal-title":"Physics"},{"key":"ref_107","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1103\/RevModPhys.38.447","article-title":"On the problem of hidden variables in quantum mechanics","volume":"38","author":"Bell","year":"1966","journal-title":"Rev. Mod. Phys."},{"key":"ref_108","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1063\/1.880968","article-title":"Is the Moon there when nobody looks? Reality and the quantum theory","volume":"38","author":"Mermin","year":"1985","journal-title":"Phys. Today"},{"key":"ref_109","doi-asserted-by":"crossref","first-page":"2057","DOI":"10.1038\/ncomms3057","article-title":"Connection between Bell nonlocality and Bayesian game theory","volume":"4","author":"Brunner","year":"2013","journal-title":"Nat. Commun."},{"key":"ref_110","doi-asserted-by":"crossref","first-page":"880","DOI":"10.1103\/PhysRevLett.23.880","article-title":"Proposed experiment to test local hidden-variable theories","volume":"23","author":"Clauser","year":"1969","journal-title":"Phys. Rev. Lett."},{"key":"ref_111","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1007\/BF00417500","article-title":"Quantum generalizations of Bell\u2019s inequality","volume":"4","author":"Cirelson","year":"1980","journal-title":"Lett. Math. Phys."},{"key":"ref_112","first-page":"59","article-title":"The problem of hidden variables in quantum mechanics","volume":"17","author":"Kochen","year":"1967","journal-title":"J. Math. Mech."},{"key":"ref_113","doi-asserted-by":"crossref","first-page":"803","DOI":"10.1103\/RevModPhys.65.803","article-title":"Hidden variables and the two theorems of John Bell","volume":"65","author":"Mermin","year":"1993","journal-title":"Rev. Mod. Phys."},{"key":"ref_114","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1038\/nature13460","article-title":"Contextuality supplies the \u2018magic\u2019 for quantum computation","volume":"510","author":"Howard","year":"2014","journal-title":"Nature"},{"key":"ref_115","doi-asserted-by":"crossref","first-page":"1380","DOI":"10.3390\/e24101380","article-title":"Contextuality, complementarity, signaling, and Bell tests","volume":"24","author":"Khrennikov","year":"2022","journal-title":"Entropy"},{"key":"ref_116","unstructured":"Fourny, G. (2024). On the interpretation of quantum theory as games between physicists and nature played in Minkowski spacetime. arXiv."},{"key":"ref_117","doi-asserted-by":"crossref","first-page":"12848","DOI":"10.1073\/pnas.211534898","article-title":"On the impossibility of prediction of the behavior of rational agents","volume":"98","author":"Foster","year":"2001","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_118","doi-asserted-by":"crossref","first-page":"1404","DOI":"10.1016\/j.amc.2009.04.051","article-title":"Uncomputability and undecidability in economic theory","volume":"215","author":"Velupillai","year":"2009","journal-title":"Appl. Math. Comput."},{"key":"ref_119","unstructured":"Ewerhart, C. (1997). On Strategic Reasoning and Theories of Rational Behavior. [Ph.D. Thesis, University of Bonn]."},{"key":"ref_120","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1007\/BF01243194","article-title":"Rationality and the definition of consistent pairs","volume":"27","author":"Ewerhart","year":"1998","journal-title":"Int. J. Game Theory"},{"key":"ref_121","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.geb.2024.02.004","article-title":"An undecidable statement regarding zero-sum games","volume":"145","author":"Fey","year":"2024","journal-title":"Games Econ. Behav."},{"key":"ref_122","unstructured":"G\u00f6del, K. (1940). The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory, Princeton University Press. Annals of Mathematics Studies."},{"key":"ref_123","doi-asserted-by":"crossref","first-page":"1143","DOI":"10.1073\/pnas.50.6.1143","article-title":"The independence of the Continuum Hypothesis, Part I","volume":"50","author":"Cohen","year":"1963","journal-title":"Proc. Nat. Acad. Sci. USA"},{"key":"ref_124","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1073\/pnas.51.1.105","article-title":"The independence of the Continuum Hypothesis, Part II","volume":"51","author":"Cohen","year":"1964","journal-title":"Proc. Nat. Acad. Sci. USA"},{"key":"ref_125","unstructured":"Hu, T.-W., and Kaneko, M. (2015). Game Theoretic Decidability and Undecidability, Social Science Research Network (SSRN)."},{"key":"ref_126","doi-asserted-by":"crossref","first-page":"4748","DOI":"10.1073\/pnas.032086299","article-title":"Chaos in learning a simple two-person game","volume":"99","author":"Sato","year":"2002","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_127","doi-asserted-by":"crossref","unstructured":"Kahneman, D., Slovic, P., and Tversky, A. (1982). Judgement under Uncertainty: Heuristics and Biases, Cambridge University Press.","DOI":"10.1017\/CBO9780511809477"},{"key":"ref_128","first-page":"273","article-title":"Thermodynamik quantummechanischer Geshameiten","volume":"1","year":"1927","journal-title":"G\u00f6tt. Nach."},{"key":"ref_129","doi-asserted-by":"crossref","first-page":"797","DOI":"10.1016\/j.physa.2007.05.001","article-title":"Quantum games entropy","volume":"383","author":"Hidalgo","year":"2017","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_130","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1006\/game.1995.1023","article-title":"Quantal response equilbria for normal form games","volume":"10","author":"McKelvey","year":"1995","journal-title":"Games Econ. Behav."},{"key":"ref_131","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1257\/mic.20190013","article-title":"Stochastic equilibria: Noise in actions or beliefs?","volume":"14","author":"Friedman","year":"2022","journal-title":"Am. Econ. J. Microeconomics"},{"key":"ref_132","unstructured":"Bland, J.R. Bayesian inference for Quantal Response Equilibrium in normal-form games, Games Econ. Behav., in press."},{"key":"ref_133","unstructured":"Durlauf, S.N., and Blume, L.E. (2010). Quantal response equilibria. Behavioural and Experimental Economics, Palgrave Macmillan London. The New Palgrave Economics Collection."},{"key":"ref_134","first-page":"89","article-title":"Mixed strategy Nash Equilibrium and Quantal Response Equilibrium: An experimental comparison using RPS games","volume":"XX1","author":"Volacu","year":"2014","journal-title":"Theor. Appl. Econ."},{"key":"ref_135","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1103\/PhysRevLett.54.735","article-title":"Undecidability and intractability in theoretical physics","volume":"54","author":"Wolfram","year":"1985","journal-title":"Phys. Rev. Lett."},{"key":"ref_136","unstructured":"Hawking, S. (2024, January 13). G\u00f6del and the End of Physics. Lecture at the Dirac Centennial Celebration. Centre for Mathematical Sciences, University of Cambridge, Cambridge, UK. Available online: https:\/\/www.damtp.cam.ac.uk\/events\/strings02\/dirac\/hawking.html."},{"key":"ref_137","first-page":"226","article-title":"The strong free will theorem","volume":"56","author":"Conway","year":"2009","journal-title":"Not. AMS"}],"container-title":["Games"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-4336\/15\/5\/30\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:43:00Z","timestamp":1760110980000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-4336\/15\/5\/30"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,26]]},"references-count":137,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2024,10]]}},"alternative-id":["g15050030"],"URL":"https:\/\/doi.org\/10.3390\/g15050030","relation":{},"ISSN":["2073-4336"],"issn-type":[{"type":"electronic","value":"2073-4336"}],"subject":[],"published":{"date-parts":[[2024,8,26]]}}}