{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,20]],"date-time":"2026-03-20T16:20:53Z","timestamp":1774023653133,"version":"3.50.1"},"reference-count":36,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2017,11,6]],"date-time":"2017-11-06T00:00:00Z","timestamp":1509926400000},"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>An interaction system has a finite set of agents that interact pairwise, depending on the current state of the system. Symmetric decomposition of the matrix of interaction coefficients yields the representation of states by self-adjoint matrices and hence a spectral representation. As a result, cooperation systems, decision systems and quantum systems all become visible as manifestations of special interaction systems. The treatment of the theory is purely mathematical and does not require any special knowledge of physics. It is shown how standard notions in cooperative game theory arise naturally in this context. In particular, states of general interaction systems are seen to arise as linear superpositions of pure quantum states and Fourier transformation to become meaningful. Moreover, quantum games fall into this framework. Finally, a theory of Markov evolution of interaction states is presented that generalizes classical homogeneous Markov chains to the present context.<\/jats:p>","DOI":"10.3390\/g8040048","type":"journal-article","created":{"date-parts":[[2017,11,6]],"date-time":"2017-11-06T11:39:38Z","timestamp":1509968378000},"page":"48","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Game Theoretic Interaction and Decision: A Quantum Analysis"],"prefix":"10.3390","volume":"8","author":[{"given":"Ulrich","family":"Faigle","sequence":"first","affiliation":[{"name":"Mathematisches Institut, Universit\u00e4t zu K\u00f6ln, Weyertal 80, 50931 K\u00f6ln, Germany"}]},{"given":"Michel","family":"Grabisch","sequence":"additional","affiliation":[{"name":"Paris School of Economics, University of Paris I, 106-112, Bd. de l\u2019H\u00f4pital, 75013 Paris, France"}]}],"member":"1968","published-online":{"date-parts":[[2017,11,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1287\/mnsc.18.5.64","article-title":"Multilinear extensions of games","volume":"18","author":"Owen","year":"1972","journal-title":"Manag. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1007\/s001820050125","article-title":"An axiomatic approach to the concept of interaction among players in cooperative games","volume":"28","author":"Grabisch","year":"1999","journal-title":"Int. J. Game Theory"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"875","DOI":"10.1007\/s00182-015-0490-x","article-title":"Bases and linear transforms of TU-games and cooperation systems","volume":"45","author":"Faigle","year":"2016","journal-title":"Int. J. Game Theory"},{"key":"ref_4","unstructured":"Penrose, R. (1994). Shadows of the Mind, Oxford University Press."},{"key":"ref_5","unstructured":"Grabbe, J.O. (2005). An Introduction to Quantum Game Theory. arXiv, 69."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1016\/j.dss.2008.07.001","article-title":"A survey of quantum games","volume":"46","author":"Guo","year":"2008","journal-title":"Decis. Support Syst."},{"key":"ref_7","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_8","doi-asserted-by":"crossref","first-page":"145002","DOI":"10.1088\/1751-8113\/49\/14\/145002","article-title":"Comonotonicity and Choquet integrals of Hermitian operators and their applications","volume":"49","author":"Vourdas","year":"2016","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Zhang, Q., Saad, W., Bennis, M., and Debbah, M. (2016, January 4\u20138). Quantum Game Theory for Beam Alignment in Millimeter Wave Device-to-Device Communications. Proceedings of the IEEE Global Communications Conference (GLOBECOM), Next Generation Networking Symposium, Washington, DC, USA.","DOI":"10.1109\/GLOCOM.2016.7842190"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/S0375-9601(02)00003-8","article-title":"Quantum cooperative games","volume":"293","author":"Iqbal","year":"2002","journal-title":"Phys. Lett. A"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"7645","DOI":"10.3390\/e17117645","article-title":"Quantum game beats classical odds\u2013Thermodynamics implications","volume":"17","author":"Levy","year":"2017","journal-title":"Entropy"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Wolpert, D.H. (2006). Information theory\u2013The bridge connecting bounded rational game theory and statistical physics. Complex Engineered Systems, Springer.","DOI":"10.1007\/3-540-32834-3_12"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1287\/moor.6.1.1","article-title":"Cooperative fuzzy games","volume":"6","author":"Aubin","year":"1981","journal-title":"Math. Oper. Res."},{"key":"ref_14","unstructured":"Nielsen, M., and Chuang, I. (2000). Quantum Computation, Cambrigde University Press."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1090\/S0002-9904-1960-10418-1","article-title":"Von Neumann-Morgenstern solutions to cooperative games without side payments","volume":"66","author":"Aumann","year":"1960","journal-title":"Bull. Amer. Math. Soc."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Bilbao, J.M. (2000). Bicooperative games. Cooperative Games on Combinatorial Structures, Kluwer Academic Publishers.","DOI":"10.1007\/978-1-4615-4393-0"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1007\/s00182-008-0126-5","article-title":"A value for bi-cooperative games","volume":"37","author":"Labreuche","year":"2008","journal-title":"Int. J. Game Theory"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Faigle, U., Kern, W., and Still, G. (2002). Algorithmic Principles of Mathematical Programming, Springer.","DOI":"10.1007\/978-94-015-9896-5"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/j.geb.2010.06.003","article-title":"Influence functions, followers and command games","volume":"72","author":"Grabisch","year":"2011","journal-title":"Games Econ. Behav."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"316","DOI":"10.1016\/j.mathsocsci.2013.07.003","article-title":"A model of influence based on aggregation functions","volume":"66","author":"Grabisch","year":"2013","journal-title":"Math. Soc. Sci."},{"key":"ref_21","first-page":"307","article-title":"A value for n-person games","volume":"Volume II","author":"Kuhn","year":"1953","journal-title":"Contributions to the Theory of Games"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Jackson, M.O. (2008). Social and Economic Networks, Princeton University Press.","DOI":"10.1515\/9781400833993"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1007\/BF01258281","article-title":"Monotonicity and dummy free property for multi-choice games","volume":"21","author":"Hsiao","year":"1992","journal-title":"Int. J. Game Theory"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"814","DOI":"10.1016\/j.disc.2008.01.019","article-title":"Weighted lattice polynomials","volume":"309","author":"Marichal","year":"2009","journal-title":"Discret. Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"340","DOI":"10.1007\/BF00531932","article-title":"On the foundations of combinatorial theory I. Theory of M\u00f6bius functions","volume":"2","author":"Rota","year":"1964","journal-title":"Zeitschrift f\u00fcr Wahrscheinlichkeitstheorie und Verwandte Gebiete"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"5","DOI":"10.2307\/2387224","article-title":"A closed set of normal orthogonal functions","volume":"45","author":"Walsh","year":"1923","journal-title":"Am. J. Math."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Hammer, P.L., and Rudeanu, S. (1968). Boolean Methods in Operations Research and Related Areas, Springer.","DOI":"10.1007\/978-3-642-85823-9"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"412","DOI":"10.1016\/S0196-8858(02)00023-4","article-title":"A Fourier-theoretic perspective on the Condorcet paradox and Arrow\u2019s theorem","volume":"29","author":"Kalai","year":"2002","journal-title":"Adv. Appl. Math."},{"key":"ref_29","unstructured":"O\u2019Donnell, R. (2014). Analysis of Boolean Functions, Cambridge University Press."},{"key":"ref_30","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_31","doi-asserted-by":"crossref","first-page":"2342","DOI":"10.1109\/TIT.2007.899514","article-title":"Asymptotic Mean Stationarity of Sources with Finite Evolution Dimension","volume":"53","author":"Faigle","year":"2007","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1007\/s00199-011-0617-7","article-title":"Values for Markovian coalition processes","volume":"51","author":"Faigle","year":"2012","journal-title":"Econ. Theory"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1109\/MSP.2015.2451994","article-title":"Game Theory for Networks: A tutorial on game-theoretic tools for emerging signal processing applications","volume":"33","author":"Bacci","year":"2016","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Faigle, U., and Gierz, G. (2017). Markovian statistics on evolving systems. Evol. Syst.","DOI":"10.1007\/s12530-017-9186-8"},{"key":"ref_35","unstructured":"Puterman, M. (2005). Markov Decision Processes: Discrete Stochastic Dynamic Programming, John Wiley."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Lozovanu, D., and Pickl, S. (2015). Optimization of Stochastic Discrete Systems and Control on Complex Networks, Springer.","DOI":"10.1007\/978-3-319-11833-8"}],"container-title":["Games"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-4336\/8\/4\/48\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:48:22Z","timestamp":1760208502000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-4336\/8\/4\/48"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,11,6]]},"references-count":36,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2017,12]]}},"alternative-id":["g8040048"],"URL":"https:\/\/doi.org\/10.3390\/g8040048","relation":{},"ISSN":["2073-4336"],"issn-type":[{"value":"2073-4336","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,11,6]]}}}