{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:42:39Z","timestamp":1760208159556,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2017,1,3]],"date-time":"2017-01-03T00:00:00Z","timestamp":1483401600000},"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>The outcome of many social and economic interactions, such as stock-market transactions, is strongly determined by the predictions that agents make about the behavior of other individuals. Cognitive hierarchy theory provides a framework to model the consequences of forecasting accuracy that has proven to fit data from certain types of game theory experiments, such as Keynesian beauty contests and entry games. Here, we focus on symmetric two-player-two-action games and establish an algorithm to find the players\u2019 strategies according to the cognitive hierarchy approach. We show that the snowdrift game exhibits a pattern of behavior whose complexity grows as the cognitive levels of players increases. In addition to finding the solutions up to the third cognitive level, we demonstrate, in this theoretical frame, two new properties of snowdrift games: (i) any snowdrift game can be characterized by only a parameter, its class; (ii) they are anti-symmetric with respect to the diagonal of the pay-off\u2019s space. Finally, we propose a model based on an evolutionary dynamics that captures the main features of the cognitive hierarchy theory.<\/jats:p>","DOI":"10.3390\/g8010001","type":"journal-article","created":{"date-parts":[[2017,1,3]],"date-time":"2017-01-03T10:26:49Z","timestamp":1483439209000},"page":"1","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Cognitive Hierarchy Theory and Two-Person Games"],"prefix":"10.3390","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9769-8796","authenticated-orcid":false,"given":"Carlos","family":"Gracia-L\u00e1zaro","sequence":"first","affiliation":[{"name":"Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza, 50018 Zaragoza, Spain"}]},{"given":"Luis","family":"Flor\u00eda","sequence":"additional","affiliation":[{"name":"Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza, 50018 Zaragoza, Spain"},{"name":"Departamento de F\u00edsica de la Materia Condensada, University of Zaragoza, 50009 Zaragoza, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0895-1893","authenticated-orcid":false,"given":"Yamir","family":"Moreno","sequence":"additional","affiliation":[{"name":"Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza, 50018 Zaragoza, Spain"},{"name":"Departamento de F\u00edsica Te\u00f3rica, University of Zaragoza, 50009 Zaragoza, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2017,1,3]]},"reference":[{"key":"ref_1","unstructured":"Keynes, J.M. (1936). The General Theory of Employment, Interest and Money, Harcourt Brace and Co."},{"key":"ref_2","unstructured":"Moulin, H. (1986). Game Theory for the Social Sciences, NYU Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"34","DOI":"10.2307\/2555533","article-title":"Cheap talk, coordination, and entry","volume":"18","author":"Farrell","year":"1987","journal-title":"RAND J. Econ."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/0165-4101(90)90048-9","article-title":"Financial disclosure policy in an entry game","volume":"12","author":"Darrough","year":"1990","journal-title":"J. Account. Econ."},{"key":"ref_5","unstructured":"Krugman, P. (1995). Peddling Prosperity: Economic Sense and Nonsense in the Age of Diminished Expectations, W. W. Norton and Co."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1006\/game.1997.0619","article-title":"Coordination, \u201cmagic,\u201d and reinforcement learning in a market entry game","volume":"23","author":"Erev","year":"1998","journal-title":"Games Econ. Behav."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1007\/PL00020947","article-title":"An experimental study of coordination and learning in iterated two-market entry games","volume":"16","author":"Rapoport","year":"2000","journal-title":"Econ. Theory"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1006\/game.1995.1031","article-title":"On players\u2019 models of other players: Theory and experimental evidence","volume":"10","author":"Stahl","year":"1995","journal-title":"Games Econ. Behav."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1193","DOI":"10.1111\/1468-0262.00239","article-title":"Cognition and behavior in normal-form games: An experimental study","volume":"69","author":"Crawford","year":"2001","journal-title":"Econometrica"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"861","DOI":"10.1162\/0033553041502225","article-title":"A cognitive hierarchy model of games","volume":"119","author":"Camerer","year":"2004","journal-title":"Q. J. Econ."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1731","DOI":"10.1257\/aer.97.5.1731","article-title":"Fatal attraction: Salience, naivete, and sophistication in experimental \u201chide-and-seek\u201d games","volume":"97","author":"Crawford","year":"2007","journal-title":"Am. Econ. Rev."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1721","DOI":"10.1111\/j.1468-0262.2007.00810.x","article-title":"Level-k Auctions: Can a Nonequilibrium Model of Strategic Thinking Explain the Winner\u2019s Curse and Overbidding in Private-Value Auctions?","volume":"75","author":"Crawford","year":"2007","journal-title":"Econometrica"},{"key":"ref_13","first-page":"729","article-title":"Stated beliefs and play in normal-form games","volume":"75","year":"2011","journal-title":"Rev. Econ. Stud."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1287\/mksc.1120.0770","article-title":"When do markets tip? A cognitive hierarchy approach","volume":"32","author":"Hossain","year":"2013","journal-title":"Market. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1146\/annurev.ps.31.020180.001125","article-title":"Social dilemmas","volume":"31","author":"Dawes","year":"1980","journal-title":"Annu. Rev. Psychol."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1146\/annurev.soc.24.1.183","article-title":"Social dilemmas: The anatomy of cooperation","volume":"24","author":"Kollock","year":"1998","journal-title":"Annu. Rev. Soc."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/j.obhdp.2012.11.003","article-title":"The psychology of social dilemmas: A review","volume":"120","author":"Joireman","year":"2013","journal-title":"Organ. Behav. Hum. Dec. Process."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Smith, J.M. (1982). Evolution and the Theory of Games, Cambridge University Press.","DOI":"10.1017\/CBO9780511806292"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Cooper, R. (1998). Coordination Games, Cambridge University Press.","DOI":"10.1017\/CBO9780511609428"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Skyrms, B. (2003). The Stag Hunt and the Evolution of Social Structure, Cambridge University Press.","DOI":"10.1017\/CBO9781139165228"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Sugden, R. (2005). The Economics of Rights, Cooperation and Welfare, Palgrave Macmillan. [2nd ed.].","DOI":"10.1057\/9780230536791"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1016\/j.geb.2005.12.006","article-title":"Anti-coordination and social interactions","volume":"58","year":"2007","journal-title":"Games Econ. Behav."},{"key":"ref_23","unstructured":"Myerson, R.B. (1991). Game Theory-Analysis of Conflict, Harvard University Press."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Gintis, H. (2009). Game Theory Evolving: A Problem-Centered Introduction to Evolutionary Game Theory, Princeton University Press. [2nd ed.].","DOI":"10.1515\/9781400830077"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Sigmund, K. (2010). The Calculus of Selfishness, Princeton University Press.","DOI":"10.1515\/9781400832255"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"011904","DOI":"10.1103\/PhysRevE.77.011904","article-title":"Social diversity and promotion of cooperation in the spatial prisoner\u2019s dilemma game","volume":"77","author":"Perc","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_27","unstructured":"Kagel, J.H., and Roth, A.E. (1997). The Handbook of Experimental Economics, Princeton University Press."},{"key":"ref_28","unstructured":"Camerer, C.F. (2003). Behavioral Game Theory: Experiments in Strategic Interaction, Princeton University Press."},{"key":"ref_29","first-page":"203","article-title":"A taxonomy of 2 \u00d7 2 games","volume":"11","author":"Rapoport","year":"1966","journal-title":"Gen. Syst."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1390","DOI":"10.1126\/science.7466396","article-title":"The evolution of cooperation","volume":"211","author":"Axelrod","year":"1981","journal-title":"Science"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"e1600451","DOI":"10.1126\/sciadv.1600451","article-title":"Humans display a reduced set of consistent behavioral phenotypes in dyadic games","volume":"2","author":"Vicens","year":"2016","journal-title":"Sci. Adv."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Hofbauer, J., and Sigmund, K. (1998). Evolutionary Games and Population Dynamics, Cambridge University Press.","DOI":"10.1017\/CBO9781139173179"},{"key":"ref_33","unstructured":"Gintis, H. (2000). Game, Theory Evolving, Princeton University Press."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1090\/S0273-0979-03-00988-1","article-title":"Evolutionary game dynamics","volume":"40","author":"Hofbauer","year":"2003","journal-title":"Bull. Am. Math. Soc."}],"container-title":["Games"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-4336\/8\/1\/1\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:25:23Z","timestamp":1760207123000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-4336\/8\/1\/1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1,3]]},"references-count":34,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2017,3]]}},"alternative-id":["g8010001"],"URL":"https:\/\/doi.org\/10.3390\/g8010001","relation":{},"ISSN":["2073-4336"],"issn-type":[{"type":"electronic","value":"2073-4336"}],"subject":[],"published":{"date-parts":[[2017,1,3]]}}}