{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T14:33:41Z","timestamp":1774967621387,"version":"3.50.1"},"reference-count":84,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2023,8,26]],"date-time":"2023-08-26T00:00:00Z","timestamp":1693008000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,8,26]],"date-time":"2023-08-26T00:00:00Z","timestamp":1693008000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100007768","name":"Max Planck Institute for Evolutionary Anthropology","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100007768","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Dyn Games Appl"],"published-print":{"date-parts":[[2023,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The prisoner\u2019s dilemma, the snowdrift game, and the stag hunt are two-player symmetric games that are often considered as prototypical examples of cooperative dilemmas across disciplines. However, surprisingly little consensus exists about the precise mathematical meaning of the words \u201ccooperation\u201d and \u201ccooperative dilemma\u201d for these and other binary-action symmetric games, in particular when considering interactions among more than two players. Here, we propose definitions of these terms and explore their evolutionary consequences on the equilibrium structure of cooperative dilemmas in relation to social optimality. We show that our definition of cooperative dilemma encompasses a large class of collective action games often discussed in the literature, including congestion games, games with participation synergies, and public goods games. One of our main results is that regardless of the number of players, all cooperative dilemmas\u2014including multi-player generalizations of the prisoner\u2019s dilemma, the snowdrift game, and the stag hunt\u2014feature inefficient equilibria where cooperation is underprovided, but cannot have equilibria in which cooperation is overprovided. We also find simple conditions for full cooperation to be socially optimal in a cooperative dilemma. Our framework and results unify, simplify, and extend previous work on the structure and properties of cooperative dilemmas with binary actions and two or more players.<\/jats:p>","DOI":"10.1007\/s13235-023-00524-9","type":"journal-article","created":{"date-parts":[[2023,8,26]],"date-time":"2023-08-26T15:02:10Z","timestamp":1693062130000},"page":"1156-1193","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Cooperative Dilemmas with Binary Actions and Multiple Players"],"prefix":"10.1007","volume":"13","author":[{"given":"Jorge","family":"Pe\u00f1a","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Georg","family":"N\u00f6ldeke","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,8,26]]},"reference":[{"key":"524_CR1","doi-asserted-by":"publisher","first-page":"120","DOI":"10.1016\/j.jebo.2005.05.006","volume":"63","author":"SP Anderson","year":"2007","unstructured":"Anderson SP, Engers M (2007) Participation games: market entry, coordination, and the beautiful blonde. J Econ Behav Org 63:120\u2013137","journal-title":"J Econ Behav Org"},{"key":"524_CR2","doi-asserted-by":"publisher","first-page":"1140","DOI":"10.1111\/j.1558-5646.2010.01185.x","volume":"65","author":"M Archetti","year":"2011","unstructured":"Archetti M, Scheuring I (2011) Coexistence of cooperation and defection in public goods games. Evolution 65:1140\u20131148","journal-title":"Evolution"},{"key":"524_CR3","first-page":"406","volume":"84","author":"WB Arthur","year":"1994","unstructured":"Arthur WB (1994) Inductive reasoning and bounded rationality. Am Econ Rev 84:406\u2013411","journal-title":"Am Econ Rev"},{"key":"524_CR4","doi-asserted-by":"publisher","first-page":"1390","DOI":"10.1126\/science.7466396","volume":"211","author":"R Axelrod","year":"1981","unstructured":"Axelrod R, Hamilton WD (1981) The evolution of cooperation. Science 211:1390\u20131396","journal-title":"Science"},{"key":"524_CR5","doi-asserted-by":"publisher","first-page":"426","DOI":"10.1016\/j.jtbi.2005.06.007","volume":"238","author":"LA Bach","year":"2006","unstructured":"Bach LA, Helvik T, Christiansen FB (2006) The evolution of n-player cooperation-threshold games and ESS bifurcations. J Theor Biol 238:426\u2013434","journal-title":"J Theor Biol"},{"key":"524_CR6","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1016\/0047-2727(86)90024-1","volume":"29","author":"T Bergstrom","year":"1986","unstructured":"Bergstrom T, Blume L, Varian H (1986) On the private provision of public goods. J Public Econ 29:25\u201349","journal-title":"J Public Econ"},{"key":"524_CR7","doi-asserted-by":"publisher","first-page":"687","DOI":"10.1177\/002200277602000406","volume":"20","author":"P Bonacich","year":"1976","unstructured":"Bonacich P, Shure GH, Kahan JP, Meeker RJ (1976) Cooperation and group size in the n-person prisoners\u2019 dilemma. J Conflict Resolut 20:687\u2013706","journal-title":"J Conflict Resolut"},{"key":"524_CR8","doi-asserted-by":"publisher","first-page":"931","DOI":"10.1007\/BF02460000","volume":"59","author":"M Broom","year":"1997","unstructured":"Broom M, Cannings C, Vickers GT (1997) Multi-player matrix games. Bull Math Biol 59:931\u2013952","journal-title":"Bull Math Biol"},{"key":"524_CR9","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1007\/s001820400183","volume":"33","author":"M Bukowski","year":"2004","unstructured":"Bukowski M, Miekisz J (2004) Evolutionary and asymptotic stability in symmetric multi-player games. Int J Game Theory 33:41\u201354","journal-title":"Int J Game Theory"},{"key":"524_CR10","volume-title":"Frontiers of game theory","author":"H Carlsson","year":"1993","unstructured":"Carlsson H, van Damme E (1993) Equilibrium selection in stag hunt games. In: Binmore KG, Tani P (eds) Frontiers of game theory. MIT Press, Cambridge"},{"key":"524_CR11","doi-asserted-by":"publisher","first-page":"1640","DOI":"10.1126\/science.284.5420.1640","volume":"284","author":"TH Clutton-Brock","year":"1999","unstructured":"Clutton-Brock TH, O\u2019riain M, Brotherton PN, Gaynor D, Kansky R, Griffin AS, Manser M (1999) Selfish sentinels in cooperative mammals. Science 284:1640\u20131644","journal-title":"Science"},{"key":"524_CR12","doi-asserted-by":"publisher","first-page":"276","DOI":"10.1016\/0040-5809(76)90020-4","volume":"10","author":"D Cohen","year":"1976","unstructured":"Cohen D, Eshel I (1976) On the founder effect and the evolution of altruistic traits. Theor Popul Biol 10:276\u2013302","journal-title":"Theor Popul Biol"},{"key":"524_CR13","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1146\/annurev.ps.31.020180.001125","volume":"31","author":"RM Dawes","year":"1980","unstructured":"Dawes RM (1980) Social dilemmas. Annu Rev Psychol 31:169\u2013193","journal-title":"Annu Rev Psychol"},{"key":"524_CR14","doi-asserted-by":"publisher","first-page":"1171","DOI":"10.2307\/1960862","volume":"80","author":"RM Dawes","year":"1986","unstructured":"Dawes RM, Orbell JM, Simmons RT, Van De Kragt AJC (1986) Organizing groups for collective action. Am Polit Sci Rev 80:1171\u20131185","journal-title":"Am Polit Sci Rev"},{"key":"524_CR15","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.tpb.2016.09.003","volume":"113","author":"K De Jaegher","year":"2017","unstructured":"De Jaegher K (2017) Harsh environments and the evolution of multi-player cooperation. Theor Popul Biol 113:1\u201312","journal-title":"Theor Popul Biol"},{"key":"524_CR16","doi-asserted-by":"publisher","first-page":"18","DOI":"10.1016\/j.jtbi.2018.10.006","volume":"460","author":"K De Jaegher","year":"2019","unstructured":"De Jaegher K (2019) Harsh environments: Multi-player cooperation with excludability and congestion. J Theor Biol 460:18\u201336","journal-title":"J Theor Biol"},{"key":"524_CR17","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02888-9","volume-title":"Constructive approximation","author":"RA DeVore","year":"1993","unstructured":"DeVore RA, Lorentz GG (1993) Constructive approximation. Springer, New York"},{"key":"524_CR18","doi-asserted-by":"publisher","first-page":"605","DOI":"10.1177\/0022002785029004003","volume":"29","author":"A Diekmann","year":"1985","unstructured":"Diekmann A (1985) Volunteer\u2019s dilemma. J Confl Resolut 29:605\u2013610","journal-title":"J Confl Resolut"},{"key":"524_CR19","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1007\/s10614-011-9253-3","volume":"37","author":"P Dindo","year":"2011","unstructured":"Dindo P, Tuinstra J (2011) A class of evolutionary models for participation games with negative feedback. Comput Econ 37:267\u2013300","journal-title":"Comput Econ"},{"key":"524_CR20","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1016\/S0047-2727(99)00089-4","volume":"76","author":"A Dixit","year":"2000","unstructured":"Dixit A, Olson M (2000) Does voluntary participation undermine the Coase theorem? J Public Econ 76:309\u2013335","journal-title":"J Public Econ"},{"key":"524_CR21","doi-asserted-by":"publisher","first-page":"748","DOI":"10.1111\/j.1461-0248.2005.00773.x","volume":"8","author":"M Doebeli","year":"2005","unstructured":"Doebeli M, Hauert C (2005) Models of cooperation based on the prisoner\u2019s dilemma and the snowdrift game. Ecol Lett 8:748\u2013766","journal-title":"Ecol Lett"},{"key":"524_CR22","doi-asserted-by":"publisher","first-page":"6212","DOI":"10.1038\/s41598-017-06063-9","volume":"7","author":"M dos Santos","year":"2017","unstructured":"dos Santos M, Pe\u00f1a J (2017) Antisocial rewarding in structured populations. Sci Rep 7:6212","journal-title":"Sci Rep"},{"key":"524_CR23","doi-asserted-by":"publisher","first-page":"1331","DOI":"10.1073\/pnas.79.4.1331","volume":"79","author":"I Eshel","year":"1982","unstructured":"Eshel I, Cavalli-Sforza LL (1982) Assortment of encounters and evolution of cooperativeness. Proc Natl Acad Sci 79:1331\u20131335","journal-title":"Proc Natl Acad Sci"},{"key":"524_CR24","doi-asserted-by":"publisher","first-page":"379","DOI":"10.1016\/j.cagd.2012.03.001","volume":"29","author":"RT Farouki","year":"2012","unstructured":"Farouki RT (2012) The Bernstein polynomial basis: a centennial retrospective. Comput Aided Geom Des 29:379\u2013419","journal-title":"Comput Aided Geom Des"},{"key":"524_CR25","doi-asserted-by":"publisher","first-page":"13","DOI":"10.1098\/rspb.2008.0829","volume":"276","author":"JA Fletcher","year":"2008","unstructured":"Fletcher JA, Doebeli M (2008) A simple and general explanation for the evolution of altruism. Proc R Soc B Biol Sci 276:13\u201319","journal-title":"Proc R Soc B Biol Sci"},{"key":"524_CR26","volume-title":"Game theory","author":"D Fudenberg","year":"1991","unstructured":"Fudenberg D, Tirole J (1991) Game theory. MIT Press, Cambridge"},{"key":"524_CR27","doi-asserted-by":"publisher","first-page":"468","DOI":"10.1007\/s13235-014-0106-2","volume":"4","author":"CS Gokhale","year":"2014","unstructured":"Gokhale CS, Traulsen A (2014) Evolutionary multiplayer games. Dyn Games Appl 4:468\u2013488","journal-title":"Dyn Games Appl"},{"key":"524_CR28","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1007\/BF01560583","volume":"7","author":"M Gradstein","year":"1990","unstructured":"Gradstein M, Nitzan S (1990) Binary participation and incremental provision of public goods. Soc Choice Welf 7:171\u2013192","journal-title":"Soc Choice Welf"},{"key":"524_CR29","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0022-5193(64)90038-4","volume":"7","author":"WD Hamilton","year":"1964","unstructured":"Hamilton WD (1964) The genetical evolution of social behaviour. I. J Theor Biol 7:1\u201316","journal-title":"J Theor Biol"},{"key":"524_CR30","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1016\/0022-5193(64)90039-6","volume":"7","author":"WD Hamilton","year":"1964","unstructured":"Hamilton WD (1964) The genetical evolution of social behaviour. II. J Theor Biol 7:17\u201352","journal-title":"J Theor Biol"},{"key":"524_CR31","doi-asserted-by":"publisher","first-page":"1243","DOI":"10.1126\/science.162.3859.1243","volume":"162","author":"G Hardin","year":"1968","unstructured":"Hardin G (1968) The tragedy of the commons. Science 162:1243\u20131248","journal-title":"Science"},{"key":"524_CR32","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1016\/j.jtbi.2005.08.040","volume":"239","author":"C Hauert","year":"2006","unstructured":"Hauert C, Michor F, Nowak MA, Doebeli M (2006) Synergy and discounting of cooperation in social dilemmas. J Theor Biol 239:195\u2013202","journal-title":"J Theor Biol"},{"key":"524_CR33","doi-asserted-by":"publisher","first-page":"16425","DOI":"10.1073\/pnas.1407887111","volume":"111","author":"C Hilbe","year":"2014","unstructured":"Hilbe C, Wu B, Traulsen A, Nowak MA (2014) Cooperation and control in multiplayer social dilemmas. Proc Natl Acad Sci 111:16425\u201316430","journal-title":"Proc Natl Acad Sci"},{"key":"524_CR34","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1016\/j.tree.2003.10.004","volume":"19","author":"B Kerr","year":"2004","unstructured":"Kerr B, Godfrey-Smith P, Feldman MW (2004) What is altruism? Trends Ecol Evol 19:135\u2013140","journal-title":"Trends Ecol Evol"},{"key":"524_CR35","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1146\/annurev.soc.24.1.183","volume":"24","author":"P Kollock","year":"1998","unstructured":"Kollock P (1998) Social dilemmas: the anatomy of cooperation. Ann Rev Sociol 24:183\u2013214","journal-title":"Ann Rev Sociol"},{"key":"524_CR36","doi-asserted-by":"publisher","first-page":"1365","DOI":"10.1111\/j.1420-9101.2006.01119.x","volume":"19","author":"L Lehmann","year":"2006","unstructured":"Lehmann L, Keller L (2006) The evolution of cooperation and altruism\u2014a general framework and a classification of models. J Evol Biol 19:1365\u20131376","journal-title":"J Evol Biol"},{"key":"524_CR37","first-page":"61","volume":"24","author":"AN Licht","year":"1999","unstructured":"Licht AN (1999) Games commissions play: 2$$\\times $$2 games of international securities regulation. Yale J Int Law 24:61","journal-title":"Yale J Int Law"},{"key":"524_CR38","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2021.132943","volume":"424","author":"Q Luo","year":"2021","unstructured":"Luo Q, Liu L, Chen X (2021) Evolutionary dynamics of cooperation in the n-person stag hunt game. Physica D 424:132943","journal-title":"Physica D"},{"key":"524_CR39","doi-asserted-by":"publisher","first-page":"292","DOI":"10.1016\/j.geb.2008.11.003","volume":"67","author":"M Makris","year":"2009","unstructured":"Makris M (2009) Private provision of discrete public goods. Games Econom Behav 67:292\u2013299","journal-title":"Games Econom Behav"},{"key":"524_CR40","doi-asserted-by":"publisher","first-page":"360","DOI":"10.1016\/0040-5809(76)90053-8","volume":"9","author":"C Matessi","year":"1976","unstructured":"Matessi C, Jayakar SD (1976) Conditions for the evolution of altruism under Darwinian selection. Theor Popul Biol 9:360\u2013387","journal-title":"Theor Popul Biol"},{"key":"524_CR41","doi-asserted-by":"publisher","first-page":"1754","DOI":"10.1073\/pnas.81.6.1754","volume":"81","author":"C Matessi","year":"1984","unstructured":"Matessi C, Karlin S (1984) On the evolution of altruism by kin selection. Proc Natl Acad Sci U S A 81:1754\u20131758","journal-title":"Proc Natl Acad Sci U S A"},{"key":"524_CR42","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1038\/246015a0","volume":"246","author":"J Maynard Smith","year":"1973","unstructured":"Maynard Smith J, Price GR (1973) The logic of animal conflict. Nature 246:15\u201318","journal-title":"Nature"},{"key":"524_CR43","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780198815778.001.0001","volume-title":"Game theory in biology: concepts and frontiers","author":"JM McNamara","year":"2020","unstructured":"McNamara JM, Leimar O (2020) Game theory in biology: concepts and frontiers. Oxford University Press, Oxford"},{"key":"524_CR44","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1007\/s00199-005-0611-z","volume":"28","author":"FM Menezes","year":"2006","unstructured":"Menezes FM, Pitchford R (2006) Binary games with many players. Econ Theor 28:125\u2013143","journal-title":"Econ Theor"},{"key":"524_CR45","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1016\/S0022-5193(05)80358-3","volume":"151","author":"U Motro","year":"1991","unstructured":"Motro U (1991) Co-operation and defection: playing the field and the ESS. J Theor Biol 151:145\u2013154","journal-title":"J Theor Biol"},{"key":"524_CR46","doi-asserted-by":"publisher","first-page":"499","DOI":"10.1111\/j.1467-937X.2008.00482.x","volume":"75","author":"DP Myatt","year":"2008","unstructured":"Myatt DP, Wallace C (2008) When does one bad apple spoil the barrel? An evolutionary analysis of collective action. Rev Econ Stud 75:499\u2013527","journal-title":"Rev Econ Stud"},{"key":"524_CR47","doi-asserted-by":"publisher","first-page":"1560","DOI":"10.1126\/science.1133755","volume":"314","author":"MA Nowak","year":"2006","unstructured":"Nowak MA (2006) Five rules for the evolution of cooperation. Science 314:1560\u20131563","journal-title":"Science"},{"key":"524_CR48","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.jtbi.2012.01.014","volume":"299","author":"MA Nowak","year":"2012","unstructured":"Nowak MA (2012) Evolving cooperation. J Theor Biol 299:1\u20138","journal-title":"J Theor Biol"},{"key":"524_CR49","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1016\/j.geb.2016.06.016","volume":"99","author":"G N\u00f6ldeke","year":"2016","unstructured":"N\u00f6ldeke G, Pe\u00f1a J (2016) The symmetric equilibria of symmetric voter participation games with complete information. Games Econ Behav 99:71\u201381","journal-title":"Games Econ Behav"},{"key":"524_CR50","doi-asserted-by":"publisher","first-page":"42","DOI":"10.1016\/j.jmateco.2020.02.003","volume":"88","author":"G N\u00f6ldeke","year":"2020","unstructured":"N\u00f6ldeke G, Pe\u00f1a J (2020) Group size and collective action in a binary contribution game. J Math Econ 88:42\u201351","journal-title":"J Math Econ"},{"key":"524_CR51","doi-asserted-by":"crossref","DOI":"10.4159\/9780674041660","volume-title":"The logic of collective action: public goods and the theory of groups","author":"M Olson","year":"1965","unstructured":"Olson M (1965) The logic of collective action: public goods and the theory of groups. Harvard University Press, Cambridge"},{"key":"524_CR52","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511807763","volume-title":"Governing the commons: the evolution of institutions for collective action","author":"E Ostrom","year":"1990","unstructured":"Ostrom E (1990) Governing the commons: the evolution of institutions for collective action. Cambridge University Press, Cambridge"},{"key":"524_CR53","doi-asserted-by":"publisher","first-page":"315","DOI":"10.1098\/rspb.2008.1126","volume":"276","author":"JM Pacheco","year":"2009","unstructured":"Pacheco JM, Santos FC, Souza MO, Skyrms B (2009) Evolutionary dynamics of collective action in n-person stag hunt dilemmas. Proc R Soc B Biol Sci 276:315\u2013321","journal-title":"Proc R Soc B Biol Sci"},{"key":"524_CR54","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1016\/0047-2727(84)90023-9","volume":"24","author":"T Palfrey","year":"1984","unstructured":"Palfrey T, Rosenthal H (1984) Participation and the provision of discrete public goods: a strategic analysis. J Public Econ 24:171\u2013193","journal-title":"J Public Econ"},{"key":"524_CR55","doi-asserted-by":"publisher","first-page":"7","DOI":"10.1007\/BF00124048","volume":"41","author":"TR Palfrey","year":"1983","unstructured":"Palfrey TR, Rosenthal H (1983) A strategic calculus of voting. Public Choice 41:7\u201353","journal-title":"Public Choice"},{"key":"524_CR56","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1016\/j.jtbi.2018.08.004","volume":"457","author":"J Pe\u00f1a","year":"2018","unstructured":"Pe\u00f1a J, N\u00f6ldeke G (2018) Group size effects in social evolution. J Theor Biol 457:211\u2013220","journal-title":"J Theor Biol"},{"key":"524_CR57","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1016\/j.jtbi.2013.12.016","volume":"346","author":"J Pe\u00f1a","year":"2014","unstructured":"Pe\u00f1a J, Lehmann L, N\u00f6ldeke G (2014) Gains from switching and evolutionary stability in multi-player matrix games. J Theor Biol 346:23\u201333","journal-title":"J Theor Biol"},{"key":"524_CR58","doi-asserted-by":"publisher","first-page":"72","DOI":"10.1016\/j.jtbi.2015.10.023","volume":"389","author":"J Pe\u00f1a","year":"2016","unstructured":"Pe\u00f1a J, N\u00f6ldeke G (2016) Variability in group size and the evolution of collective action. J Theor Biol 389:72\u201382","journal-title":"J Theor Biol"},{"key":"524_CR59","doi-asserted-by":"publisher","first-page":"122","DOI":"10.1016\/j.jtbi.2015.06.039","volume":"382","author":"J Pe\u00f1a","year":"2015","unstructured":"Pe\u00f1a J, N\u00f6ldeke G, Lehmann L (2015) Evolutionary dynamics of collective action in spatially structured populations. J Theor Biol 382:122\u2013136","journal-title":"J Theor Biol"},{"key":"524_CR60","doi-asserted-by":"publisher","first-page":"20150881","DOI":"10.1098\/rsif.2015.0881","volume":"13","author":"J Pe\u00f1a","year":"2016","unstructured":"Pe\u00f1a J, Wu B, Traulsen A (2016) Ordering structured populations in multiplayer cooperation games. J R Soc Interface 13:20150881","journal-title":"J R Soc Interface"},{"key":"524_CR61","doi-asserted-by":"publisher","first-page":"67","DOI":"10.1007\/s13235-015-0174-y","volume":"7","author":"T P\u0142atkowski","year":"2017","unstructured":"P\u0142atkowski T (2017) On derivation and evolutionary classification of social dilemma games. Dyn Games Appl 7:67\u201375","journal-title":"Dyn Games Appl"},{"key":"524_CR62","doi-asserted-by":"publisher","first-page":"413","DOI":"10.1016\/j.tics.2013.06.003","volume":"17","author":"DG Rand","year":"2013","unstructured":"Rand DG, Nowak MA (2013) Human cooperation. Trends Cogn Sci 17:413\u2013425","journal-title":"Trends Cogn Sci"},{"key":"524_CR63","doi-asserted-by":"publisher","first-page":"74","DOI":"10.1037\/0033-295X.94.1.74","volume":"94","author":"A Rapoport","year":"1987","unstructured":"Rapoport A (1987) Research paradigms and expected utility models for the provision of step-level public goods. Psychol Rev 94:74\u201383","journal-title":"Psychol Rev"},{"key":"524_CR64","doi-asserted-by":"publisher","DOI":"10.3998\/mpub.20269","volume-title":"Prisoner\u2019s dilemma: a study in conflict and cooperation","author":"A Rapoport","year":"1965","unstructured":"Rapoport A, Chammah AM (1965) Prisoner\u2019s dilemma: a study in conflict and cooperation, vol 165. University of Michigan Press, Ann Arbor"},{"key":"524_CR65","doi-asserted-by":"publisher","first-page":"10","DOI":"10.1177\/000276426601000303","volume":"10","author":"A Rapoport","year":"1966","unstructured":"Rapoport A, Chammah AM (1966) The game of chicken. Am Behav Sci 10:10\u201328","journal-title":"Am Behav Sci"},{"key":"524_CR66","doi-asserted-by":"publisher","DOI":"10.1515\/9781400847242","volume-title":"Genetic structure and selection in subdivided populations","author":"F Rousset","year":"2004","unstructured":"Rousset F (2004) Genetic structure and selection in subdivided populations, vol 40. Princeton University Press, Princeton"},{"key":"524_CR67","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1086\/383541","volume":"79","author":"J Sachs","year":"2004","unstructured":"Sachs J, Mueller U, Wilcox T, Bull J (2004) The evolution of cooperation. Q Rev Biol 79:135\u2013160","journal-title":"Q Rev Biol"},{"key":"524_CR68","doi-asserted-by":"publisher","first-page":"582","DOI":"10.1086\/261768","volume":"99","author":"RK Sah","year":"1991","unstructured":"Sah RK (1991) The effects of child mortality changes on fertility choice and parental welfare. J Polit Econ 99:582\u2013606","journal-title":"J Polit Econ"},{"key":"524_CR69","doi-asserted-by":"publisher","first-page":"10421","DOI":"10.1073\/pnas.1015648108","volume":"108","author":"FC Santos","year":"2011","unstructured":"Santos FC, Pacheco JM (2011) Risk of collective failure provides an escape from the tragedy of the commons. Proc Natl Acad Sci 108:10421\u201310425","journal-title":"Proc Natl Acad Sci"},{"key":"524_CR70","doi-asserted-by":"publisher","first-page":"1583","DOI":"10.2105\/AJPH.2008.152629","volume":"99","author":"G Siegal","year":"2009","unstructured":"Siegal G, Siegal N, Bonnie RJ (2009) An account of collective actions in public health. Am J Public Health 99:1583\u20131587","journal-title":"Am J Public Health"},{"key":"524_CR71","volume-title":"The stag hunt and the evolution of social structure","author":"B Skyrms","year":"2004","unstructured":"Skyrms B (2004) The stag hunt and the evolution of social structure. Cambridge University Press, Cambridge"},{"key":"524_CR72","doi-asserted-by":"publisher","first-page":"698","DOI":"10.1111\/j.1558-5646.2012.01831.x","volume":"67","author":"R Smead","year":"2013","unstructured":"Smead R, Forber P (2013) The evolutionary dynamics of spite in finite populations. Evolution 67:698\u2013707","journal-title":"Evolution"},{"key":"524_CR73","doi-asserted-by":"publisher","first-page":"581","DOI":"10.1016\/j.jtbi.2009.07.010","volume":"260","author":"MO Souza","year":"2009","unstructured":"Souza MO, Pacheco JM, Santos FC (2009) Evolution of cooperation under n-person snowdrift games. J Theor Biol 260:581\u2013588","journal-title":"J Theor Biol"},{"key":"524_CR74","doi-asserted-by":"publisher","first-page":"350","DOI":"10.1111\/j.1467-9248.1982.tb00545.x","volume":"30","author":"M Taylor","year":"1982","unstructured":"Taylor M, Ward H (1982) Chickens, whales, and lumpy goods: alternative models of public-goods provision. Polit Stud 30:350\u2013370","journal-title":"Polit Stud"},{"key":"524_CR75","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1016\/0025-5564(78)90077-9","volume":"40","author":"PD Taylor","year":"1978","unstructured":"Taylor PD, Jonker LB (1978) Evolutionary stable strategies and game dynamics. Math Biosci 40:145\u2013156","journal-title":"Math Biosci"},{"key":"524_CR76","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.2303546120","volume":"120","author":"A Traulsen","year":"2023","unstructured":"Traulsen A, Levin SA, Saad-Roy CM (2023) Individual costs and societal benefits of interventions during the COVID-19 pandemic. Proc Natl Acad Sci 120:e2303546120","journal-title":"Proc Natl Acad Sci"},{"key":"524_CR77","doi-asserted-by":"publisher","first-page":"380","DOI":"10.1016\/0040-5809(80)90033-7","volume":"17","author":"M Uyenoyama","year":"1980","unstructured":"Uyenoyama M, Feldman MW (1980) Theories of kin and group selection: a population genetics perspective. Theor Popul Biol 17:380\u2013414","journal-title":"Theor Popul Biol"},{"key":"524_CR78","doi-asserted-by":"publisher","first-page":"2","DOI":"10.1016\/j.tpb.2015.05.002","volume":"103","author":"J Van Cleve","year":"2015","unstructured":"Van Cleve J (2015) Social evolution and genetic interactions in the short and long term. Theor Popul Biol 103:2\u201326","journal-title":"Theor Popul Biol"},{"key":"524_CR79","first-page":"2245","volume":"68","author":"J Van Cleve","year":"2014","unstructured":"Van Cleve J, Ak\u00e7ay E (2014) Pathways to social evolution: reciprocity, relatedness, and synergy. Evolution 68:2245\u20132258","journal-title":"Evolution"},{"key":"524_CR80","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1016\/j.obhdp.2012.11.003","volume":"120","author":"PAM Van Lange","year":"2013","unstructured":"Van Lange PAM, Joireman J, Parks CD, Van Dijk E (2013) The psychology of social dilemmas: a review. Organ Behav Hum Decis Process 120:125\u2013141","journal-title":"Organ Behav Hum Decis Process"},{"key":"524_CR81","doi-asserted-by":"publisher","first-page":"600","DOI":"10.1177\/0022002798042005004","volume":"42","author":"J Weesie","year":"1998","unstructured":"Weesie J, Franzen A (1998) Cost sharing in a Volunteer\u2019s dilemma. J Conflict Resolut 42:600\u2013618","journal-title":"J Conflict Resolut"},{"key":"524_CR82","volume-title":"Evolutionary game theory","author":"JW Weibull","year":"1995","unstructured":"Weibull JW (1995) Evolutionary game theory. MIT Press, Cambridge"},{"key":"524_CR83","doi-asserted-by":"publisher","first-page":"R661","DOI":"10.1016\/j.cub.2007.06.004","volume":"17","author":"SA West","year":"2007","unstructured":"West SA, Griffin AS, Gardner A (2007) Evolutionary explanations for cooperation. Curr Biol 17:R661\u2013R672","journal-title":"Curr Biol"},{"key":"524_CR84","doi-asserted-by":"publisher","first-page":"382","DOI":"10.1016\/j.jtbi.2007.03.015","volume":"247","author":"G Wild","year":"2007","unstructured":"Wild G, Traulsen A (2007) The different limits of weak selection and the evolutionary dynamics of finite populations. J Theor Biol 247:382\u2013390","journal-title":"J Theor Biol"}],"container-title":["Dynamic Games and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s13235-023-00524-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s13235-023-00524-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s13235-023-00524-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,26]],"date-time":"2024-10-26T22:25:56Z","timestamp":1729981556000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s13235-023-00524-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,26]]},"references-count":84,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2023,12]]}},"alternative-id":["524"],"URL":"https:\/\/doi.org\/10.1007\/s13235-023-00524-9","relation":{},"ISSN":["2153-0785","2153-0793"],"issn-type":[{"value":"2153-0785","type":"print"},{"value":"2153-0793","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,8,26]]},"assertion":[{"value":"3 August 2023","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 August 2023","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no competing interests to declare.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval."}}]}}