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Effectiveness of conditional punishment for the evolution of public cooperation
Open accessAbstract
Collective actions, from city marathons to labor strikes, are often mass-driven and subject to the snowball effect. Motivated by this, we study evolutionary advantages of conditional punishment in the spatial public goods game. Unlike unconditional punishers who always impose the same fines on defectors, conditional punishers do so proportionally with the number of other punishers in the group. Phase diagrams in dependence on the punishment fine and cost reveal that the two types of punishers cannot coexist. Spontaneous coarsening of the two strategies leads to an indirect territorial competition with the defectors, which is won by unconditional punishers only if the sanctioning is inexpensive. Otherwise conditional punishers are the victors of the indirect competition, indicating that under more realistic conditions they are indeed the more effective strategy. Both continuous and discontinuous phase transitions as well as tricritical points characterize the complex evolutionary dynamics, which is due to multipoint interactions that are introduced by conditional punishment. We propose indirect territorial competition as a generally applicable mechanism relying on pattern formation, by means of which spatial structure can be utilized by seemingly subordinate strategies to avoid evolutionary extinction.
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“However, this solution is ineffective when counter-punishments are possible and when social interactions are noisy.”
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Citations
Cites
Hardin, G. (1968) · Science · doi:10.1126/science.162.3859.1243 · Cited by 25,167
Hardin, G. (2009) · Journal of Natural Resources Policy Research · doi:10.1080/19390450903037302 · Cited by 6,461
Altruistic punishment in humans
Fehr, E., & Gächter, S. (2002) · Nature · doi:10.1038/415137a · Cited by 5,616
Evolutionary games and spatial chaos
Nowak, M. A., & May, R. M. (1992) · Nature · doi:10.1038/359826a0 · Cited by 4,445
House, D., & Keyser, J. C. (2016) · A K Peters/CRC Press eBooks · doi:10.1201/9781315373140-16 · Cited by 3,332
Szabó, G., & Fáth, G. (2007) · Physics Reports · doi:10.1016/j.physrep.2007.04.004 · Cited by 2,976
Liggett, T. M. (1985) · Grundlehren der mathematischen Wissenschaften · doi:10.1007/978-1-4613-8542-4 · Cited by 2,935
(2011) · Springer eBooks · doi:10.1007/springerreference_60360 · Cited by 2,894
Antisocial Punishment Across Societies
Herrmann, B., Thöni, C., & Gächter, S. (2008) · Science · doi:10.1126/science.1153808 · Cited by 2,038
Coevolutionary games—A mini review
Perc, M., & Szolnoki, A. (2009) · Biosystems · doi:10.1016/j.biosystems.2009.10.003 · Cited by 1,963
Self-citationThe Neural Basis of Altruistic Punishment
de Quervain, D. J.-F., Fischbacher, U., Treyer, V., Schellhammer, M., Schnyder, U., Buck, A., & Fehr, E. (2004) · Science · doi:10.1126/science.1100735 · Cited by 1,768
Mothers and others: the evolutionary origins of mutual understanding
(2010) · Choice Reviews Online · doi:10.5860/choice.47-3250 · Cited by 1,735
A Cooperative Species: Human Reciprocity and Its Evolution
Bowles, S., & Gintis, H. M. (2011) · Economics Books · Cited by 1,431
Social diversity promotes the emergence of cooperation in public goods games
Santos, F. C., Santos, M. D., & Pacheco, J. M. (2008) · Nature · doi:10.1038/nature06940 · Cited by 1,430
Volunteering as Red Queen Mechanism for Cooperation in Public Goods Games
Hauert, C., Monte, S. D., Hofbauer, J., & Sigmund, K. (2002) · Science · doi:10.1126/science.1070582 · Cited by 1,109
Punishment in animal societies
Clutton-Brock, T. H., & Parker, G. A. (1995) · Nature · doi:10.1038/373209a0 · Cited by 1,043
Indirect reciprocity can stabilize cooperation without the second-order free rider problem
Panchanathan, K., & Boyd, R. (2004) · Nature · doi:10.1038/nature02978 · Cited by 865
(2001) · doi:10.1006/jtbi.2000.2202 · Cited by 803
Detrimental effects of sanctions on human altruism
Fehr, E., & Rockenbach, B. (2003) · Nature · doi:10.1038/nature01474 · Cited by 800
Evolutionary game theory: Temporal and spatial effects beyond replicator dynamics
Roca, C. P., Cuesta, J. A., & Sánchez, A. (2009) · Physics of Life Reviews · doi:10.1016/j.plrev.2009.08.001 · Cited by 766
Via Freedom to Coercion: The Emergence of Costly Punishment
Hauert, C., Traulsen, A., Brandt, H., Nowak, M. A., & Sigmund, K. (2007) · Science · doi:10.1126/science.1141588 · Cited by 721
Dreber, A., Rand, D. G., Fudenberg, D., & Nowak, M. A. (2008) · Nature · doi:10.1038/nature06723 · Cited by 718
Positive Interactions Promote Public Cooperation
Rand, D. G., Dreber, A., Ellingsen, T., Fudenberg, D., & Nowak, M. A. (2009) · Science · doi:10.1126/science.1177418 · Cited by 715
Reward and cooperation in the spatial public goods game
Szolnoki, A., & Perc, M. (2010) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/92/38003 · Cited by 714
Self-citationThe evolution of strong reciprocity: cooperation in heterogeneous populations
Bowles, S., & Gintis, H. M. (2003) · Theoretical Population Biology · doi:10.1016/j.tpb.2003.07.001 · Cited by 711
Sigmund, K., Hauert, C., & Nowak, M. A. (2001) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.161155698 · Cited by 644
Coordinated Punishment of Defectors Sustains Cooperation and Can Proliferate When Rare
Boyd, R., Gintis, H., & Bowles, S. (2010) · Science · doi:10.1126/science.1183665 · Cited by 613
The Long-Run Benefits of Punishment
Gächter, S., Renner, E., & Sefton, M. (2008) · Science · doi:10.1126/science.1164744 · Cited by 611
Social learning promotes institutions for governing the commons
Sigmund, K., De Silva, H., Traulsen, A., & Hauert, C. (2010) · Nature · doi:10.1038/nature09203 · Cited by 580
Altruistic punishment and the origin of cooperation
Fowler, J. H. (2005) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.0500938102 · Cited by 473
Topology-independent impact of noise on cooperation in spatial public goods games
Szolnoki, A., Perc, M., & Szabó, G. (2009) · Physical Review E · doi:10.1103/physreve.80.056109 · Cited by 448
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Dornic, I., Chaté, H., Chave, J., & Haye Hinrichsen, B. (2001) · Physical Review Letters · doi:10.1103/physrevlett.87.045701 · Cited by 375
Punishment sustains large-scale cooperation in prestate warfare
Mathew, S., & Boyd, R. T. (2011) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1105604108 · Cited by 373
Volunteering leads to rock–paper–scissors dynamics in a public goods game
Semmann, D., Krambeck, H.-J., & Milinski, M. (2003) · Nature · doi:10.1038/nature01986 · Cited by 360
The evolution of antisocial punishment in optional public goods games
Rand, D. G., & Nowak, M. A. (2011) · Nature Communications · doi:10.1038/ncomms1442 · Cited by 350
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Baldassarri, D., & Grossman, G. (2011) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1105456108 · Cited by 337
Indirect reciprocity provides only a narrow margin of efficiency for costly punishment
Ohtsuki, H., Iwasa, Y., & Nowak, M. A. (2008) · Nature · doi:10.1038/nature07601 · Cited by 336
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Gracia-Lázaro, C., Ferrer, A., Ruiz, G., Tarancón, A., Cuesta, J. A., Sánchez, Á., & Moreno, Y. (2012) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1206681109 · Cited by 335
Punish or perish? Retaliation and collaboration among humans
Sigmund, K. (2007) · Trends in Ecology & Evolution · doi:10.1016/j.tree.2007.06.012 · Cited by 309
Risk of collective failure provides an escape from the tragedy of the commons
Santos, F. C., & Pacheco, J. M. (2011) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1015648108 · Cited by 286
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Wakano, J. Y., Nowak, M. A., & Hauert, C. (2009) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.0812644106 · Cited by 220
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Perc, M. (2012) · Scientific Reports · doi:10.1038/srep00344 · Cited by 172
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Lozano, S., Arenas, À., & Sánchez, Á. (2008) · PLoS ONE · doi:10.1371/journal.pone.0001892 · Cited by 152
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Sasaki, T., Brännström, Å., Dieckmann, U., & Sigmund, K. (2012) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1115219109 · Cited by 151
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Bowles, S., & Gintis, H. M. (2004) · SSRN Electronic Journal · doi:10.2139/ssrn.480761 · Cited by 100
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The Evolution of Strong ReciprocityPreprint
Bowles, S. (2021) · RePEc: Research Papers in Economics · doi:10.7275/1072044 · Cited by 94
Effects of heterogeneous wealth distribution on public cooperation with collective risk
Wang, J., Fu, F., & Wang, L. (2010) · Physical Review E · doi:10.1103/physreve.82.016102 · Cited by 87
Nakamaru, M., & Iwasa, Y. (2005) · Institutional Repositories DataBase (IRDB) · Cited by 84
Partner selections in public goods games with constant group size
Wu, T., Fu, F., & Wang, L. (2009) · Physical Review E · doi:10.1103/physreve.80.026121 · Cited by 74
Evolution of cooperation driven by zealots
Masuda, N. (2012) · Scientific Reports · doi:10.1038/srep00646 · Cited by 69
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Fowler, J. H. (2005) · Nature · doi:10.1038/nature04201 · Cited by 65
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Disentangling social and group heterogeneities: Public Goods games on complex networks
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Mobilia, M. (2012) · Physical Review E · doi:10.1103/physreve.86.011134 · Cited by 47
Evolution of cooperation driven by zealotsPreprint
Masuda, N. (2012) · Cited by 41
Selection pressure transforms the nature of social dilemmas in adaptive networks
Segbroeck, S. V., Santos, F. C., Lenaerts, T., & Pacheco, J. M. (2011) · New Journal of Physics · doi:10.1088/1367-2630/13/1/013007 · Cited by 35
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Peña, J., & Rochat, Y. (2012) · PLoS ONE · doi:10.1371/journal.pone.0044514 · Cited by 35
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Vukov, J., Santos, F. C., & Pacheco, J. M. (2011) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2011.07.022 · Cited by 34
A minimal model for the evolution of cooperation through evolving heterogeneous games
Fort, H. (2008) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/81/48008 · Cited by 28
麻由子, 中. (2005) · Institutional Repositories DataBase (IRDB) · Cited by 27
Requejo, R. J., Camacho, J., Cuesta, J. A., & Arenas, À. (2012) · Physical Review E · doi:10.1103/physreve.86.026105 · Cited by 25
Reward and cooperation in the spatial public goods game
Szolnoki, A., & Perc, M. (2010) · Springer Link (Chiba Institute of Technology) · doi:10.48550/arxiv.1010.5771 · Cited by 22
Self-citationHow to treat those of ill repute
Rockenbach, B., & Milinski, M. (2008) · Nature · doi:10.1038/457039a · Cited by 19
PREFERENTIAL OPPONENT SELECTION IN PUBLIC GOODS GAMES
Brede, M. (2012) · Advances in Complex Systems · doi:10.1142/s0219525912500749 · Cited by 5
Kodaka, F., Takahashi, H., Yamada, M., Takano, H., Nakayama, K., Ito, H., & Suhara, T. (2012) · PLoS ONE · doi:10.1371/journal.pone.0041338 · Cited by 5
Research works printed but not linked: 1
- Nakamaru, 2005, Evolution of altruism by costly punishment in lattice-structured population, Evol. Ecol. Res., 7, 853
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Metadata
- Authors
- Szolnoki, A., & Perc, M.
- Affiliations
- Institute for Technical Physics and Materials Science (HU)
- University of Maribor (SI)
- Corresponding author
- Szolnoki, Attila
- Venue
- Journal of Theoretical Biology
- Publisher
- Elsevier BV
- Published
- 2013-05
- Volume
- 325
- Pages
- 34-41
- ISSN
- 0022-5193 (ISSN-L)
- OpenAlex
- W2100165388
- Cited by
- 173
- References
- 91
- Citations by year
- 2026: 13 · 2025: 16 · 2024: 15 · 2023: 13 · 2022: 13 · 2021: 10 · 2020: 5 · 2019: 16 · 2018: 17 · 2017: 14 · 2016: 12 · 2015: 11 · 2014: 10 · 2013: 8
- FWCI
- 42.18
- Type
- journal article
- Language
- en
- Open access
- green
- real.mtak.hu
- Licence
- Locations
- Journal of Theoretical Biology (publishedVersion)
- PubMed (publishedVersion)
- Repository of the Academy's Library (Library of the Hungarian Academy of Sciences) (acceptedVersion, other-oa)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Social Sciences › Sociology and Political Science › Evolutionary Game Theory and Cooperation
- Social Sciences › Economics, Econometrics and Finance › Economics and Econometrics › Experimental Behavioral Economics Studies
- Life Sciences › Biochemistry, Genetics and Molecular Biology › Genetics › Evolution and Genetic Dynamics
- Keywords
- public goods game, evolution of cooperation, spatial games, phase transitions, tricritical point, pattern formation, collective action, coexistence of strategies
- MeSH terms
- Cooperative Behavior, Biological Evolution, Game Theory, Humans, Models, Genetic, Models, Psychological, Punishment, Social Justice
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