Evolutionary dynamics of cooperation in a population with probabilistic corrupt enforcers and violators

Pro-social punishment is a key driver of harmonious and stable society. However, this institution is vulnerable to corruption since law-violators can avoid sanctioning by paying bribes to corrupt law-enforcers. Consequently, to understand how altruistic behavior survives in a corrupt environment is an open question. To reveal potential explanations here, we introduce corrupt enforcers and violators into the public goods game with pool punishment, and assume that punishers, as corrupt enforcers, may select defectors probabilistically to take a bribe from, and meanwhile defectors, as corrupt violators, may select punishers stochastically to be corrupted. By means of mathematical analysis, we aim to study the necessary conditions for the evolution of cooperation in such corrupt environment. We find that cooperation can be maintained in the population in two distinct ways. First, cooperators, defectors, and punishers can coexist by all keeping a steady fraction of the population. Second, these three strategies can form a cyclic dominance that resembles a rock-scissors-paper cycle or a heteroclinic cycle. We theoretically identify conditions when the competing strategies coexist in a stationary way or they dominate each other in a cyclic way. These predictions are confirmed numerically.

[1]  Tatsuya Sasaki,et al.  The evolution of cooperation by social exclusion , 2012, Proceedings of the Royal Society B: Biological Sciences.

[2]  M. Nowak,et al.  Evolutionary Dynamics of Biological Games , 2004, Science.

[3]  Junpyo Park,et al.  Balancedness among competitions for biodiversity in the cyclic structured three species system , 2018, Appl. Math. Comput..

[4]  G. Hardin,et al.  The Tragedy of the Commons , 1968, Green Planet Blues.

[5]  M. Perc,et al.  Social diversity and promotion of cooperation in the spatial prisoner's dilemma game. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Attila Szolnoki,et al.  Coevolutionary Games - A Mini Review , 2009, Biosyst..

[7]  Attila Szolnoki,et al.  Alliance formation with exclusion in the spatial public goods game , 2017, Physical review. E.

[8]  Attila Szolnoki,et al.  Competitions between prosocial exclusions and punishments in finite populations , 2017, Scientific Reports.

[9]  Francisco C. Santos,et al.  Cooperation dynamics of polycentric climate governance , 2015 .

[10]  Francisco C. Santos,et al.  EVOLUTIONARY DYNAMICS OF CLIMATE CHANGE UNDER COLLECTIVE-RISK DILEMMAS , 2012 .

[11]  U. Fischbacher,et al.  The nature of human altruism , 2003, Nature.

[12]  Lorenzo Pareschi,et al.  Reviews , 2014 .

[13]  Attila Szolnoki,et al.  Resolving social dilemmas on evolving random networks , 2009, 0910.1905.

[14]  Supratim Sengupta,et al.  Bribery games on inter-dependent regular networks , 2017, Scientific Reports.

[15]  Attila Szolnoki,et al.  Central governance based on monitoring and reporting solves the collective-risk social dilemma , 2018, Appl. Math. Comput..

[16]  Mirosław Lachowicz,et al.  Modeling altruism and selfishness in welfare dynamics: The role of nonlinear interactions , 2014 .

[17]  D. Burini,et al.  Collective learning modeling based on the kinetic theory of active particles. , 2016, Physics of life reviews.

[18]  M Dolfin,et al.  Modeling human behavior in economics and social science. , 2017, Physics of life reviews.

[19]  Arne Traulsen,et al.  Social learning promotes institutions for governing the commons , 2010, Nature.

[20]  Attila Szolnoki,et al.  Statistical Physics of Human Cooperation , 2017, ArXiv.

[21]  Jun Tanimoto,et al.  Relationship between dilemma occurrence and the existence of a weakly dominant strategy in a two-player symmetric game , 2007, Biosyst..

[22]  F. C. Santos,et al.  The role of diversity in the evolution of cooperation. , 2012, Journal of theoretical biology.

[23]  Attila Szolnoki,et al.  Risk-driven migration and the collective-risk social dilemma , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  F. C. Santos,et al.  Risk of collective failure provides an escape from the tragedy of the commons , 2011, Proceedings of the National Academy of Sciences.

[25]  Supratim Sengupta,et al.  Bribe and Punishment: An Evolutionary Game-Theoretic Analysis of Bribery , 2015, PloS one.

[26]  M. A. Herrero,et al.  Cooperation, competition, organization: The dynamics of interacting living populations , 2015 .

[27]  Bettina Rockenbach,et al.  The efficient interaction of indirect reciprocity and costly punishment , 2006, Nature.

[28]  F. C. Santos,et al.  Social diversity promotes the emergence of cooperation in public goods games , 2008, Nature.

[29]  Feng Fu,et al.  Invasion and expansion of cooperators in lattice populations: prisoner's dilemma vs. snowdrift games. , 2010, Journal of theoretical biology.

[30]  Krishnendu Chatterjee,et al.  Indirect reciprocity with private, noisy, and incomplete information , 2018, Proceedings of the National Academy of Sciences.

[31]  J. Carr.  Applications of Centre Manifold Theory , 1981 .

[32]  M. Milinski,et al.  Reputation helps solve the ‘tragedy of the commons’ , 2002, Nature.

[33]  G. Szabó,et al.  Cooperation enhanced by inhomogeneous activity of teaching for evolutionary Prisoner's Dilemma games , 2006, q-bio/0610001.

[34]  C. Hauert,et al.  Reputation-based partner choice promotes cooperation in social networks. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[35]  Francisco C. Santos,et al.  A bottom-up institutional approach to cooperative governance of risky commons , 2013 .

[36]  Arne Traulsen,et al.  Social Control and the Social Contract: The Emergence of Sanctioning Systems for Collective Action , 2011, Dyn. Games Appl..

[37]  Han-Xin Yang,et al.  Promoting cooperation by punishing minority , 2018, Appl. Math. Comput..

[38]  Rebecca B. Hoyle,et al.  Social influence preserves cooperative strategies in the conditional cooperator public goods game on a multiplex network , 2018, Physical Review E.

[39]  M. Nowak.  Five Rules for the Evolution of Cooperation , 2006, Science.

[40]  W. Hamilton.  The Evolution of Altruistic Behavior , 1963, The American Naturalist.

[41]  Stéphane Côté,et al.  Higher social class predicts increased unethical behavior , 2012, Proceedings of the National Academy of Sciences.

[42]  Qiang Wang,et al.  Replicator dynamics for public goods game with resource allocation in large populations , 2018, Appl. Math. Comput..

[43]  Nicola Bellomo,et al.  Modeling behavioral social systems , 2017 .

[44]  C. Hauert,et al.  Volunteering as Red Queen Mechanism for Cooperation in Public Goods Games , 2002, Science.

[45]  Seung‐Yeal Ha,et al.  A quest toward a mathematical theory of the dynamics of swarms , 2017 .

[46]  Ulf Dieckmann,et al.  The take-it-or-leave-it option allows small penalties to overcome social dilemmas , 2012, Proceedings of the National Academy of Sciences.

[47]  William T. Harbaugh,et al.  The Carrot or the Stick: Rewards, Punishments and Cooperation , 2002 .

[48]  Attila Szolnoki,et al.  Probabilistic sharing solves the problem of costly punishment , 2014, ArXiv.

[49]  Shaul Shalvi,et al.  Behavioural economics: Corruption corrupts , 2016, Nature.

[50]  Attila Szolnoki,et al.  Phase Diagrams for the Spatial Public Goods Game with Pool-Punishment , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[51]  Xiaojie Chen,et al.  First carrot, then stick: how the adaptive hybridization of incentives promotes cooperation , 2015, Journal of The Royal Society Interface.

[52]  J. Henrich,et al.  Corrupting cooperation and how anti-corruption strategies may backfire , 2017, Nature Human Behaviour.

[53]  Long Wang,et al.  Evolution of cooperation in a hierarchical society with corruption control. , 2018, Journal of theoretical biology.

[54]  Attila Szolnoki,et al.  Effectiveness of conditional punishment for the evolution of public cooperation. , 2013, Journal of theoretical biology.

[55]  Attila Szolnoki,et al.  Evolutionary advantages of adaptive rewarding , 2012, 1208.3457.

[56]  Attila Szolnoki,et al.  Reciprocity-based cooperative phalanx maintained by overconfident players , 2018, Physical review. E.

[57]  Nicola Bellomo,et al.  Stochastic Evolving Differential Games Toward a Systems Theory of Behavioral Social Dynamics , 2015, 1506.05699.

[58]  C. Hauert,et al.  Replicator dynamics for optional public good games. , 2002, Journal of theoretical biology.

[59]  Iyad Rahwan,et al.  Corruption drives the emergence of civil society , 2013, Journal of The Royal Society Interface.

[60]  Nicola Bellomo,et al.  MATHEMATICS AND COMPLEXITY IN BIOLOGICAL SCIENCES , 2011 .

[61]  Josef Hofbauer,et al.  Evolutionary Games and Population Dynamics , 1998 .

[62]  Nicola Bellomo,et al.  MATHEMATICS AND COMPLEXITY IN LIFE AND HUMAN SCIENCES , 2009 .

[63]  Long Wang,et al.  Coevolutionary dynamics of aspiration and strategy in spatial repeated public goods games , 2018, New Journal of Physics.

[64]  Attila Szolnoki,et al.  Competition and partnership between conformity and payoff-based imitations in social dilemmas , 2018, New Journal of Physics.

[65]  Yoh Iwasa,et al.  Games of corruption in preventing the overuse of common-pool resources. , 2017, Journal of theoretical biology.

[66]  E. Fehr,et al.  Altruistic punishment in humans , 2002, Nature.

[67]  Matjaz Perc,et al.  Evolutionary dynamics in the public goods games with switching between punishment and exclusion , 2018, Chaos.

[68]  Attila Szolnoki,et al.  Punishment and inspection for governing the commons in a feedback-evolving game , 2018, PLoS Comput. Biol..