Physica A 436 (2015) 327–337 Contents lists available at ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa Imitating winner or sympathizing loser? Quadratic effects on cooperative behavior in prisoners’ dilemma games Peng Lu * Department of Automation, Tsinghua University, China Department of Sociology, Tsinghua University, China highlights • Investigates the effect of sympathy on cooperation, which is not adequately revealed before. • Sympathy has a quadratic effect on cooperation, and it promotes cooperation beyond its threshold. • Temptation has a quadratic effect as well, and it even promotes cooperation beyond a threshold. • Although cooperation falls at earlier stages, the resilience is strong enough to promote cooperation later on. article info Article history: Received 21 January 2015 Received in revised form 28 March 2015 Available online 17 May 2015 Keywords: Winner Loser Sympathy Quadratic effects Cooperation Irrationalism abstract Cooperation is vital in human societies and therefore is widely investigated in the evolu- tionary game theory. Varieties of mechanisms have been proposed to overcome temptation and promote cooperation. Existing studies usually believe that agents are rational, but ir- rationalism such as emotions and feelings matters as well. Winner and loser are defined by their payoffs. In addition to admiring and imitating winners, the mechanism of sympa- thizing and imitating losers is introduced into the model as an alternative action rule, and each one plays the prisoners’ dilemma game with eight neighbors under the influence of both irrationalism and rationalism. Rationalism refers to imitating winner to get highest payoff, and irrationalism means that people sympathize and adopt the actions of losers. As it is widely recognized that temptation reduces cooperation, this study focuses on the effect of sympathy on cooperation within a certain group or society. If it overcomes temp- tation that leads to defection, sympathy will be a powerful mechanism to promote coop- erative behavior. Simulation results indicate that sympathy and temptation shares similar quadratic relationships with cooperation. Both sympathy and temptation undermine coop- eration below their thresholds, and they both promote cooperation above their thresholds. Temptation not only reduces cooperation but also promote it as temptation goes beyond the threshold. Although sympathy is a good merit or human nature that is beneficial to society, a crisis or collapse of cooperation is inevitable when the sympathy propensity is relatively smaller. After cooperation reaches a minimal bottom, it then rises increasingly and dramatically, which brings a much brighter future of the society. © 2015 Elsevier B.V. All rights reserved. 1. Introduction Cooperation is vital and therefore widespread in the real world, which can be observed at biological groups as well as human society [1–3]. Human society is based on cooperation, but the puzzle of cooperation [3] indicates that cooperators * Correspondence to: Department of Automation, Tsinghua University, China. E-mail address: lvpeng.tsinghua@hotmail.com. http://dx.doi.org/10.1016/j.physa.2015.05.029 0378-4371/© 2015 Elsevier B.V. All rights reserved.