Opting out against defection leads to stable coexistence with cooperation

www.nature.com/scientificreports OPEN Opting out against defection leads to stable coexistence with cooperation received: 04 March 2016 Bo-Yu Zhang1,2,*, Song-Jia Fan1,*, Cong Li1,3,*, Xiu-Deng Zheng1,*, Jian-Zhang Bao4, Ross Cressman5 & Yi Tao1 accepted: 05 October 2016 Published: 24 October 2016 Cooperation coexisting with defection is a common phenomenon in nature and human society. Previous studies for promoting cooperation based on kin selection, direct and indirect reciprocity, graph selection and group selection have provided conditions that cooperators outcompete defectors. However, a simple mechanism of the long-term stable coexistence of cooperation and defection is still lacking. To reveal the effect of direct reciprocity on the coexistence of cooperation and defection, we conducted a simple experiment based on the Prisoner’s Dilemma (PD) game, where the basic idea behind our experiment is that all players in a PD game should prefer a cooperator as an opponent. Our experimental and theoretical results show clearly that the strategies allowing opting out against defection are able to maintain this stable coexistence. A great deal of research has been devoted to explain how the evolution of cooperation can be favored by natural selection. Five rules for promoting cooperation based on kin selection1, direct and indirect reciprocity2–5, graph selection6,7 and group selection8 have been summarized9, and these models provided simple conditions that natural selection can lead to full cooperation. However, few literatures have considered how cooperation and defection can coexist in the long-term even though this phenomenon is common in nature and human society10. Other studies11,12 have shown ongoing oscillations between cooperative and defective societies can evolve in theoretical models, possibly explaining such phenomena as the alternate appearance of war and peace11. However, these models still do not provide a simple mechanism to drive the long-term stable coexistence of cooperation and defection. Cooperation means that a donor pays a cost, c, for a recipient to get a benefit, b, where b >  c11,12. In the corresponding one-shot PD game, defection is the only Nash equilibrium (NE)11,12. On the other hand, for the repeated PD game with two strategies TFT (tit-for-tat) and AllD (always defect), TFT is a NE if the expected number of iterated interactions between a pair of individuals is larger than the critical value b/(b −  c)3,4,9,11,12. However, the stable coexistence of TFT and AllD is impossible in the TFT-AllD game. Clearly, the success of TFT is mainly due to the increased chance of interactions between cooperators4,13. That is, TFT provides a mechanism whereby cooperators preferentially interact among themselves. Similarly, assortative matching among cooperators has been used to explain why altruism can emerge14–18, although the evolutionary origin of the non-uniform interaction rates among cooperators has not been explained17,18. For the repeated PD game, one of the key assumptions is that the interaction between a pair of individuals will be repeated for several rounds, and no player in the game is able to stop the interaction with his/her opponent4,11–13. However, based on individual self-interest in the PD game, both cooperators and defectors prefer an opponent who cooperates (i.e. only cooperator is always welcome). Thus, if players are able to unilaterally terminate the interactions with their opponents, then a simple rule will be followed by all individuals: I would like to keep my opponent if he/she is a cooperator; and if my opponent is a defector, I will stop the interaction with him/ her and seek a new partner instead. 1 Key Lab of Animal Ecology and Conservation Biology, Chinese Academy of Science, Beijing, China. 2Laboratory of Mathematics and Complex Systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University, Beijing, China. 3Department of Mathematics and Statistics, University of Montreal, Montreal, Canada. 4 School of Complex Systems, Beijing Normal University, Beijing, China. 5Department of Mathematics, Wilfrid Laurier University, Waterloo, Canada. *These authors contributed equally to this work. Correspondence and requests for materials should be addressed to R.C. (email: ) or Y.T. (email: ) Scientific Reports | 6:35902 | DOI: 10.1038/srep35902 1 www.nature.com/scientificreports/ Figure 1. Cooperation levels per round for treatment compared to control experiments. Panel (a) shows the time evolution of cooperation levels per round in C1, C2 and T respectively, with dashed line at round 60. Panel (b) shows the average cooperation levels over 60 rounds with standard errors in C1, C2 and T, respectively, which are: 0.72 ± 0.0808 in C1; 0.32 ± 0.0876 in C2; and 0.56 ± 0.0287 in T. Mann-Whitney U-test shows that the differences between C1 and C2, between C1 and T and between T and C2 are significant with p-value <  0.01 (after Bonferroni correction) (SI, Table S3). Recently, an interesting study based on the concept of conditional dissociation found that a strategy called “out-for-tat” (OFT) is important for the coexistence of cooperation and defection19–26. Since OFT means that an individual displaying cooperation (C) will respond to defection (D) by merely leaving, OFT will not tolerate defection but, unlike TFT, it does not seek revenge. To reveal the fundamental evolutionary force driving the coexistence of C and D, we conduct a simple experiment based on the repeated PD game, where, unlike the classic repeated game, each player can unilaterally break off the pairwise interaction with his/her opponent according to his/her own volition. On the other hand, different from previous experiments on repeated PD game with outside option19–26, the expected number of rounds between a pair of individuals is still limited in our experimental design even if these two individuals would like to continue their interaction4,11–13. Results A total of 264 university students were divided into five groups, including two control groups (C1 and C2) and three treatment groups (T1, T2 and T3) (Supplementary Information (SI), Section 1.1). Note that the experimental settings in all three treatment groups T1, T2 and T3 are exactly the same, therefore in the data analysis we treat them as one group, denoted by T (SI, Section 1.2). The basic payoff matrix in our experiment is C D C D 4 1 , 5 2 ( ) [1] where this payoff matrix can be normalized as a simplified PD game with b =  3 and c = 1. Each subject participated in 65 to 80 rounds of interactions between pairs of individuals playing this game over about 40 minutes. Participants were told that the experiment would be randomly stopped at 60–80 rounds. Thus, to avoid end-round effects and to keep the comparison unbiased, we only used data in the first 60 rounds in all groups in later statistical analysis. The control experiments C1 and C2 are the classic repeated PD g (...truncated)