Opting out against defection leads to stable coexistence with cooperation

Abstract

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.

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).

Figure 2. Individuals’ responses to the behavior of their opponents in the first 60 rounds.

Panel (a) shows the probability that, at the end of each round, a player chooses to keep, or break, the interaction with his/her opponent who uses C (D). Panel (b) shows the probabilities that, at the end of each round, a player using C (D) chooses to keep, or break, the interaction when his/her opponent uses C and when his/her opponent uses D.

Figure 3. The setup of the evolutionary model.

OFT-cooperators and OFT-defectors are marked by blue angels and red fiends, respectively. At the end of a round, C-D pairs and D-D pairs will be broken since all individuals immediately stop the interaction with a defector, and a C-C pair will be terminated with probability ρ even though both individuals are willing to continue. These single individuals will be paired with a new partner through random meeting in the next round.

Figure 4. Evolutionary dynamics of Eq. [2] with b = 3 and c = 1.

(a) Blue, yellow and pink regions represent respectively the proportions of C-C, C-D and D-D pairs for all possible 0 < x < 1 at the temporal equilibrium with (Eq. [S3] in SI, Section 2.1), where the parameter ρ is taken as ρ = 1/6; and the blue line denotes the stable interior equilibrium . (b) Phase portrait of the dynamics Eq. [2] for different ρ. The red line denotes the stable boundary x = 0, the solid blue curve denotes the stable interior equilibrium (which is bigger than 1/2), and the dashed curve denotes the unstable interior equilibrium . The population evolves to the boundary x = 0 for initial x in the pink region, and the dynamics leads to a stable coexistence of C and D for initial x in the blue region. The inverse 1/ρ represents the expected number of interactions of a C-C pair, where the vertical dash line denotes 1/ρ = 6.