Cooperation in an Assortative Matching Prisoners Dilemma Experiment with Pro-Social Dummies

www.nature.com/scientificreports OPEN Received: 26 June 2019 Accepted: 3 September 2019 Published: xx xx xxxx Cooperation in an Assortative Matching Prisoners Dilemma Experiment with Pro-Social Dummies Chun-Lei Yang1 & Ching-Syang Jack Yue2 Assortative matching (AM) can be theoretically an effective means to facilitate cooperation. We designed a controlled lab experiment with three treatments on multi-round prisoner’s dilemma. With matching based on weighted history (WH) as surrogate for AM, we show that adding pro-social dummies to the WH treatment may significantly improve cooperation, compared to both the random matching and the WH treatment. In society where assortative matching is effective and promoted by the underlying culture, institutional promotion of virtue role models can be interpreted as generating additional pro-social dummies, so as to move the initial state of cooperators into the basin of attraction for a highly cooperative polymorphic equilibrium. Cooperation in social dilemma games has attracted the attention of many scientific disciplines. Various settings have been investigated that induce significantly higher level of cooperation than the baseline of random matching one-shot encounters without community information. Models of repeated partnership resort to trigger and other strategies (direct reciprocity) to achieve full cooperation1–3; and recent studies offer various approaches to estimate strategy distributions based on experimental data4–7. Indirect reciprocity may be of positive impact, both in theory and experiments8–14. Options of punishment15–20 and reward21–23 may also greatly improve cooperation in experimental and theoretical studies, amidst the strong reciprocity debate24,25. We study the issue of cooperation using prisoner’s dilemma (PD) as the basic game within the general assortative matching (AM) framework, as e.g. studied in26–29. Note that30 allows for type recognition to generate AM effect, and public goods games with endogenous group formation may induce assortative outcomes in favor of cooperation31,32. Assortative matching has been identified as a universal principle in human and non-human ecologies as a mechanism for promotion of pro-social behavior. Ref.28 illustrates stylized facts and anecdotal stories in human societies and points out characteristic behavioral features underneath a functioning condition of assortative matching. In general, honest and trustworthy persons often display non-imitable behavioral and biological traits or labels, which enable people to be selective in partner choice for joint endeavor such as the PD game, resulting in like-minded people to be more likely matched that it would be random that reflects the saying “birds of a feather flock together”. In this environment, cooperation has a chance to proliferate for the betterment of society. References33,34 provide a thorough discussion of the common evolutionary models of assortative matching. Consider the generic PD game with T > R > P > S, where one player’s payoff is R, P, T, or S, if both cooperate (C), both defect (D), the player is the exploiter, or he the sucker, respectively. Consider a population of x ∈ [0, 1] cooperators and 1 − x defectors. Define a(x ) = P(C|C ) − P(C|D) as the index of assortativity, where P(X|Y) denotes the conditional probability that a type-Y player expects to be matched with a type-X player. Assume replicator dynamics for population change x , it is perfectly aligned with the sign of fitness difference between C and D types, δ(x ) = πC(x ) − πD(x ), given the population state x and the index a( ⋅ ). Depending on application environments, various forms of index of assortativity are conceivable as detailed in34. For the most simple case, it can be constant with a(x) = a. The equilibrium analysis is straightforward in this case. With a sufficiently high, cooperation may survive in the long run, either in form of a stable mixed population equilibrium or as one of the stable pure population equilibria. However, envision the physical realization 1 Economics Experimental Lab, Nanjing Audit University, 86 Yushanxi Road, Nanjing, 211815, China. 2Department of Statistics, National Chengchi University, Taipei, Taiwan, ROC. Correspondence and requests for materials should be addressed to C.-L.Y. (email: ) Scientific Reports | (2019) 9:13609 | https://doi.org/10.1038/s41598-019-50083-6 1 www.nature.com/scientificreports www.nature.com/scientificreports/ of a real-world sorting mechanism, it is reasonable to assume that individuals need to spend some small but positive search cost in pursuit of a proper match. In fact, the existence of such cost is exactly the very reason for the reasonable assumption of imperfect assortativity a(x) < 1. This implies that in the pure population states, the matching result must be equivalent to random matching due to zero likelihood to meet the lacking type, i.e. a(0) = a(1) = 0, where only defection prevails in the long run. Fortunately,34 demonstrates that non-constant AM that satisfies a(0) = a(1) = 0 exists, such as in the so-called stranger-in-the-night model, where cooperation survives in form of a locally stable mixed population equilibrium. Stranger-In-The-Night Model: Imagine each of two different types has distinctive appearance characteristics that are not perfectly recognizable in the night when strangers are supposed meet and decide whether go home with each other. Let s and m with 0 < m < s ≤ 1 denote the match success rates if the randomly encountered counterpart is of the same and different type, respectively, then the resulting index of assortativity is a(x ): = x(1 − x )(s 2 − m2) xs xm − = . (1 − x )s + xm xs + (1 − x )m x(1 − x )(s − m)2 + sm Straightforward calculation yields a(0) = a(1) = 0. Note that max xa(x ) = a(1/2)= (s − m)/(s + m), i.e. the assortativity effect is strongest when C and D have equal shares in the population. With δ(x) properly spelled out, it is obvious that δ(1/2) > 0 with m sufficiently small. In this specific model, δ(x) = 0 has three solutions {0, xmin , xmax}, 0 < xmin < xmax < 1, where 0 and xmax are locally stable equilibria with respective basins of attraction (0, xmin) and (xmin, 1]. The general insight from this theoretical discussion is as follows. While pure defection is the only stable population equilibrium in RM, AM potentially may admit additional stable equilibria with a higher share of cooperators, depending on the specifics of the mechanism. Moreover, in the latter case, the initial state of population is crucial at determining whether the dynamics converges to the bad pure defectors equilibrium of x* = 0 or some better ones such as x* = xmax. This, henceforth, opens up the gate for culture, social conventions or state actions to be shaped and constructed in a way so as to positively affect either the effectiveness of AM mechanism, for example by increasing s or decreasing m in the strangers-in-the-night model above; (...truncated)