Static network structure can stabilize human cooperation

Abstract

The evolution of cooperation in network-structured populations has been a major focus of theoretical work in recent years. When players are embedded in fixed networks, cooperators are more likely to interact with, and benefit from, other cooperators. In theory, this clustering can foster cooperation on fixed networks under certain circumstances. Laboratory experiments with humans, however, have thus far found no evidence that fixed network structure actually promotes cooperation. Here, we provide such evidence and help to explain why others failed to find it. First, we show that static networks can lead to a stable high level of cooperation, outperforming well-mixed populations. We then systematically vary the benefit that cooperating provides to one's neighbors relative to the cost required to cooperate (b/c), as well as the average number of neighbors in the network (k). When b/c > k, we observe high and stable levels of cooperation. Conversely, when b/c ≤ k or players are randomly shuffled, cooperation decays. Our results are consistent with a quantitative evolutionary game theoretic prediction for when cooperation should succeed on networks and, for the first time to our knowledge, provide an experimental demonstration of the power of static network structure for stabilizing human cooperation.

Keywords: Prisoner’s Dilemma; assortment; economic games; evolutionary game theory; structured populations.

Fig. 1.

Examples of the network structure for the k = 2 (A), k = 4 (B), and k = 6 (C) cases. Consider the topmost player as the ego (in dark blue); her links are highlighted in blue, and her neighbors are colored light blue.

Fig. 2.

Networked interactions promote cooperation when b/c = 6 and k = 2 in experiment 1, run in the physical laboratory. Shown is the fraction of subjects choosing cooperation in each round, for network (dark green circles) and well-mixed (light green diamonds) conditions.

Fig. 3.

Stable cooperation emerges when b/c > k, but not b/c ≤ k, in experiment 2, run online. (A) Fraction of subjects choosing cooperation in each round, for b/c > k (green circles) and b/c ≤ k (orange triangles). Observations from the first half of the game are open, and observations from the second half are filled. (B) Fraction of subjects choosing cooperation in the final round, for each [b/c,k] combination. Bars are grouped by [b/c < k, b/c = k, b/c > k] and are sorted in decreasing order of final cooperation level within each grouping.

Fig. 4.

Substantial assortment emerges when b/c > k, but not b/c ≤ k. (A) Shown is the average level of assortment (cooperators’ average number of cooperative neighbors minus defectors’ average number of cooperative neighbors) by round for b/c > k (green circles) and b/c ≤ k (yellow triangles). Also shown are two sample networks across the first five rounds (which are labeled), where cooperators are shown in blue and defectors in red; low clustering with b/c ≤ k (B; b/c = 2, k = 2) and high clustering with b/c > k (C; b/c = 6, k = 2). Despite similar initial levels of cooperation, clustering maintains cooperation when b/c > k, whereas cooperation decays when b/c ≤ k.

Fig. 5.

Defectors (D) significantly out-earn cooperators (C) when b/c ≤ k (A), but not when b/c > k (B). Shown is the distribution of payoffs relative to average session payoff, with one observation per subject per round, for C (blue) and D (red). To make payoffs comparable across conditions, they are normalized by the maximum possible relative payoff [bk − ck(N − 1)]/N, where N is the number of players in the session.