Information Cascades and the Collapse of Cooperation.

In various types of structured communities newcomers choose their interaction partners by selecting a role-model and copying their social networks. Participants in these networks may be cooperators who contribute to the prosperity of the community, or cheaters who do not and simply exploit the cooperators. For newcomers it is beneficial to interact with cooperators but detrimental to interact with cheaters. However, cheaters and cooperators usually cannot be identified unambiguously and newcomers' decisions are often based on a combination of private and public information. We use evolutionary game theory and dynamical networks to demonstrate how the specificity and sensitivity of those decisions can dramatically affect the resilience of cooperation in the community. We show that promiscuous decisions (high sensitivity, low specificity) are advantageous for cooperation when the strength of competition is weak; however, if competition is strong then the best decisions for cooperation are risk-adverse (low sensitivity, high specificity). Opportune decisions based on private and public information can still support cooperation but suffer of the presence of information cascades that damage cooperation, especially in the case of strong competition. Our research sheds light on the way the interplay of specificity and sensitivity in individual decision-making affects the resilience of cooperation in dynamical structured communities.

When decision-making is based on private information, for a given benefit-to-cost ratio b/c and a given selection strength δ, the amount of long term cooperation and connectivity is dependent on the threshold τ , see Figure 1. With the increase of the decision-threshold the network gets more connected due to the decrease of specificity, but can be invaded by cheaters more easily. As discussed in the main text, long term cooperation is maximized at some intermediate τ s, with an opportune level of specificity and sensitivity, which facilitate the spreading of cooperation meanwhile inhibit the growth of defection. In Figure 1 we plot the typical trajectories of the network, with switches between configurations of all cooperators and of all cheaters.
Figure 1: Typical trajectories of the network. We plot the amount of cooperation (normalized number of cooperators present in the network) and connectivity (average connectivity of the network) for τ = −1, τ = 0 and τ = 1. The simulations have been obtained using δ = 0.01 and b/c = 10/9. 1.2 Low benefit-to-cost ratio: typical networks Figure 2 illustrates the typical networks for low benefit-to-cost ratio; as discussed in the main text, the collapse of cooperation is led by the introduction of cheaters connected to many cooperators which can happen more frequently at high values of τ and for strong selection.

Low benefit-to-cost ratio: cooperation
In Figure 3 we present the amount of long term cooperation, connectivity, prosperity and transitions for low benefit-to-cost ratio. As in the case of higher benefit-to-cost-ratio (presented in the main text) we can see that the collapse of cooperation starts at a lower τ as the selection becomes stronger.  2 Decision-making based on private and public information As discussed in the main text, long term cooperation is strongly affected (and possibly damaged) by decisions based on the combination of private and public information. In Figure 3 we show the amount of long term

Long term cooperation and connectivity
When decisions are based on private and public information the amount of long term cooperation is highly dependent on the parameters p and q (which regulates the weight of the public information, as discussed in the main text) as well as by the decision-threshold τ (Figure 4).
With the increase of p and the decrease of q the public information has a stronger weight in the decision. As discussed in the main text, this can lead to erroneous connections to cheaters (and information cascades) which damage long term cooperation.

Typical networks
In Figure 5 we illustrate the typical networks for the case when private information has a stronger weight in the decision and the case in which public information has a stronger weight in the decision.

Cooperation and information cascades
In Figure 6 we present the amount of long term cooperation, connectivity, prosperity and transitions for low benefit-to-cost-ratio.  /c = 10/9.9 and weak/strong selection. We show the typical networks with all-cooperators (top row), cooperators-to-defectors (second row), all-defectors (third row) and defectors-to-cooperators (bottom row). The decisions where private information is stronger (p = 0.25 and q = 0.75) and the decisions where public information is stronger (p = 0.75 and q = 0.25) are compared for weak selection (δ = 0.001) and strong selection (δ = 0.1).
(a) More private information (b) More public information Figure 6: Long term cooperation, connectivity, prosperity and transitions as function of τ s. The case of decisions with stronger public information is presented in the lower panel (p = 0.75 and q = 0.25). The case for stronger private information is presented in the upper panel (p = 0.25 and q = 0.75). The benefit-to-cost ratio is b/c = 10/9.9 and we consider weak selection (δ = 0.001, left panel) and strong selection (δ = 0.1, right panel). Results are obtained with simulations of 10 8 steps.