Top panel: time courses depicting the evolution of cooperation for different values of under the random distribution of strategies.

<p>Note that the arrows denote the end of enduring (END) period and the beginning of expanding (EXP) period. Bottom panel: the evolution snapshots of different values of . From top to bottom, the values of are 0, 0.5, 1.0 and 2.0, respectively. The colore code is the same with <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0091012#pone-0091012-g002" target="_blank">Fig.2:</a> cooperator (red) and defector (blue). From left to right, the time steps are 0, 10, 100, 1000 for each value of .</p

Characteristic snapshots of cooperators (red) and defectors (blue) under the prepared initial state for different times steps.

<p>From top row to bottom panel, the tunable parameter is set to be 0, 0.5 and 1.0, respectively. In all dynamical patterns, the synergy factor is 0.6, lattice size is 200 and strategy adoption uncertainty is 0.1.</p

Fraction of cooperators as a function of normalized enhancement factor on triangular lattices in which and as that in Ref.[52].

<p>Other parameters are identical with those in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0091012#pone-0091012-g001" target="_blank">Fig.1</a>.</p

Full normalized phase diagrams for different tunable parameters .

<p>From top to bottom, panels (a), (b), (c) correspond to , respectively. The lattice size is and PGG group size is fixed to be .</p

Supporting Information from Heterogeneous resource allocation can change social hierarchy in public goods games

Public goods games (PGGs) represent one of the most useful tools to study group interactions. However, even if they could provide an explanation for the emergence and stability of cooperation in modern societies, they are not able to reproduce some key features observed in social and economical interactions. The typical shape of wealth distribution—known as Pareto Law—and the microscopic organization of wealth production are two of them. Here, we introduce a modification to the classical formulation of PGGs that allows for the emergence of both of these features from first principles. Unlike traditional PGGs, where players contribute equally to all the games in which they participate, we allow individuals to redistribute their contribution according to what they earned in previous rounds. Results from numerical simulations show that not only a Pareto distribution for the pay-offs naturally emerges but also that if players do not invest enough in one round they can act as defectors even if they are formally cooperators. Our results not only give an explanation for wealth heterogeneity observed in real data but also points to a conceptual change on cooperation in collective dilemmas

Stationary fraction of cooperators in the whole population (<i><b><i>F</i>_<i>C</i></b></i><b>) as a function of defection parameter (</b><i><b>b</b></i><b>) under different amplitude of coupling strength (</b><i><b>A</b></i>).

<p>On the top panel (a), corresponding players on two networks always have the same coupling strength which is taken from the interval [−<i>A</i>, <i>A</i>] according to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0129542#pone.0129542.e002" target="_blank">Eq (2)</a>, and the cooperation can be greatly promoted as <i>A</i> increases; On the bottom panel (b), corresponding players on two networks hold distinct coupling strengths which are independently derived from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0129542#pone.0129542.e002" target="_blank">Eq (2)</a>, and the cooperation is also promoted but the enhancement level is much smaller than that in the top panel. Other parameters are set to be <i>L</i> = 200 and <i>K</i> = 0.1.</p

Relationship between the fraction of cooperators and the amplitude of coupling strength at a fixed defection parameter.

<p>In the whole, the tendency is that the cooperation will be elevated as <i>A</i> increases. However, under the symmetric case (<i>i.e</i>., case <b>I</b>), the optimal amplitude of coupling strength exists, but the asymmetric scenario (case <b>II</b>) leads to the monotonic variation of cooperation as coupling strength grows. The parameter setup is still set to be: MCS = 50000, <i>b</i> = 1.05, <i>L</i> = 200 and <i>K</i> = 0.1.</p

Characteristic patterns between cooperators and defectors on two interdependent lattices at MCS = 50000 when <i>A</i> is changed from 0 to 1.0.

<p>The panels for the first two rows denote the distribution of cooperators and defectors for the upper lattice and lower lattice in the case <b>II</b>; while the bottom two rows of panels represent the evolutionary patterns of players for the case <b>I</b>. In all these panels, red dots stand for cooperators and green dots represent defectors. From these figures, it is clearly shown that the fraction of cooperators is largely enhanced as <i>A</i> increases, meanwhile the cooperative pairs (C-C type coupling way) have the predominant advantage and dominate the evolutionary behavior on interdependent lattices. In particular, the designating way of the coupling strength between players on interdependent lattices can also affect the evolution of cooperation. The simulation parameters are set as follows: MCS = 50000, <i>b</i> = 1.05, <i>L</i> = 200 and <i>K</i> = 0.1, and <i>A</i> is varied from 0 to 1.0 with the step length 0.25 (from the left panels to the right ones).</p

For a specific initial setup, characteristic patterns between cooperators and defectors on two interdependent lattices at MCS = 1 and MCS = 50000 under case I.

<p>In all panels, red dots represent cooperators and green dot denote defectors. For the left four panels, central cooperators are initially set on upper and lower lattices at the same time; only cooperators exist on the center of upper lattices in the middle four panels; while cooperators merely exist on the center of lower lattices in the right four panels. However, the coupling strength for each pair of players on two lattices is set to be equal, in which the coupling strength is taken from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0129542#pone.0129542.e002" target="_blank">Eq (2)</a>. It is clearly indicated that cooperative pairs support the emergence of cooperation on interdependent networks. From left to right, the only difference lies that initial conditions are set as different parameter deployments between cooperators and defectors. The simulation parameters are set as follows: MCS = 50000, <i>b</i> = 1.05, <i>L</i> = 200, <i>K</i> = 0.1 and <i>A</i> = 0.5.</p

Time evolution of fraction of cooperators and cooperative pairs between two interdependent networks.

<p>The left three panels (a, b, c) depict the time course of evolution under the case <b>I</b>, where the corresponding players hold the same interdependency taken from the interval [−1,1] (i.e., <i>A</i> = 1) according to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0129542#pone.0129542.e002" target="_blank">Eq (2)</a>. Panel (a) and (b) denote the fraction of cooperators on the upper lattice and lower one, respectively, and panel (c) represents the fraction of cooperative pairs which means the corresponding players are both cooperators on these two lattices. While for the case <b>II</b> in which each individual takes the interdependency value between −1 and 1, the right three panels, from panel (d) to (f), describe the time evolution of corresponding quantities. The defection parameter <i>b</i> is fixed to be <i>b</i> = 1.05, other parameters are set to be <i>L</i> = 200 and <i>K</i> = 0.1.</p