Fraction of behavior 1 in population which changes over time when the game parameters are constant.

<p>In the figure, the five curves correspond to five different fractions of small group 1, namely 0.1, 0.3, 0.5, 0.7, 0.9. The used payoff parameters are [<i>B</i>₁, <i>B</i>₂, <i>B</i>₃, <i>B</i>₄, <i>C</i>₁, <i>C</i>₂, <i>C</i>₃, <i>C</i>₄] = [2, 2, –1, –1, 1, 1, 2, 2].</p

The Payoff Matrix of the Interactions in Small Group 2(Simplified Form).

<p>The Payoff Matrix of the Interactions in Small Group 2(Simplified Form).</p

Effect of Heterogeneous Investments on the Evolution of Cooperation in Spatial Public Goods Game

<div><p>Understanding the emergence of cooperation in spatial public goods game remains a grand challenge across disciplines. In most previous studies, it is assumed that the investments of all the cooperators are identical, and often equal to 1. However, it is worth mentioning that players are diverse and heterogeneous when choosing actions in the rapidly developing modern society and researchers have shown more interest to the heterogeneity of players recently. For modeling the heterogeneous players without loss of generality, it is assumed in this work that the investment of a cooperator is a random variable with uniform distribution, the mean value of which is equal to 1. The results of extensive numerical simulations convincingly indicate that heterogeneous investments can promote cooperation. Specifically, a large value of the variance of the random variable can decrease the two critical values for the result of behavioral evolution effectively. Moreover, the larger the variance is, the better the promotion effect will be. In addition, this article has discussed the impact of heterogeneous investments when the coevolution of both strategy and investment is taken into account. Comparing the promotion effect of coevolution of strategy and investment with that of strategy imitation only, we can conclude that the coevolution of strategy and investment decreases the asymptotic fraction of cooperators by weakening the heterogeneity of investments, which further demonstrates that heterogeneous investments can promote cooperation in spatial public goods game.</p></div

Direct observations of behavioral evolution for the whole population when <i>f</i> = 0.1 and <i>q</i> = 1.

<p>Snapshots are taken at <i>t</i> = 1, 3, 100 and 500 MC steps. In the figure blue, yellow, green and red are used to represent behavior 1 in group 1, behavior 2 in group 1, behavior 1 in group 2 and behavior 2 in group 2 respectively. The game parameters in the numerical simulation are set as [<i>B</i>₁, <i>B</i>₂, <i>B</i>₃, <i>B</i>₄, <i>C</i>₁, <i>C</i>₂, <i>C</i>₃, <i>C</i>₄] = [2, 2, –1, –1, 1, 1, 2, 2].</p

Time series of the fraction of cooperators for different values of <i>σ</i> when <i>r</i> equals to 5.

<p>The value of <i>σ</i> is equal to 0.0,0.2,0.5,0.8,1.0 respectively. While <i>σ</i> = 0, which is exactly the traditional case, the investment of all the cooperators is 1. With the increment of <i>σ</i>, the asymptotic fraction of cooperators will increase.</p

The steady distribution of investment corresponding to strategy evolution only (Left panel) and that corresponding to coevolution of strategy and investment (Right panel), when <i>σ</i> = 0.8, <i>r</i> = 5.2.

<p>From the figure, it can be found that when only strategy is updated, the distribution of the investment in steady state is still a uniform distribution [0.2,1.8], however, when the coevolution of strategy and investment is taken into account, the previous uniform distribution is severely distorted, and the steady investment satisfies a discrete distribution valued [1.786,1.790,1.792,1.793,1.795,1.797,1.798,1.799]. In conclusion, compared to the former case, as the heterogeneity of investment is weakened, the asymptotic fraction of cooperators <i>ρ</i>_<i>C</i> decreases in the case of coevolution.</p

The Payoff Matrix of the Interactions between Small Groups 1 and 2(Simplified Form).

<p>The Payoff Matrix of the Interactions between Small Groups 1 and 2(Simplified Form).</p

Typical snapshots of strategy distributions on the square lattice when <i>σ</i> = 0.5 and <i>r</i> = 5.

<p>Cooperators and defectors are colored red and blue, and the MCS of (a)-(d) is 1, 10, 100, 50000 respectively. The figure shows that the fraction of cooperators decreases at the beginning of the evolution, but as the evolution proceeds, the cooperators form into clusters to restrain the invasion of the defectors, and spread to the defectors reversely. At the end of the evolution all the players in the population hold the cooperation strategy.</p

Fraction of behavior 1 in population which changes over time, when the fraction of small group 1 is fixed.

<p>In this figure, the five curves correspond to five different values of <i>q</i>. In detail, in the simulations <i>f</i> = 0.5, and the game parameters are selected as [<i>B</i>₁, <i>B</i>₂, <i>B</i>₃, <i>B</i>₄, <i>C</i>₁, <i>C</i>₃] = [2, 2, –1, –1, 1, 2], <i>C</i>₂ = <i>C</i>₁/<i>q</i>, <i>C</i>₄ = <i>C</i>₃/<i>q</i> and values of <i>q</i> are chosen as 1/9, 1/3, 1, 3, 9, respectively.</p

The Payoff Matrix of the Interactions in Small Group 1(Simplified Form).

<p>The Payoff Matrix of the Interactions in Small Group 1(Simplified Form).</p