Fig. 3: Undemanding philanthropic and demanding aspirational behavioral motivations promote cooperation.

We consider four types of games, namely two-action donation games (a, e), three-action linear donation games (b, f), three-action nonlinear donation games with benefit factor ω = 1.5 (c, g), and with benefit factor ω = 0.5 (d, h). Dots in (a–d) illustrate the optional actions in the game, with the action cost shown in the x-axis and the generated benefit in the y-axis. e–h presents the abundance of cooperation as the average need threshold \(\bar{\alpha }\) varies from undemanding to demanding levels. In each game, we investigate both philanthropic (blue, λ = 0.01) and aspirational (red, λ = −0.01) behavioral motivations on six classes of networks (random regular networks (RR), Erdös-Rényi networks (ER)⁵⁷, Watts-Strogatz small-world networks (SW)⁵⁸ with rewiring probability 0.1, Barabási-Albert scale-free networks (BA-SF)⁵⁹, Goh-Kahng-Kim scale-free networks (GKK-SF)⁶⁰ with exponent 2.5, and Holme-Kim scale-free networks (HK-SF)⁶¹ with triad formation probability 0.1). Dots in (e–h) indicate the results of Monte Carlo simulations, and lines are analytical results. The results show that in a population of undemanding philanthropic and demanding aspirational behavioral motivations, cooperation is favored, consistent in all population structures. Each dot in (e–h) is the result averaged over 5000 simulations, and each simulation lasts for 10⁵ rounds. We consider two initial configurations: individuals occupying the largest 50% of highly connected nodes have a higher threshold, α + ϵ, while those in the least 50% have a lower threshold, α − ϵ (positive correlation, solid dots); individuals in the largest 50% of highly connected nodes have a lower threshold, α − ϵ, and those in the least 50% have a higher threshold, α + ϵ (negative correlation, open dots). Parameter values: network size N = 100, average degree d = 6, benefit b = 6, cost c = 1, and ϵ = 0.5.