Increasing returns to scale: The solution to the second-order social dilemma

Abstract

Humans benefit from extensive cooperation; however, the existence of free-riders may cause cooperation to collapse. This is called the social dilemma. It has been shown that punishing free-riders is an effective way of resolving this problem. Because punishment is costly, this gives rise to the second-order social dilemma. Without exception, existing solutions rely on some stringent assumptions. This paper proposes, under very mild conditions, a simple model of a public goods game featuring increasing returns to scale. We find that punishers stand out and even dominate the population provided that the degree of increasing returns to scale is large enough; consequently, the second-order social dilemma dissipates. Historical evidence shows that people are more willing to cooperate with others and punish defectors when they suffer from either internal or external menaces. During the prehistoric age, the abundance of contributors was decisive in joint endeavours such as fighting floods, defending territory, and hunting. These situations serve as favourable examples of public goods games in which the degrees of increasing returns to scale are undoubtedly very large. Our findings show that natural selection has endowed human kind with a tendency to pursue justice and punish defection that deviates from social norms.

Figure 1. Collective hunting in a primitive society.

Figure 2. The relative time of strategies in homogenous states.

In the calculations, the parameter values are M = 100, N = 5, c = 1, r = 3, δ = 1, γ = 0.3, ω = 0.1, μ → 0.

Figure 3. Evolution of cooperation in PGG.

In the simulations, the parameter values are M = 100, N = 5, X = 30, Y = 40, Z = 30, c = 1, r = 3, α = 1.0 or α = 1.8, δ = 1, γ = 0.3, ω = 0.5, μ = 0.001. (a)With constant returns to scale (α = 1). (b) With increasing returns to scale (α = 1.8).

Figure 4. The effect of different parameter values on the evolution of cooperation in a PGG.

(a ~ c) coefficient of increasing returns to scale α in different initial composition: (a) 100% cooperators, (b) 100% defectors, (c) 100% punishers; (d) multiplier of return r; (e) contribution cost c; (f) strength of punishment δ; (g) cost of punishment γ; (h) selection strength ω; (i) mutation rate μ. The parameter values are X = 30, Y = 40, Z = 30 (except for figure a ~ c), r = 3 (except for figure d), c = 1 (except for figure e), α = 1.0 (except for figure d), α = 1.8 (except for figure a ~ c), δ = 1 (except for figure f), γ = 0.3 (except for figure g), ω = 0.5 (except for figure h), μ = 0.001 (except for figure i).