Cheating is evolutionarily assimilated with cooperation in the continuous snowdrift game

Abstract

It is well known that in contrast to the Prisoner's Dilemma, the snowdrift game can lead to a stable coexistence of cooperators and cheaters. Recent theoretical evidence on the snowdrift game suggests that gradual evolution for individuals choosing to contribute in continuous degrees can result in the social diversification to a 100% contribution and 0% contribution through so-called evolutionary branching. Until now, however, game-theoretical studies have shed little light on the evolutionary dynamics and consequences of the loss of diversity in strategy. Here, we analyze continuous snowdrift games with quadratic payoff functions in dimorphic populations. Subsequently, conditions are clarified under which gradual evolution can lead a population consisting of those with 100% contribution and those with 0% contribution to merge into one species with an intermediate contribution level. The key finding is that the continuous snowdrift game is more likely to lead to assimilation of different cooperation levels rather than maintenance of diversity. Importantly, this implies that allowing the gradual evolution of cooperative behavior can facilitate social inequity aversion in joint ventures that otherwise could cause conflicts that are based on commonly accepted notions of fairness.

Keywords: Adaptive dynamics; Evolution of cooperation; Evolutionary branching; Replicator dynamics; Snowdrift game; Speciation in reverse.

Fig. 1

Evolution of cooperation in snowdrift games. For discrete strategies, on the one hand, the evolution of the strategy frequencies can lead to the coexistence of cooperators and cheaters (upper arrows, X₀ to B and X₁ to B), yet do not help in understanding whether or not the resultant mixture is stable against continuously small mutations. For continuous strategies, on the other hand, the population converges to an intermediate level of cooperation (lower arrows, X₀ to A and X₁ to A) and can further undergo evolutionary branching (vertical arrow, A to B). In this case, the population splits into diverging clusters across an evolutionary-branching point $x={\displaystyle \hat{x}}$ and eventually evolves to an evolutionarily stable mixture of full- and non-contributors (B). Otherwise, it is possible that a point where $x={\displaystyle \hat{x}}$ has already become evolutionarily stable. In this case, the initially dimorphic population across a point $x={\displaystyle \hat{x}}$ can be evolutionarily unstable, and thus the population will approach each other and finally merge into one cluster at the point (“evolutionary merging”; vertical arrow, B to A).

Fig. 2

Classification diagrams of evolutionary scenarios in snowdrift games. We employ (d₁,d₂) = (R − T,S − P) as the coordinate system for parameterization. Parameter sets in the fourth quadrant, {d₁ > 0, d₂ < 0}, lead to the classical snowdrift game. However, parameters by which the diversified population of cooperators and cheaters can stabilize against continuously small mutations are restricted in the triangle OQR for decelerating costs c₂ < 0 (b), and do not exist for accelerating costs c₂ > 0 (a). Moreover, the sub-region for evolutionary branching to occur is sub-triangle PQR (iv-B). Compared to stabilization of the strategic diversity, its destabilization can happen within a wider region of parameters. Indeed, in region (iv-A) of (a) and (b), the mixed equilibrium in the classical snowdrift game is no longer stable under the continuous game. The two strategies will eventually converge to an evolutionarily stable state with an intermediate level of cooperation. In (b), these regions (iv-A) and (iv-B) are divided by line QR given by b₂ − c₂ = (d₂ − d₁)/2 − c₂ < 0. Lines PQ and PR are given by D(0) = 0 and D(1) = 0, respectively. In the shaded regions one of the natural assumptions, Eq. (18), does not hold: the benefit function B(x) is not increasing. Parameters: c₁ = 4.6, c₂ = 1 (a) or −1 (b).

Fig. 3

Pairwise invisibility plots (PIPs) for the continuous snowdrift game. Each panel shows a sign plot of invasion fitness S(x,y) in Eq. (11). Due to the linearity of the payoff difference with respect to the strategy frequency, the sign pair (S(x₂,x₁),S(x₁,x₂)) can indicate the frequency dynamics between the strategies with x₁ and x₂. Panel (x) exemplifies the case of (S(x₂,x₁),S(x₁,x₂)) = (+,−) which leads to a unilateral evolution: x₂ dominates x₁. The five sign plots are representative corresponding to the five cases of adaptive dynamics in the continuous snowdrift game: (a), (b), (c), (d), and (e) are for (ii-B), (i-B), (iii-B), (iv-B), and (iv-A), respectively. Parameters: c₁ = 4.6, c₂ = −1; (d₁,d₂) = (0.3, −0.3) for (a), (0.7, −0.3) for (b), (0.3, −0.7) for (c), (0.7, −0.7) for (d), and (1.7, −1.7) for (e).

Fig. 4

Individual-based simulations of (a) merging and (b) branching in the continuous snowdrift game. Panels show evolutionary changes in the frequency distribution of cooperative investment levels over the population (from high to low: red, orange, yellow, green, blue, white (for 0)). At the outset of each tree, for (a) the population is at a traditionally acknowledged, mixed equilibrium with full-investment (x = 1) or non-investment (x = 0) and for (b) all have no investment (x = 0). In (a), the dimorphic population will eventually merge into a single branch. In (b), in contrast to this, the monomorphic population will first converge to an intermediate level and then diverge into double branches moving to the extreme states, respectively. Parameters: population size N = 10,000, mutation rate μ = 0.01, mutation variance σ = 0.005; for (a), b₁ = 7, b₂ = −1.7, c₁ = 4.6, c₂ = −1 (d₁ = 1.7, d₂ = −1.7); for (b), b₁ = 6, b₂ = −1.4, c₁ = 4.8, c₂ = −1.6 (d₁ = 1.4, d₂ = −1.4). In both cases the interior singular strategy is with x = 0.5. The scaling factor for proportional selection is set so as to be greater than the maximal difference over all possibilities of two samples. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 5

Sign plots of differences in the average cooperation level and payoff over the populations with a classical mixed equilibrium with ${\displaystyle \hat{n}}$ in Eq. (18) and the interior singular strategy with ${\displaystyle \hat{x}}$ in Eq. (13). Parameters are as in Fig. 2. For each index, the sign is “+”, if the value in the singular-strategy case is greater than that in the mixed-equilibrium case; otherwise, “−”.