Evolution of cooperation with shared costs and benefits

Abstract

The quest to determine how cooperation evolves can be based on evolutionary game theory, in spite of the fact that evolutionarily stable strategies (ESS) for most non-zero-sum games are not cooperative. We analyse the evolution of cooperation for a family of evolutionary games involving shared costs and benefits with a continuum of strategies from non-cooperation to total cooperation. This cost-benefit game allows the cooperator to share in the benefit of a cooperative act, and the recipient to be burdened with a share of the cooperator's cost. The cost-benefit game encompasses the Prisoner's Dilemma, Snowdrift game and Partial Altruism. The models produce ESS solutions of total cooperation, partial cooperation, non-cooperation and coexistence between cooperation and non-cooperation. Cooperation emerges from an interplay between the nonlinearities in the cost and benefit functions. If benefits increase at a decelerating rate and costs increase at an accelerating rate with the degree of cooperation, then the ESS has an intermediate level of cooperation. The game also exhibits non-ESS points such as unstable minima, convergent-stable minima and unstable maxima. The emergence of cooperative behaviour in this game represents enlightened self-interest, whereas non-cooperative solutions illustrate the Tragedy of the Commons. Games having either a stable maximum or a stable minimum have the property that small changes in the incentive structure (model parameter values) or culture (starting frequencies of strategies) result in correspondingly small changes in the degree of cooperation. Conversely, with unstable maxima or unstable minima, small changes in the incentive structure or culture can result in a switch from non-cooperation to total cooperation (and vice versa). These solutions identify when human or animal societies have the potential for cooperation and whether cooperation is robust or fragile.

Figure 1

Nonlinearities in the cost and benefit functions affect the convergence stability and resistance to invasion of a candidate solution to the cost–benefit game. In (a,b), the x-axis lies on ${{\mathcal{C}}_i(v)|}_{v=u_i}=0$ and the y-axis lies on ${{\mathcal{B}}_i(v)|}_{v=u_i}=0$. Thus, each of these functions is positive in the regions shown, satisfying the conditions that benefits are positive and costs are negative. For a given b₂ and c₂, a candidate solution is given by equation (3.4). The line with circles (S₁=0) separates the space into regions where the candidate solution for u₁ is either a maximum (S₁<0) or a minimum (S₁>0) on its adaptive landscape. The line with stars (S₂=0) separates the space into regions where the candidate solution for u₁ is either convergent stable (S₂<0) or convergent unstable (S₂>0). Arrows indicate the direction that values for S₁ or S₂ are positive. The solid line indicates the values for b₂ and c₂ that will generate a candidate solution of u₁=0.6 when (i) b₁=6 and c₁=4 or (ii) b₁=2 and c₁=4. Moving from lower left to upper right along the candidate solution curve u₁=0.6 in (a), the solution shifts from being a convergent-stable minimum (not an ESS) to a convergent-stable maximum (an ESS). Moving from lower left to upper right along the candidate solution curve u₁=0.6 in (b), the solution shifts from being an unstable minimum to an unstable maximum (neither are an ESS).

Figure 2

Depicted are the strategy dynamics and resulting ESS for a convergent-stable maximum for the Snowdrift game. (a) Starting with two very similar strategies (u₁=0.1 and u₂=0.101), the two strategies evolve in tandem and converge on the same value of u₁=0.6 and u₂=0.6. (b) Both strategies evolve towards the same maximum on the adaptive landscape. The strategy of 0.6 is an ESS, but collective fitness is not maximized at the ESS. We used b₁=7, b₂=−1.5, c₁=4.6 and c₂=−1 as in Doebeli et al. (2004). (i) t=0, (ii) t=2, (iii) t=4, (iv) t=6, (v) t=8 and (vi) t=20.

Figure 3

Depicted are the strategy dynamics and resulting ESS for a convergent-stable minimum for the Snowdrift game. (a) Starting with two very similar strategies (u₁=0.1 and u₂=0.101), the strategies evolve in tandem towards the convergent-stable minimum of 0.6, at which point the two strategies diverge and evolve to their ESS values of u₁=0 and u₂=1. The frequencies of the two strategies stay very close to their starting values of 0.5, with changes manifest only after the two strategies have diverged considerably from each other. (b) The adaptive landscape changes shape as shown at different time intervals, while the strategies first evolve towards the minimum and then diverge to the ESS. Near the convergent-stable minimum, a valley appears to the left of the strategies and then moves under and between the two strategies. With respect to collective pay-offs, the strategy u₁=0 is at minimum fitness and the strategy u₂=1 is at maximum fitness. We used b₁=6, b₂=−1.4, c₁=4.56 and c₂=−1.6 as in Doebeli et al. (2004). (i) t=0, (ii) t=2, (iii) t=5, (iv) t=15, (v) t=30 and (vi) t=60.

Figure 4

Depicted are the strategy dynamics and resulting ESS for an unstable minimum in the Snowdrift game. (a) Starting with two strategies (u₁=0.5 and u₂=0.7), the strategies immediately diverge from the unstable minimum of 0.6 Unlike the convergent-stable minimum (figure 3), the strategies do not initially evolve to the critical value of 0.6. Similar to the convergent-stable minimum, the strategies diverge and evolve to their ESS values of u₁=0 and u₂=1. Note, had the initial strategies been at u₁=0.1 and u₂=0.101, they both would have evolved to u=0, a non-ESS local maximum of the adaptive landscape. In fact, any two starting strategies that are to the left (or right) of u₁=0.6 will evolve to the non-ESS solution of u=0 (or u=1). Compare this result with the convergent-stable minimum case, where the two strategies can have very similar values and still evolve to the ESS, whereas for the unstable minimum the initial strategy values must be sufficiently different for the two strategies to evolve to the ESS. (b) By starting the strategy values on either side of the unstable minimum, the adaptive landscape begins with a valley between the two strategies, which allows them to evolve towards the ESS. For this unstable minimum case, we used b₁=3.4, b₂=−0.5, c₁=4 and c₂=−1.5 as in Doebeli et al. (2004). (i) t=0, (ii) t=3, (iii) t=6, (iv) t=10, (v) t=15 and (vi) t=40.

Figure 5

The Snowdrift game with an unstable maximum has two ESS configurations represented by non-cooperation or complete cooperation. (a) By starting with two strategies (u₁=0.3 and u₂=0.7) at values averaging less than the unstable maximum of 0.6, strategy dynamics result in convergent evolution along the adaptive landscape to the ESS of u₁=u₂=0. Collective fitness is minimized at this ESS of non-cooperation. (b) By starting with two strategies (u₁=0.5 and u₂=0.8) at values averaging greater than the unstable maximum, strategy dynamics result in convergent evolution towards the alternative ESS of u₁=u₂=1. At this ESS, collective fitness is maximized. For the unstable maximum, we used b₁=2.8, b₂=2, c₁=4 and c₂=3. One might wonder about the case of starting values that have an average equal to 0.6. In this case, evolution will drive both strategies to u=0.6; however, this solution is unstable. The slightest change in one of the strategies at this point will result in the system further evolving to one boundary or the other. (i) t=0, (ii) t=2, (iii) t=4, (iv) t=6, (v) t=8 and (vi) t=50.