The durability of public goods changes the dynamics and nature of social dilemmas

Abstract

An implicit assumption underpins basic models of the evolution of cooperation, mutualism and altruism: The benefits (or pay-offs) of cooperation and defection are defined by the current frequency or distribution of cooperators. In social dilemmas involving durable public goods (group resources that can persist in the environment-ubiquitous from microbes to humans) this assumption is violated. Here, we examine the consequences of relaxing this assumption, allowing pay-offs to depend on both current and past numbers of cooperators. We explicitly trace the dynamic of a public good created by cooperators, and define pay-offs in terms of the current public good. By raising the importance of cooperative history in determining the current fate of cooperators, durable public goods cause novel dynamics (e.g., transient increases in cooperation in Prisoner's Dilemmas, oscillations in Snowdrift Games, or shifts in invasion thresholds in Stag-hunt Games), while changes in durability can transform one game into another, by moving invasion thresholds for cooperation or conditions for coexistence with defectors. This enlarged view challenges our understanding of social cheats. For instance, groups of cooperators can do worse than groups of defectors, if they inherit fewer public goods, while a rise in defectors no longer entails a loss of social benefits, at least not in the present moment (as highlighted by concerns over environmental lags). Wherever durable public goods have yet to reach a steady state (for instance due to external perturbations), the history of cooperation will define the ongoing dynamics of cooperators.

Figure 1. Social dilemmas and payoffs to cooperators and defectors.

a, Common social dilemmas organized on the ‘sucker’, ‘temptation’ (S, T) plane (R and P normalised to 1 and 0; see text for details). b–d, expected relative payoff of cooperators as a function of public goods e is f_c-f_d = S+e(1-T-S), for 0≤e<∞. Cooperator and defector payoffs are equal on the dashed lines (f_c = f_d). b, Dominance games. Red line, Prisoner's Dilemma, S = −0.5, T = 1.6. Blue line, cooperator dominance, S = 0.3, T = 0.4. c, Coexistence games. Red line, Snowdrift Game, S = 0.5, T = 1.5. Blue line, cooperator dominance, S = 0.5, T = 0.9 (but becomes snowdrift if u/c decreases sufficiently to allow e*_{p  = 1}>S/(S+T-1), here if u/c<4/5). d, Bistability games. Blue line, Stag-hunt Game, S = −0.5, T = 0.5. Red line, Prisoner's Dilemma, S = −0.5, T = 1.1 (but becomes Stag-hunt if u/c decreases sufficiently to allow e*_{p  = 1}>S/(S+T-1), here if u/c<4/5). Unless otherwise stated, game identities are consistent with u = c. See methods for more details.

Figure 2. Impact of durability on Snowdrift (coexistence) game dynamics.

Snowdrift game (T = 1.5, S = 0.5, c = u = x), stable coexistence of cooperators and defectors at p* = e* = S/(S+T-1) = 0.5, threshold to oscillations x<1 (see methods). a, b Temporal dynamics of cooperators p (black) and public good e (grey). Initial values of p range from 0.1 to 0.9. Initial value of e is zero. a, x = 10. b, x = 0.01. c public good (e)–cooperator (p) phase plane. Lines illustrate simulated trajectories for differing values of x (10, 0.1, 0.01) from initial position p ₀ = e ₀ = 0.05.

Figure 3. Impact of durability on Stag-hunt (bistable) game dynamics.

Staghunt game (T = 0.5, S = −0.5, c = u = x), repellor at p* = e* = S/(S+T-1) = 0.5. a,b Temporal dynamics of cooperator p (black) and public good e (grey). Initial values of p range from 0.1 to 0.9. Initial value of e is zero. a, x = 10. b, x = 0.1. c public good (e)–cooperator (p) phase plane. Lines illustrate simulated separatrices demarcating the basins of attraction for pure cooperator and pure defector equilibria (closed circles). Unstable equilibrium (open circle) at p* = e* = S/(S+T-1) = 0.5. Lines represent differing values of x (10, 1, 0.1, 0.01).

Figure 4. Prisoner's dilemma (defector dominance) game dynamics given durable public goods.

Public good (e)–cooperator (p) phase plane. T = 1.1, S = −0.5, c = u = x = 0.1. Sole stable equilibrium, p* = e* = 0. Lines illustrate simulated trajectories for differing initial values of e (0.2 to 1.2 in red; 1.3 to 1.9 in black) for initial p = 0.5