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 fc-fd = S+e(1-T-S), for 0≤e<∞. Cooperator and defector payoffs are equal on the dashed lines (fc = fd). 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 p0 = e0 = 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