Figure 2.

Threshold public good games with reward funds. The simplex represents the state space. The three nodes, D: 100% defectors, C: 100% cooperators and R: 100% rewarders, are trivial equilibria. (a) Risk zero (p = 0). The unique interior equilibrium Q is surrounded by closed orbits, along which the three strategies dynamically coexist. Boundary orbits form a cycle connecting the three nodes. (b,c) Partial agreement (1 < k < n). For a small risk p, there can exist a stable closed orbit (bold, black line) (b). When p goes beyond a critical value p*, a mix among the three strategies is no longer sustainable, and only cooperators can stably coexist with defectors at point X₂. All interior population states evolve to this state (c). (d,e) Unanimous agreement (k = n). When p increases beyond p*, all individuals end up with the all-cooperation equilibrium C. Parameters: n = 5, c = c′ = 0.1, r′ = 2.5, and for (b,c), k = 3. (Online version in colour.)