![]() Trajectories either converge to R, now a sink, or else to N. ( c) For J − < I < J +, an interior saddle point Q moves, with increasing I, along the dashed line from the CD-edge to N. All trajectories in the interior of the triangle Δ lead to N. Rewarding: ( b) For I − < I < J −, the equilibrium R on the CD-edge is a saddle point. For the range of incentives in between a and h, the impacts of rewards and penalties differ. All trajectories converge to C, the state of full cooperation. ( h) If I is larger than I + = c, the incentives alter the outcome and cooperation prevails. In that state, arbitrarily small random perturbations lead to short bursts of cooperation, immediately subverted by defection and followed by a return to nonparticipation. The interior of the triangle Δ is filled with trajectories issuing from and converging to the vertex N of nonparticipation in the joint enterprise. ![]() ( a) If I is smaller than I − = c/ n, the incentives have no effect on the outcome of the public good game. The three vertices C, D, and N correspond to the three homogeneous states in which the population consists exclusively of cooperators ( x = 1), defectors ( y = 1), or nonparticipants ( z = 1). The triangles Δ represent the state space Δ =, where x, y, and z are the frequencies of cooperators, defectors, and nonparticipants, respectively. Combining punishing with optional participation enables full cooperation for a small fraction of the cost needed in the compulsory case. We also demonstrate that punishing provides a "lighter touch" than rewarding, guaranteeing full cooperation at considerably lower cost.Įffects of institutional rewarding and punishing on the optional public good game for different per capita incentives I. This removes the social trap and implies that whenever a society of cooperators cannot be invaded by free-riders, it will necessarily become established in the long run, through social learning, irrespective of the initial number of cooperators. In particular, optional participation allows institutions punishing free-riders to overcome the social dilemma at a much lower cost, and to promote a globally stable regime of cooperation. We show that this combination fundamentally improves the efficiency of incentives. Here, we analyze the interplay of (a) incentives provided by institutions and (b) the effects of voluntary participation. A society of free-riders is then caught in a social trap, and the institution is unable to provide an escape, except at a high, possibly prohibitive cost. In the latter case, both norms, cooperation and defection, are stable: To avoid a collapse to full defection, cooperators must be sufficiently numerous initially. Even though an incentive-providing institution may protect a cooperative society from invasion by free-riders, it cannot always convert a society of free-riders to cooperation. Such social dilemmas can be overcome by positive or negative incentives. Free-riders can invade a society of cooperators, causing a tragedy of the commons. Self-interest frequently causes individuals engaged in joint enterprises to choose actions that are counterproductive.
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