Figure 1.
Phase diagrams showing the remaining strategies in the spatial public goods game with cooperators (C), defectors (D), moralists (M) and immoralists (I), after a sufficiently long transient time.
Initially, each of the four strategies occupies 25% of the sites of the square lattice, and their distribution is uniform in space. However, due to their evolutionary competition, two or three strategies die out after some time. The finally resulting state depends on the synergy of cooperation, the punishment cost
, and the punishment fine
. The displayed phase diagrams are for (a)
, (b)
, and (d)
. (d) Enlargement of the small-cost area for
. Solid separating lines indicate that the resulting fractions of all strategies change continuously with a modification of the model parameters
and
, while broken lines correspond to discontinuous changes. All diagrams show that cooperators cannot stop the spreading of moralists, if only the fine-to-cost ratio is large enough. Furthermore, there are parameter regions where moralist can crowd out cooperators in the presence of defectors. Note that the spreading of moralists is extremely slow and follows a voter model kind of dynamics [46], if their competition with cooperators occurs in the absence of defectors. Therefore, computer simulations had to be run over extremely long times (up to
iterations for a systems size of
). For similar reasons, a small level of strategy mutations (which permanently creates a small number of strategies of all kinds, in particular defectors) can largely accelerate the spreading of moralists in the M phase, while it does not significantly change the resulting fractions of the four strategies [51]. The existence of immoralists is usually not relevant for the outcome of the evolutionary dynamics. Apart from a very small parameter area, where immoralists and moralists coexist, immoralists are quickly extinct. Therefore, the 4-strategy model usually behaves like a model with the three strategies C, D, and M only. As a consequence, the phase diagrams for the latter look almost the same like the ones presented here [58].
Figure 2.
Elimination of second-order free-riders (non-punishing cooperators) in the spatial public goods game with costly punishment for ,
, and
.
(a) Initially, at time , cooperators (blue), defectors (red), moralists (green) and immoralists (yellow) are uniformly distributed over the spatial lattice. (b) After a short time period (here, at
), defectors prevail. (c) After 100 iterations, immoralists have almost disappeared, and cooperators prevail, since cooperators earn high payoffs when organized in clusters. (d) At
, there is a segregation of moralists and cooperators, with defectors in between. (e) The evolutionary battle continues between cooperators and defectors on the one hand, and defectors and moralists on the other hand (here at
). (f) At
, cooperators have been eliminated by defectors, and a small fraction of defectors survives among a large majority of moralists. Interestingly, each strategy (apart from I) has a time period during which it prevails, but only moralists can maintain their majority. While moralists perform poorly in the beginning, they are doing well in the end. We refer to this as the “who laughs last laughs best” effect.
Figure 3.
Coexistence of moralists and immoralists for ,
, and
, supporting the occurrence of individuals with ‘double moral standards’ (who punish defectors, while defecting themselves).
(a) Initially, at time , cooperators (blue), defectors (red), moralists (green) and immoralists (yellow) are uniformly distributed over the spatial lattice. (b) After 250 iterations, cooperators have been eliminated in the competition with defectors (as the synergy effect
of cooperation is not large enough), and defectors are prevailing. (c–e) The snapshots at
,
, and
show the interdependence of moralists and immoralists, which appears like a tacit collaboration. It is visible that the two punishing strategies win the struggle with defectors by staying together. On the one hand, due to the additional punishment cost, immoralists can survive the competition with defectors only by exploiting moralists. On the other hand, immoralists support moralists in fighting defectors. (f) After 12000 iterations, defectors have disappeared completely, leading to a coexistence of clusters of moralists with immoralists.
Figure 4.
Dependence of cluster shapes on the punishment fine in the stationary state, supporting an adaptive balance between the payoffs of two different strategies at the interface between competing clusters.
Snapshots in the top row were obtained for low punishment fines, while the bottom row depicts results obtained for higher values of . (a) Coexistence of moralists and defectors for a synergy factor
, punishment cost
, and punishment fine
. (b) Same parameters, apart from
. (c) Coexistence of moralists and immoralists for
,
, and
. (d) Same parameters, apart from
. A similar change in the cluster shapes is found for the coexistence of cooperators and defectors, if the synergy factor
is varied.
Figure 5.
Resulting fractions of the four strategies C, D, I, and M, for random regular graphs as a function of the punishment fine .
The graphs were constructed by rewiring links of a square lattice of size with probability
, thereby preserving the degree distribution (i.e. every player has 4 nearest neighbors) [59]. For small values of
, small-world properties result, while for
, we have a random regular graph. By keeping the degree distribution fixed, we can study the impact of randomness in the network structure independently of other effects. An inhomogeneous degree distribution can further promote cooperation [37]. The results displayed here are averages over 10 simulation runs for the model parameters
,
, and
. Similar results can be obtained also for other parameter combinations.