Different-size world dynamics: a 5 | 5 world, a 25 | 25 world, and 100 | 100 world. Several simulation runs (each on a separate row are depicted. The initial population contains 50% D -players randomly distributed. S = 2 and p is selected in the middle of the p -transition interval for each N | N world. 0,1,2,3,4,5,10,50,150 , and 500 rounds of the SH game are captured. The color code is: blue - honest/was honest; red dishonest/was dishonest; green - honest/was dishonest; yellow - dishonest/was honest. doi:10.1371/journal.pone.0087471.g010 

Context in source publication

Context 1

... found. S -transition intervals are depicted in Fig. 6. For lower p values, the S -transition intervals become significantly wider and translated to higher values. This means that punishment severity is ineffective when punishment probability is very low. Higher punishment probability makes it possible to reduce significantly punishment severity, with the same effect on the D and H population rate. If D -player’s advantage when playing against an H -player is double ( a is set to 6 instead of 3) the effect is a significant change in the p -transition interval: from (0 : 12,0 : 18) to (0 : 4,0 : 47) , indicating a large increase in the required punishment probability. This indicates that D -player’s advantage in a ( D , H ) interaction is a sensitive parameter of the model. If p 2 is set to 0 , meaning that a D -player is punished only when playing against an H -player, the effect is a translation of the p 1 transition interval, which also becomes slightly narrower. For S = 2 and p 2 = 0 , the p 1 -transition interval changes from (0 : 12,0 : 18) to (0 : 23,0 : 26) . This indicates that a double punishment probability is necessary when, for some reasons, p 2 is close to zero. This corresponds to a case when both D -players are difficult to expose. Such a form of cooperation between D -players is clearly unfavorable to H -player spreading. For the rest of experiments we set p 1 = p 2 = p . Different rates of randomly distributed D and H -players in the initial population are considered. Fig. 7 illustrates the effect of the initial D -player rate on the p -transition interval. The p -transition interval changes when the initial rate of D players changes. The difference is significant for a D rate superior to 50% ( H rate below 50%) and less significant when D -players are a minority. Between one D -player and 50% D -players the p transition interval does not change much. When D -player rate is high ( w 70 % ) the p -transition interval becomes more nosy. An explanation may be found if we correlate these results with the observations about cluster dynamics (see Experiment 1, Fig. 1 and 2). When the initial rate of D -players is high, the H cluster formation probability is low. If no H cluster appears in the first rounds, then D -players will spread all over the population. This suggests the fact that the initial cluster structure is more important than the initial proportion of H and D -players. The importance of the initial cluster structure is investigated in the next experiment. In this experiment we start from a situation where clusters of D and H -players already exist (in all other experiments we start from H and D -players randomly spread). Two world states are generated by letting the population evolve: one with 95% and the second with 12% D -players. In both cases clusters already exist (similar to what is seen in Fig. 4). The granularity is measured by counting the strategy changes for each lattice row. The value is averaged and normalized. The granularity is similar for the two cases (about 0.9, whereas in the case of randomly spread players it is about 0.5). The results are depicted in Fig. 8. It may be observed that p -transition intervals are very similar for the two different initial world states. This fact indicates that cluster existence is much more important than the initial H / D population rate. In this experiment we study the effect of the population size. Fig. 9 depicts the average p -transition intervals for different population sizes: 3 | 3,4 | 4,5 | 5,10 | 10,20 | 20,30 | 30,50 | 50,100 | 100 , and 250 | 250 . It may be observed that, for small size worlds, the p -transition intervals are wider, translated to higher values, and also exhibit some noise. Significant changes appear when the world size is smaller than 50 | 50 . Since the values depicted in Fig. 9 are averages, the noise indicates a high dispersion of the H rate dynamic. The noise observed for the small size worlds indicate that their dynamic is less stable. These results may be explained by the initial cluster formation: in a small size world, the cluster formation probability is lower than in a large world. If H clusters do not appear, the situation converges rapidly to a pure D dominance. A similarity may be observed between the p -transition interval for a small population (Fig. 9) and the p -transition interval for an initial world state with numerous D -players (Fig. 8). This similarity may be explained by a common cause: the probability of H cluster formation in the first rounds depends on the initial distribution but also on the population size (probability of cluster formation is higher in larger populations). We notice that higher punishment probability and severity are needed in small-size worlds (e.g. 3 | 3 or 20 | 20 ) in order to obtain the same effect as in larger size worlds (e.g. 50 | 50 or 200 | 200 ). As we already noticed, cluster formation is the main driver for spreading honest/dishonest behavior. Fig. 10 depicts three different-size world dynamics. Very small size worlds tend to converge towards a pure distribution (100% D or 100% H ). In medium and large ...