Figure 7: Relative fraction of players belonging to rank for different values of when : (a) ; (b) ; (c) ; (d) . For , most players still occupy the three middle ranks, which indicates that the feedback mechanism in (b) is similar to that in (a). Nevertheless, the empty sites prohibit defectors to exploit cooperators, and thus the level of cooperation is substantially enhanced in the region of when , as shown in Figure 1. However, for , most players occupy three lower ranks, while only a few players emerge in two higher ranks as shown in (c), which drastically weaken the feedback mechanism. Despite that cooperators may still benefit from the dilution, the level of cooperation remarkably decreases by comparison with that for , as shown in Figure 1. When , as shown in (d), almost all players occupy three lower ranks and nearly none occupy two higher ranks. This kind of rank distribution results in a further decrease of , as shown in Figure 1.