  Cooperation in the evolutionary snowdrift game with a self-questioning
updating mechanism is studied on annealed and quenched small-world networks
with directed couplings. Around the payoff parameter value $r=0.5$, we find a
size-invariant symmetrical cooperation effect. While generally suppressing
cooperation for $r>0.5$ payoffs, rewired networks facilitated cooperative
behavior for $r<0.5$. Fair amounts of noise were found to break the observed
symmetry and further weaken cooperation at relatively large values of $r$.
However, in the absence of noise, the self-questioning mechanism recovers
symmetrical behavior and elevates altruism even under large-reward conditions.
Our results suggest that an updating mechanism of this type is necessary to
stabilize cooperation in a spatially structured environment which is otherwise
detrimental to cooperative behavior, especially at high cost-to-benefit ratios.
Additionally, we employ component and local stability analyses to better
understand the nature of the manifested dynamics.

Physics and Society;Computer Science and Game Theory;Computational Physics;Populations and Evolution