Involution game with spatio-temporal heterogeneity of social resources

When group members claim a portion of limited resources, it is tempting to invest more effort to get a larger share. However, if everyone acts similarly, they all get the same piece they would obtain without extra effort. This is the involution game dilemma that can be detected in several real-life situations. It is also a realistic assumption that resources are not uniform in space and time, which may influence the system's resulting involution level. We here introduce spatio-temporal heterogeneity of social resources and explore their consequences on involution. When spatial heterogeneity is applied, network reciprocity can mitigate the involution for rich resources, which would be critical otherwise in a homogeneous population. Interestingly, when the resource level is modest, spatial heterogeneity causes more intensive involution in a system where most cooperator agents, who want to keep investment at a low level, are present. This picture is partly the opposite in the extreme case when more investment is less effective. Spatial heterogeneity can also produce a counterintuitive effect when the presence of alternative resource levels cannot explain the emergence of involution. If we apply temporal heterogeneity additionally, then the impact of spatial heterogeneity practically vanishes, and we turn back to the behavior observed in a homogeneous population earlier. Our observations are also supported by solving the corresponding replicator equations numerically.