  Cooperation is ubiquitous ranging from multicellular organisms to human
societies. Population structures indicating individuals' limited interaction
ranges are crucial to understand this issue. But it is still at large to what
extend multiple interactions involving nonlinearity in payoff play a role on
cooperation in structured populations. Here we show a rule, which determines
the emergence and stabilization of cooperation, under multiple discounted,
linear, and synergistic interactions. The rule is validated by simulations in
homogenous and heterogenous structured populations. We find that the more
neighbors there are the harder for cooperation to evolve for multiple
interactions with linearity and discounting. For synergistic scenario, however,
distinct from its pairwise counterpart, moderate number of neighbors can be the
worst, indicating that synergistic interactions work with strangers but not
with neighbors. Our results suggest that the combination of different factors
which promotes cooperation alone can be worse than that with every single
factor.
