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Cooperation is ubiquitous ranging from multicellular organisms to humansocieties. Population structures indicating individuals' limited interactionranges are crucial to understand this issue. But it is still at large to whatextend multiple interactions involving nonlinearity in payoff play a role oncooperation in structured populations. Here we show a rule, which determinesthe emergence and stabilization of cooperation, under multiple discounted,linear, and synergistic interactions. The rule is validated by simulations inhomogenous and heterogenous structured populations. We find that the moreneighbors there are the harder for cooperation to evolve for multipleinteractions with linearity and discounting. For synergistic scenario, however,distinct from its pairwise counterpart, moderate number of neighbors can be theworst, indicating that synergistic interactions work with strangers but notwith neighbors. Our results suggest that the combination of different factorswhich promotes cooperation alone can be worse than that with every singlefactor.

Cooperation with both synergistic and local interactions can be worse than each alone