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Appendix S1. Dynamics of obligatory strategies at extremely weak selection.

Appendix S2. Demonstration of the evolutionarily stable switching density for M = 3 at extremely weak selection.

Appendix S3. The steady-state relative abundance of the entire species at extremely weak selection.

Figure S1. The tragedy of the commons in sub-divided populations.

Figure S2. Persistence of low-density cooperators.

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