Fig. 8: Phase diagrams of the system behavior with pool punishment are qualitatively similar under non-marginal and weak selection strength.
From: Evolutionary dynamics of any multiplayer game on regular graphs
a and c (the data are in agreement with those published in Figs. 5 and 10 from ref. 8), Numerical simulations under non-marginal selection (δ = 2). The phases are defined as follows: D–only D exists; O–only O exists; (D+C+O)C–cyclic dominance among D, C, and O; FO⇔D–fixation of either O or D; DO–D and O coexist; (D+C+DO)C–cyclic dominance among D, C, and DO. b and d, The phase diagram is divided by analytical β0 and β⋆ under weak selection (δ → 0+). Here, β0 divides the D and DO⇔D phases, while β⋆ separates the DO⇔D and (D+C+O)C phases. Specifically, in b, β0 = 81/134 + (135/134)α (red), β⋆ = 3/2 + (5/2)α (blue); in d, β0 = 81/268 + (135/134)α, β⋆ = 3/4 + (5/2)α. The definition of the DO⇔D phase–the system finally evolves to full D if cooperation is initially present, or to the fixation of either O or D in the absence of initial cooperators. Other parameters: c = 1, k = 4.
