Figure 2

Phase portraits for cases where the threshold public-good game is a Volunteer’s Dilemma (\(k=1\)), a Stag Hunt \((k=n)\), and a Hybrid Game (\(1 < k < n\)) (where meant are n-player versions of these game types). A filled dot indicates a stable fixed point, a non-filled dot an unstable fixed point; the arrows on the axes indicate the direction in which the population evolves. \({p}_{k}^{{\rm{II}}}\) denotes the fraction of cooperators in a stable interior fixed point, \({p}_{k}^{{\rm{I}}}\) the fraction of cooperators in an unstable interior fixed point (with \(1-{p}_{k}^{{\rm{I}}}\) equal to the size of the basin of attraction of a stable fixed point where at least a fraction of the players cooperates).