Fig. 1: The frequency, game payoffs, and the amplitude of fluctuations for different strategies.

The frequency (solid blue), game payoffs (dashed red), and amplitude of fluctuations (dotted blue) of costly cooperators (a), costly defectors (b), non-costly cooperators (c), and non-costly defectors (d), as a function of the enhancement factor, r. As r increases, above a first threshold (r* = 1 + cg), cooperation in the costly institution evolves, and above a second threshold (approximately r = 2) cooperation in both the costly and the free institutions evolves. For medium r, the system shows periodic fluctuations. Parameter values: g = 5, nu = 10−3, π0 = 2, cg = 0.398. The replicator dynamic, derived in the Methods Section, is solved for 9000 time steps, and the time averages are taken over the last 2000 time steps.