Figure 2.

The effect of s parameter on the level of cooperativeness in the population. (a) The s parameter determines the steepness, hence the shape of the sigmoid function. When s approaches ∞, the probabilistic benefit function approaches to strict deterministic step-wise function, meaning that above the threshold (T) the public good is always achieved, while below never. With decreasing s the probability of public good achievement changes from strict all or nothing to a smoother function. In the later case there is a non-zero probability of achieving the common goal even below the threshold, and also above the threshold the group can fail. (b) The position of the hysteresis point with the use of strict deterministic step-wise benefit function. (c) The effect of s (steepness of the sigmoid function) parameter on the location of the hysteresis point. The hysteresis point indicates the highest cost value for which cooperation is still a stable outcome of the game, and even a small increase in the cost would cause the collapse of this polymorphic equilibrium to defection. Below this cost value we always find cooperative equilibria. (●,○: T = 5; ■,□: T = 4; ♦,◊: T = 3; ▲, Δ: T = 2).