  Cooperation and defection may be considered as two extreme responses to a
social dilemma. Yet the reality is much less clear-cut. Between the two
extremes lies an interval of ambivalent choices, which may be captured
theoretically by means of continuous strategies defining the extent of the
contributions of each individual player to the common pool. If strategies are
chosen from the unit interval, where 0 corresponds to pure defection and 1
corresponds to the maximal contribution, the question is what is the
characteristic level of individual investments to the common pool that emerges
if the evolution is guided by different benefit functions. Here we consider the
steepness and the threshold as two parameters defining an array of generalized
benefit functions, and we show that in a structured population there exist
intermediate values of both at which the collective contributions are maximal.
However, as the cost-to-benefit ratio of cooperation increases the
characteristic threshold decreases, while the corresponding steepness
increases. Our observations remain valid if more complex sigmoid functions are
used, thus reenforcing the importance of carefully adjusted benefits for high
levels of public cooperation.
