  Rewards and penalties are common practical tools that can be used to promote
cooperation in social institutions. The evolution of cooperation under reward
and punishment incentives in joint enterprises has been formalized and
investigated, mostly by using compulsory public good games. Recently, Sasaki et
al. (2012, Proc Natl Acad Sci USA 109:1165-1169) considered optional
participation as well as institutional incentives and described how the
interplay between these mechanisms affects the evolution of cooperation in
public good games. Here, we present a full classification of these evolutionary
dynamics. Specifically, whenever penalties are large enough to cause the
bi-stability of both cooperation and defection in cases in which participation
in the public good game is compulsory, these penalties will ultimately result
in cooperation if participation in the public good game is optional. The global
stability of coercion-based cooperation in this optional case contrasts
strikingly with the bi-stability that is observed in the compulsory case. We
also argue that optional participation is not so effective at improving
cooperation under rewards.
