Rewards and penalties are common practical tools that can be used to promote cooperation in social intitutions. The evolution of cooperation under reward and punishment incentives in joint enterprises has been formalized and investigated, mostly by using ompulsory public good games. Recently, Sasaki et al. (2012, Proc Natl Acad Sci USA 109:1165.1169) considered optional participation as well as institutional incentive and described how the interplay between these mechanisms affects the evolution of cooperation in public good games. Here, we present a full clssification 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 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 imrpoving cooperation under rewards.