Punishment in optional public goods games

In this work, the optional public goods games with punishment are studied. By adopting the approximate best response dynamics, a micro model is given to explain the evolutionary process. Simultaneously, the magnitude of rationality is also considered. Under the condition of bounded rationality which provides a light to interpret phenomena in human society, the model leads to two types of equilibriums. One is the equilibrium without punishers and the other is the equilibrium including only punishers and cooperators. In addition, the effects of rationality on equilibriums are briefly investigated.