  In recent years, there has been growing interest in studying evolutionary
games with environmental feedback. Previous studies exclusively focus on
two-player games. However, extension to multi-player game is needed to study
problems such as microbial cooperation and crowdsourcing collaborations. Here,
we study coevolutionary public goods games where strategies coevolve with the
multiplication factors of group cooperation. Asymmetry can arise in such
environmental feedback, where games organized by focal cooperators may have a
different efficiency than the ones by defectors. Our analysis shows that
co-evolutionary dynamics with asymmetrical environmental feedback can yield
oscillatory convergence to persistent cooperation, if the relative changing
speed of cooperators' multiplication factor is above a certain threshold. Our
work provides useful insights into sustaining group cooperation in a changing
world.
