  In social dilemmas, individuals face a conflict between their own
self-interest and the collective interest of the group. The provision of reward
has been shown to be an effective means to drive cooperation in such
situations. However, previous research has often made the idealized assumption
that rewards are constant and do not change in response to changes in the game
environment. In this paper, we introduce reward into the public goods game and
develop a coevolutionary game model in which the strength of reward is
adaptively adjusted according to the population state. Specifically, we assume
that decreasing levels of cooperation lead to an increase in reward strength,
while increasing levels of cooperation result in a decrease in reward strength.
By investigating coevolutionary dynamics between population state and reward
systems, we find that interior stable coexistence state can emerge in our
model, where the levels of cooperation and reward strength remain at constant
levels. In addition, we also reveal that the emergence of a full cooperation
state only requires a minimal level of reward strength. Our study highlights
the potential of adaptive feedback reward as a tool for achieving long-term
stability and sustainability in social dilemmas.
