Evolution of stinginess and generosity in finite populations

In iterated continuous games, the cooperative investment in a given round is determined by the initial investment and the reciprocation rate, which describe the investment in the first round and the dependence of current investment on the partner's last move, respectively. These two traits usually intertwine during evolution. However, their coevolution is not fully explored. In this paper, we thereby study their coevolution in the iterated continuous public goods games. We find that the reciprocation rate plays a dominant role during the coevolution in both finite and infinite populations. If it exceeds a threshold, a stingy population where individuals invest no more than their partner's last investment evolves to full cooperation, and a generous population where individuals invest more than their partner's last investment decreases to a moderate cooperative state, investing a portion in the first round and then escalating investment in the following rounds. Otherwise, the stingy population evolves to full defection, and the generous one rises to another moderate cooperative state.