Effects of empathy on the evolution of fairness in group-structured populations

The ultimatum game has been a prominent paradigm in studying the evolution of fairness. It predicts that responders should accept any nonzero offer and proposers should offer the smallest possible amount according to orthodox game theory. However, the prediction strongly contradicts with experimental findings where responders usually reject low offers below $20\%$ and proposers usually make higher offers than expected. To explain the evolution of such fair behaviors, we here introduce empathy in group-structured populations by allowing a proportion $\alpha$ of the population to play empathetic strategies. Interestingly, we find that for high mutation probabilities, the mean offer decreases with $\alpha$ and the mean demand increases, implying empathy inhibits the evolution of fairness. For low mutation probabilities, the mean offer and demand approach to the fair ones with increasing $\alpha$, implying empathy promotes the evolution of fairness. Furthermore, under both weak and strong intensities of natural selection, we analytically calculate the mean offer and demand for different levels of $\alpha$. Counterintuitively, we demonstrate that although a higher mutation probability leads to a higher level of fairness under weak selection, an intermediate mutation probability corresponds to the lowest level of fairness under strong selection. Our study provides systematic insights into the evolutionary origin of fairness in group-structured populations with empathetic strategies.