  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.
