Effects of Empathy on the Evolutionary Dynamics of Fairness in Group-Structured Systems

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 the experimental behaviors where the mean offer typically ranges from 0.3 to 0.5 and the mean demand tends to lie between 0.2 and 0.35. To explain the evolution of such fair behaviors, here we introduce empathy in a mutation-selection process with group structure and find that our results quantitatively reproduce the experimental behaviors at low randomness with intermediate empathy or relatively high randomness with small empathy. Moreover, we show that with low randomness more empathy leads to a fairer outcome with a higher mean offer and demand. Counterintuitively, more empathy corresponds to a lower mean offer together with a higher mean demand for relatively high randomness. Finally, we analytically provide the mean offer and demand under both weak and strong intensities of selection when the largest or smallest level of empathy is introduced. Our study provides systematic insights into the evolutionary origin of fairness in a mutation-selection process with empathetic strategies and group structure.