The evolution of fairness in the coevolutionary ultimatum games

In the ultimatum games, two players are required to split a certain sum of money. Through the responder accepting the offer of proposer, the money will be shared and the fairness is built finally. Here, we figure out five coevolutionary protocols, where strategy (offering rate p and threshold for accepting an offer q) and underlying network topology can coevolve, to demonstrate how the link severing scenario affects the evolution of fairness. We show that the equilibrium of the games is significantly influenced by these coevolutionary protocols. The deterministic rules lead to overly lavish or overly generous result that is inconsistent with the outcome of human behavior experiment. However, the probabilistic rules produce fair division, similar to the realistic case. Moreover, we also introduce an amplitude parameter b to verify the plausibility of assumed link severing protocols. By means of enhancing b we analytically exhibit that preferable performance can be obtained in the game, since the total amount of agents increases as well. Last, we further support our conclusion by showing the so-called unrealistic severing events under these coevolution scenarios. We thus present a viable way of understanding the ubiquitous fairness in nature and hope that it will inspire further studies to resolve social division.