  The Ultimatum Game (UG) is an economic game where two players (proposer and
responder) decide how to split a certain amount of money. While traditional
economic theories based on rational decision making predict that the proposer
should make a minimal offer and the responder should accept it, human subjects
tend to behave more fairly in UG. Previous studies suggested that extra
information such as reputation, empathy, or spatial structure is needed for
fairness to evolve in UG. Here we show that fairness can evolve without
additional information if players make decisions probabilistically and may
continue interactions when the offer is rejected, which we call the Not Quite
Ultimatum Game (NQUG). Evolutionary simulations of NQUG showed that the
probabilistic decision making contributes to the increase of proposers' offer
amounts to avoid rejection, while the repetition of the game works to
responders' advantage because they can wait until a good offer comes. These
simple extensions greatly promote evolution of fairness in both proposers'
offers and responders' acceptance thresholds.
