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.
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