  Behavioral experiments on the Ultimatum Game have shown that we human beings
have remarkable preference in fair play, contradicting the predictions by the
game theory. Most of the existing models seeking for explanations, however,
strictly follow the assumption of \emph{Homo economicus} in orthodox Economics
that people are self-interested and fully rational to maximize their earnings.
Here we relax this assumption by allowing that people probabilistically choose
to be "good Samaritans", acting as fair players from time to time. For
well-mixed and homogeneously structured populations, we numerically show that
as this probability increases the level of fairness undergoes from the low
scenario abruptly to the full fairness state, where occasional fair behaviors
($\sim5\%$) are sufficient to drive the whole population to behave in the
half-half split manner. We also develop a mean-field theory, which correctly
reproduces the first-order phase transition and points out that the bistability
is an intrinsic property of this game and small fair acts lead to dramatical
change due to its bifurcation structure. Heterogeneously structured
populations, however, display continuous fairness transition; surprisingly,
very few hub nodes acting as fair players are able to entrain the whole
population to the full fairness state. Our results thus reveal the unexpected
strength of "good Samaritans", which may constitute a new explanation for the
emergence of fairness in our society.
