  Masses of experiments have shown individual preference for fairness which
seems irrational. The reason behind it remains a focus for research. The effect
of spite (individuals are only concerned with their own relative standing) on
the evolution of fairness has attracted increasing attention from experiments,
but only has been implicitly studied in one evolutionary model. The model did
not involve high-offer rejections, which have been found in the form of
non-monotonic rejections (rejecting offers that are too high or too low) in
experiments. Here, we introduce a high offer and a non-monotonic rejection in
structured populations of finite size, and use strategy intervention to
explicitly study how spite influences the evolution of fairness: five
strategies are in sequence added into the competition of a fair strategy and a
selfish strategy. We find that spite promotes fairness, altruism inhibits
fairness, and the non-monotonic rejection can cause fairness to overcome
selfishness, which cannot happen without high-offer rejections. Particularly
for the group-structured population with seven discrete strategies, we
analytically study the effect of population size, mutation, and migration on
fairness, selfishness, altruism, and spite. A larger population size cannot
change the dominance of fairness, but it promotes altruism and inhibits
selfishness and spite. Intermediate mutation maximizes selfishness and
fairness, and minimizes spite; intermediate mutation maximizes altruism for
intermediate migration and minimizes altruism otherwise. The existence of
migration inhibits selfishness and fairness, and promotes altruism; sufficient
migration promotes spite. Our study may provide important insights into the
evolutionary origin of fairness.
