  Experiments on the ultimatum game have revealed that humans are remarkably
fond of fair play. When asked to share an amount of money, unfair offers are
rare and their acceptance rate small. While empathy and spatiality may lead to
the evolution of fairness, thus far considered continuous strategies have
precluded the observation of solutions that would be driven by pattern
formation. Here we introduce a spatial ultimatum game with discrete strategies,
and we show that this simple alteration opens the gate to fascinatingly rich
dynamical behavior. Besides mixed stationary states, we report the occurrence
of traveling waves and cyclic dominance, where one strategy in the cycle can be
an alliance of two strategies. The highly webbed phase diagram, entailing
continuous and discontinuous phase transitions, reveals hidden complexity in
the pursuit of human fair play.
