  The voter model on networks is crucial to understand opinion formation.
Uni-directional social interactions are ubiquitous in real social networks
whereas undirected interactions are intensively studied. We establish a voter
model on a dynamical directed network. We show that the opinion invasion is
captured by a replicator equation of an emergent four-player two-strategy game,
and the average in(out)-degree for the two opinions is fully captured by an
emergent three-player two-strategy game. Interestingly, it is shown that the
difference between the two emergent games arises from the uni-directionality of
the network. The difference implies that the opinion with a small number of
disciples can take over the population for in-group bias, provided that the
network is directed. Our work makes an explicit connection between opinion
dynamics and evolutionary games.
