  Evolutionary games between species are known to lead to intriguing
spatio-temporal patterns in systems of diffusing agent. However, the role of
inter-species interactions is hardly studied when agents are (self-)propelled,
as is the case in many biological systems. Here, we combine aspects from
active-matter and evolutionary game theory and study a system of two species
whose individuals are (self-)propelled and interact through a snowdrift game.
We derive hydrodynamic equations for the density and velocity fields of both
species from which we identify parameter regimes in which one or both species
form macroscopic orientational order as well as regimes of propagating wave
patterns. Interestingly, we find simultaneous wave patterns in both species
that result from the interplay between alignment and snowdrift interactions - a
feedback mechanism that we call game-induced pattern formation. We test these
results in agent-based simulations and confirm the different regimes of order
and spatio-temporal patterns as well as game-induced pattern formation.
