This paper studies a two-population evolutionary game in a new setting
in between a symmetric and an asymmetric evolutionary model. It distinguishes two
types of agents: Sanchos, whose payoffs are defined by a prisoner’s dilemma game,
and Quixotes, whose payoffs are defined by a snowdrift game. Considering an imita-
tive revision protocol, a revising agent is paired with someone from his own popula-
tion or the other population. When matched, they observe payoffs, but not identities.
Thus, agents in one population interact and imitate agents from their own population
and from the other population. In this setting we prove that a unique mixed-strategy
asymptotically stable fixed point of the evolutionary dynamics exists. Taking as an
example the compliance with social norms, and depending on the parameters, two
type of equilibrium are possible, one with full compliance among Quixotes and par-
tial compliance among Sanchos, or another with partial compliance among Quixotes
and defection among Sanchos. In the former type, Sanchos comply above their Nash
equilibrium (as they imitate compliant Quixotes). In the latter type, Quixotes comply
below their Nash equilibrium (as they imitate defecting Sanchos).
