  This article studies linear-quadratic Stackelberg games between two
dominating players (or equivalently, leaders) and a large group of followers,
each of whom interacts under a mean field game (MFG) framework. Unlike the
conventional major-minor player game, the mean field term herein is
endogenously affected by the two leaders simultaneously. These homogeneous
followers are non-cooperative, whereas the two leaders can either compete or
cooperate with each other, which are respectively formulated as a Nash and a
Pareto game. The complete solutions of the leader-follower game can be
expressed in terms of the solutions of some non-symmetric Riccati equations.
Notably, our analysis suggests that both modes of interactions between leaders
has their own merits and neither of them is always more favourable to the
community of followers. In our knowledge, a comparative study of the effect of
different modes of governance on the society is relatively rare in the existing
literature, we here provide its first preliminary quantitative analysis; under
a broad class of practically relevant models, we provide sufficient conditions
to decide whether cooperation or competition between leaders is more favourable
to the followers. Being in common with modern folklore, the relative merits of
the two Stackelberg games depend on whether the interests between the two
leaders and the followers align among themselves. Representative numerical
examples are also supplemented.
