  We formulate and analyze a multi-agent model for the evolution of individual
and systemic risk in which the local agents interact with each other through a
central agent who, in turn, is influenced by the mean field of the local
agents. The central agent is stabilized by a bistable potential, the only
stabilizing force in the system. The local agents derive their stability only
from the central agent. In the mean field limit of a large number of local
agents we show that the systemic risk decreases when the strength of the
interaction of the local agents with the central agent increases. This means
that the probability of transition from one of the two stable quasi-equilibria
to the other one decreases. We also show that the systemic risk increases when
the strength of the interaction of the central agent with the mean field of the
local agents increases. Following the financial interpretation of such models
and their behavior given in our previous paper (Garnier, Papanicolaou and Yang,
SIAM J. Fin. Math. 4, 2013, 151-184), we may interpret the results of this
paper in the following way. From the point of view of systemic risk, and while
keeping the perceived risk of the local agents approximately constant, it is
better to strengthen the interaction of the local agents with the central agent
than the other way around.
