  We analyze how interdependencies between organizations in financial networks
can lead to multiple possible equilibrium outcomes. A multiplicity arises if
and only if there exists a certain type of dependency cycle in the network that
allows for self-fulfilling chains of defaults. We provide necessary and
sufficient conditions for banks' solvency in any equilibrium. Building on these
conditions, we characterize the minimum bailout payments needed to ensure
systemic solvency, as well as how solvency can be ensured by guaranteeing a
specific set of debt payments. Bailout injections needed to eliminate
self-fulfilling cycles of defaults (credit freezes) are fully recoverable,
while those needed to prevent cascading defaults outside of cycles are not. We
show that the minimum bailout problem is computationally hard, but provide an
upper bound on optimal payments and show that the problem has intuitive
solutions in specific network structures such as those with disjoint cycles or
a core-periphery structure.
