  We describe the bailout of banks by governments as a Markov Decision Process
(MDP) where the actions are equity investments. The underlying dynamics is
derived from the network of financial institutions linked by mutual exposures,
and the negative rewards are associated to the banks' default. Each node
represents a bank and is associated to a probability of default per unit time
(PD) that depends on its capital and is increased by the default of
neighbouring nodes. Governments can control the systemic risk of the network by
providing additional capital to the banks, lowering their PD at the expense of
an increased exposure in case of their failure. Considering the network of
European global systemically important institutions, we find the optimal
investment policy that solves the MDP, providing direct indications to
governments and regulators on the best way of action to limit the effects of
financial crises.
