  In our previous paper, "A Unified Approach to Systemic Risk Measures via
Acceptance Set" (\textit{Mathematical Finance, 2018}), we have introduced a
general class of systemic risk measures that allow for random allocations to
individual banks before aggregation of their risks. In the present paper, we
prove the dual representation of a particular subclass of such systemic risk
measures and the existence and uniqueness of the optimal allocation related to
them. We also introduce an associated utility maximization problem which has
the same optimal solution as the systemic risk measure. In addition, the
optimizer in the dual formulation provides a \textit{risk allocation} which is
fair from the point of view of the individual financial institutions. The case
with exponential utilities which allows for explicit computation is treated in
details.
