  We establish dual representations for systemic risk measures based on
acceptance sets in a general setting. We deal with systemic risk measures of
both "first allocate, then aggregate" and "first aggregate, then allocate"
type. In both cases, we provide a detailed analysis of the corresponding
systemic acceptance sets and their support functions. The same approach
delivers a simple and self-contained proof of the dual representation of
utility-based risk measures for univariate positions.
