  We consider achieving the rates in the capacity region of a multi-level
3-receiver broadcast channel, in which the second receiver is degraded with
respect to the first receiver, with degraded message sets. We propose a
two-level chaining strategy based on polar codes that achieves the capacity
region of the considered setting without time-sharing. We also look at a slight
variation of this problem, where the first receiver only requires to decode its
own private message and the other two receivers require to decode another
private message common to them. We observe that the capacity region does not
enlarge and so the proposed polar coding strategy achieves the capacity region
for this problem as well.
