  In the classic cake-cutting problem (Steinhaus, 1948), a heterogeneous
resource has to be divided among n agents with different valuations in a
proportional way --- giving each agent a piece with a value of at least 1/n of
the total. In many applications, such as dividing a land-estate or a
time-interval, it is also important that the pieces are connected. We propose
two additional requirements: resource-monotonicity (RM) and
population-monotonicity (PM). When either the cake or the set of agents changes
and the cake is re-divided using the same rule, the utility of all remaining
agents must change in the same direction.
  Classic cake-cutting protocols are neither RM nor PM. Moreover, we prove that
no Pareto-optimal proportional division rule can be either RM or PM. Motivated
by this negative result, we search for division rules that are
weakly-Pareto-optimal --- no other division is strictly better for all agents.
  We present two such rules. The relative-equitable rule, which assigns the
maximum possible relative value equal for all agents, is proportional and PM.
The so-called rightmost-mark rule, which is an improved version of the Cut and
Choose protocol, is proportional and RM for two agents.
