  We consider the problem of envy-free cake cutting, which is the distribution
of a continuous heterogeneous resource among self interested players such that
nobody prefers what somebody else receives to what they get. Existing work has
focused on two distinct classes of solutions to this problem - allocations
which give each player a continuous piece of cake and allocations which give
each player arbitrarily many disjoint pieces of cake. Our aim is to investigate
allocations between these two extremes by parameterizing the maximum number of
disjoint pieces each player may receive. We characterize the Price of
Indivisibility (POI) as the gain achieved in social welfare (utilitarian and
egalitarian), by moving from allocations which give each player a continuous
piece of cake to allocations that may give each player up to $k$ disjoint
pieces of cake. Our results contain bounds for the Price of Indivisibility for
utilitarian as well as egalitarian social welfare, and for envy-free cake
cutting as well as cake cutting without any fairness constraints.
