  A set of objects is to be divided fairly among agents with different tastes,
modeled by additive value functions. If the objects cannot be shared, so that
each of them must be entirely allocated to a single agent, then fair division
may not exist. How many objects must be shared between two or more agents in
order to attain a fair division? The paper studies various notions of fairness,
such as proportionality, envy-freeness and equitability. It also studies
consensus division, in which each agent assigns the same value to all bundles
--- a notion that is useful in truthful fair division mechanisms. It proves
upper bounds on the number of required sharings. However, it shows that finding
the minimum number of sharings is, in general, NP-hard even for generic
instances. Many problems remain open.

Computer Science and Game Theory;Theoretical Economics