  The problem of dividing resources fairly occurs in many practical situations
and is therefore an important topic of study in economics. In this paper, we
investigate envy-free divisions in the setting where there are multiple players
in each interested party. While all players in a party share the same set of
resources, each player has her own preferences. Under additive valuations drawn
randomly from probability distributions, we show that when all groups contain
an equal number of players, a welfare-maximizing allocation is likely to be
envy-free if the number of items exceeds the total number of players by a
logarithmic factor. On the other hand, an envy-free allocation is unlikely to
exist if the number of items is less than the total number of players. In
addition, we show that a simple truthful mechanism, namely the random
assignment mechanism, yields an allocation that satisfies the weaker notion of
approximate envy-freeness with high probability.
