Fair enough: guaranteeing approximate maximin shares

We consider the problem of fairly allocating indivisible goods, focusing on a recently-introduced notion of fairness called maximin share guarantee: Each player's value for his allocation should be at least as high as what he can guarantee by dividing the items into as many bundles as there are players and receiving his least desirable bundle. Assuming additive valuation functions, we show that such allocations may not exist, but allocations guaranteeing each player 2/3 of the above value always exist, and can be computed in polynomial time when the number of players is constant. These theoretical results have direct practical implications.