Efficient fair division: Help the worst off or avoid envy?

Two or more players rank a set of indivisible items from best to worst. An efficient allocation of items is characterized, which may satisfy such properties as maximin, Borda maximin, and envy-avoidance. Whereas the two maximin properties are in conflict with envy-avoidance, there is always an efficient allocation that does not ensure envy, but it may not be maximin or Borda maximin. Computer calculations show that maximin allocations lead to envy quite often, but Borda maximin allocations do so only rarely. Implications of the theoretical findings for real-world fair-division problems are discussed.