  We investigate the efficiency of fair allocations of indivisible goods using
the well-studied price of fairness concept. Previous work has focused on
classical fairness notions such as envy-freeness, proportionality, and
equitability. However, these notions cannot always be satisfied for indivisible
goods, leading to certain instances being ignored in the analysis. In this
paper, we focus instead on notions with guaranteed existence, including
envy-freeness up to one good (EF1), balancedness, maximum Nash welfare (MNW),
and leximin. We also introduce the concept of strong price of fairness, which
captures the efficiency loss in the worst fair allocation as opposed to that in
the best fair allocation as in the price of fairness. We mostly provide tight
or asymptotically tight bounds on the worst-case efficiency loss for
allocations satisfying these notions, for both the price of fairness and the
strong price of fairness.
