Two kinds of envy are considered: The no-envy conditions guarantee that the market is stable and that the buyers do not resent the seller.
Many authors studied the computational problem of finding a price-vector that maximizes the seller's revenue, subject to market-envy-freeness.
Guruswami, Hartline, Karlin, Kempe, Kenyon and McSherry[1] (who introduced the term envy-free pricing) studied two classes of utility functions: unit demand and single-minded.
They showed that a single price, selected at random, attains an expected revenue which is a non-trivial approximation of the maximum social welfare: Briest and Krysta[3] focused on goods with unlimited supply and single-minded buyers - each buyer wants only a single bundle of goods.
They show: Cheung and Swamy[8] present polytime approximation algorithms for single-minded agents with limited supply.
Chalermsook, Laekhanukit and Nanongkai[11] prove approximation hardness to a variant called k-hypergraph pricing.
[12] Demaine, Feige, Hajiaghayi and Salavatipour[13] show hardness-of-approximation results by reduction from the unique coverage problem.
Bilo, Flammini and Monaco[15] study EF pricing with sharp demands—where each buyer is interested in a fixed quantity of an item.