Envy-free pricing

Envy-free pricing^[1] is a kind of fair item allocation. There is a single seller that owns some items, and a set of buyers who are interested in these items. The buyers have different valuations to the items, and they have a quasilinear utility function; this means that the utility an agent gains from a bundle of items equals the agent's value for the bundle minus the total price of items in the bundle. The seller should determine a price for each item, and sell the items to some of the buyers, such that there is no envy. Two kinds of envy are considered:

• Agent envy means that some agent assigns a higher utility (a higher difference value-price) to a bundle allocated to another agent.
• Market envy means that some agent assigns a higher utility (a higher difference value-price) to any bundle.

The no-envy conditions guarantee that the market is stable and that the buyers do not resent the seller. By definition, every market envy-free allocation is also agent envy-free, but not vice versa.

There always exists a market envy-free allocation (which is also agent envy-free): if the prices of all items are very high, and no item is sold (all buyers get an empty bundle), then there is no envy, since no agent would like to get any bundle for such high prices. However, such an allocation is very inefficient. The challenge in envy-free pricing is to find envy-free prices that also maximize one of the following objectives:

• The social welfare - the sum of buyers' utilities;
• The seller's revenue (or profit) - the sum of prices paid by buyers.

Envy-free pricing is related, but not identical, to other fair allocation problems:

• In envy-free item allocation, monetary payments are not allowed.
• In the rental harmony problem, monetary payments are allowed, and the agents are quasilinear, but all objects should be allocated (and each agent should get exactly one object).