Kind of fair item allocation
Envy-free pricing 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:
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:
Envy-free pricing is related, but not identical, to other fair allocation problems:
Results.
A Walrasian equilibrium is a market-envy-free pricing with the additional requirement that all items with a positive price must be allocated (all unallocated items must have a zero price). It maximizes the social welfare. ;templatestyles src="Citation/styles.css"/ ;^ However, a Walrasian equilibrium might not exist (it is guaranteed to exist only when agents have gross substitute valuations). Moreover, even when it exists, the sellers' revenue might be low. Allowing the seller to discard some items might help the seller attain a higher revenue.
Maximizing the seller's revenue subject to market-envy-freeness.
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 (who introduced the term "envy-free pricing") studied two classes of utility functions: "unit demand" and "single-minded". They showed:
Balcan, Blum and Mansour studied two settings: goods with "unlimited supply" (e.g. digital goods), and goods with "limited supply". 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 focused on goods with unlimited supply and "single-minded" buyers - each buyer wants only a single bundle of goods. They showed that:
Briest focused on unit-demand min-pricing buyers. Each such buyer has a subset of wanted items, and he would like to purchase the cheapest affordable wanted-item given the prices. He focused on the uniform-budget case. He showed that, under some reasonable complexity assumptions:
Chen, Ghosh and Vassilvtskii focused on items with "metric substitutability" - buyer "i"’s value for item "j" is "vi" − "ci,j", and the substitution costs "ci,j", form a metric. They show that
Wang, Lu and Im study the problem with supply constraints given as an independence system over the items, such as matroid constraints. They focus on unit-demand buyers.
Chen and Deng study multi-item markets: there are "m" indivisible items with unit supply each and "n" potential buyers where each buyer wants to buy a single item. They show:
Cheung and Swamy present polytime approximation algorithms for single-minded agents with limited supply. They approximate the revenue w.r.t. the maximum social welfare.
Hartline and Yan study revenue-maximization using prior-free truthful mechanisms. They also show the simple structure of nvy-free pricing and its connection to truthful mechanism design.
Chalermsook, Chuzhoy, Kannan and Khanna study two variants of the problem. In both variants, each buyer has a set of "wanted items".
They also study the "Tollbooth Pricing" problem - a special case of single-minded pricing in which each item is an edge in a graph, and each wanted-items set is a path in this graph.
Chalermsook, Laekhanukit and Nanongkai prove approximation hardness to a variant called "k-hypergraph pricing". They also prove hardness for unit-demand min-buying and single-minded pricing.
Demaine, Feige, Hajiaghayi and Salavatipour show hardness-of-approximation results by reduction from the unique coverage problem.
Anshelevich, Kar and Sekar study EF pricing in large markets. They consider both revenue-maximization and welfare-maximization.
Bilo, Flammini and Monaco study EF pricing with sharp demands—where each buyer is interested in a fixed quantity of an item.
Colini-Baldeschi, Leonardi, Sankowski and Zhang and Feldman, Fiat, Leonardi and Sankowski study EF pricing with budgeted agents.
Monaco, Sankowski and Zhang study multi-unit markets. They study revenue maximization under both market-envy-freeness and agent-envy-freeness. They consider both item-pricing and bundle-pricing.
References.
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