Envy, Multi Envy, and Revenue Maximization

Abstract

We study the envy free pricing problem faced by a seller IntroductionWe consider the combinatorial auction setting where there are several different items for sale, not all items are identical, and agents have valuations for subsets of items. We allow the seller to have identical copies of an item. We distinguish between the case of limited supply (e.g., physical goods) and that of unlimited supply (e.g., digital goods). Agents have known valuations for subsets of items. We assume free disposal, i.e., the valuation of a superset is ≥ the valuation of a subset. Let S be a set of items, agent i has valuation v i (S) for set S. The valuation functions, v i , are public knowledge. Ergo, we are not concerned with issues of truthfulness or incentive compatible bidding. Our concern here is to maximize revenue in an envy free manner.Our goal is to determine prices for sets of items while (approximately) maximizing revenue. The output of the mechanism is a payment function p that assigns prices to sets of items and an allocation a. Although there are exponentially many such sets, we will only consider payments functions that have a concise representation. For a set of items S let p(S) be the payment required for set S. Let a i be the set assigned to agent i.In general, every agent i has valuation function v i defined over every subset of items.Given a payment function p, and a set of valuation functions v i , let z i = max S (v i (S) − p(S)), and let S i to be a collection of sets such that S ∈ S i if and only if v i (S) − p(S) = z i .We now distinguish between two notions of envy freeness.Definition 1 We say that (a, p) are envy free ifDefinition 2 If a i = ∅ then we say that agent i loses, otherwise we say that agent i wins.Definition 3 A pricing p is monotone if for each subset S and for each collection of subsets C such that S ⊆ T ∈C

Competitive Algorithms for Online Pricing

“…Envy-free is an important property in computational economy, we say a pricing is envy-free if the utility of each user is maximized. Envy-free pricing has been studied in [2,5,6,8,11]. This paper is organized as follows: Section 2 consider the variant in which the highest value h of the value functions is known in advance, we first prove the lower bound of the competitive ratio for this variant is O(log h), then give a deterministic online algorithm whose competitive ratio matches this lower bound.…”

Online pricing for multi-type of items

“…The pricing problem can be viewed as a special case of the offline combinatorial auctions problem, which have been studied in [13,15,9,4,12]. One direction studied in combinatorial auctions is envy-freeness, which means that given the pricing, no user would prefer to be assigned a different bundle of items [13,15,9].…”

Online Pricing for Multi-type of Items

“…Previous work has been focused on two supply models: the unlimited supply model [1,2,6,10,12] where the number of each type of item is unbounded and the limited supply model [1, 3-5, 7, 11, 13, 14] where the number of each type of item is bounded by some value. As for the users, there are several users' behaviors studied, including single-minded [7-10, 12, 14] (each user is only interested in a particular set of items), unit-demand [2-6, 12, 14] (each user will buy at most one item in total) and envy free [1,5,7,10,12] (after the assignment, no user would prefer to be assigned a different set of items with the designated prices, loosely speaking, each user is happy with his/her purchase). Most of the previous studies have considered a combination of the above scenarios (e.g.…”