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

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