We study the envy free pricing problem faced by a seller who wishes to maximize revenue by setting prices for bundles of items. If there is an unlimited supply of items and agents are single minded then we show that finding the revenue maximizing envy free allocation pricing can be solved in polynomial time by reducing it to an instance of weighted independent set on a perfect graph. We define an allocation pricing as if no agent wishes to replace her allocation with the union of the allocations of some set of other agents and her price with the sum of their prices. We show that it is -hard to decide if a given allocation pricing is multi envy free. We also show that revenue maximization multi envy free allocation pricing is hard. Furthermore, we give efficient algorithms and hardness results for various variants of the highway problem.
Much of the work on envy free revenue maximization is on item pricing rather than on subset pricing. @cite_13 give an @math -approximation for the general single minded problem, where @math is the number of items and @math is the number of agents. This result was extended by @cite_4 to an @math -approximation for arbitrary valuations and unlimited supply using single fixed pricing which is basically pricing all bundles with the same price. @cite_6 show that the general item pricing problem with unlimited availability of items is hard to approximate within a (semi-)logarithmic factor.
@cite_13 @cite_6 @cite_4
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We consider the problem of pricing n items to maximize revenue when faced with a series of unknown buyers with complex preferences, and show that a simple pricing scheme achieves surprisingly strong guarantees. We show that in the unlimited supply setting, a random single price achieves expected revenue within a logarithmic factor of the total social welfare for customers with general valuation functions, which may not even necessarily be monotone. This generalizes work of Guruswami et. al [18], who show a logarithmic factor for only the special cases of single-minded and unit-demand customers. In the limited supply setting, we show that for subadditive valuations, a random single price achieves revenue within a factor of 2O(√(log n loglog n) of the total social welfare, i.e., the optimal revenue the seller could hope to extract even if the seller could price each bundle differently for every buyer. This is the best approximation known for any item pricing scheme for subadditive (or even submodular) valuations, even using multiple prices. We complement this result with a lower bound showing a sequence of subadditive (in fact, XOS) buyers for which any single price has approximation ratio 2Ω(log1 4 n), thus showing that single price schemes cannot achieve a polylogarithmic ratio. This lower bound demonstrates a clear distinction between revenue maximization and social welfare maximization in this setting, for which [12,10] show that a fixed price achieves a logarithmic approximation in the case of XOS [12], and more generally subadditive [10], customers. We also consider the multi-unit case examined by [1111] in the context of social welfare, and show that so long as no buyer requires more than a 1 -- e fraction of the items, a random single price now does in fact achieve revenue within an O(log n) factor of the maximum social welfare. We consider prior-free auctions for revenue and welfare maximization when agents have a common budget. The abstract environments we consider are ones where there is a downward-closed and symmetric feasibility constraint on the probabilities of service of the agents. These environments include position auctions where slots with decreasing click-through rates are auctioned to advertisers. We generalize and characterize the envy-free benchmark from Hartline and Yan (2011) to settings with budgets and characterize the optimal envy-free outcomes for both welfare and revenue. We give prior-free mechanisms that approximate these benchmarks. A building block in our mechanism is a clinching auction for position auction environments. This auction is a generalization of the multi-unit clinching auction of (2008) and a special case of the polyhedral clinching auction of (2012). For welfare maximization, we show that this clinching auction is a good approximation to the envy-free optimal welfare for position auction environments. For profit maximization, we generalize the random sampling profit extraction auction from (2002) for digital goods to give a 10.0-approximation to the envy-free optimal revenue in symmetric, downward-closed environments. The profit maximization question is of interest even without budgets and our mechanism is a 7.5-approximation which improving on the 30.4 bound of Ha and Hartline (2012). We study envy-free (EF) mechanisms for multi-unit auctions with budgeted agents that approximately maximize revenue. In an EF auction, prices are set so that every bidder receives a bundle that maximizes her utility amongst all bundles; We show that the problem of revenue-maximizing EF auctions is NP-hard, even for the case of identical items and additive valuations (up to the budget). The main result of our paper is a novel EF auction that runs in polynomial time and provides a approximation of 1 2 with respect to the revenue-maximizing EF auction. A slight variant of our mechanism will produce an allocation and pricing that is more restrictive (so called item pricing) and gives a 1 2 approximation to the optimal revenue within this more restrictive class. Abstract Consider the revenue-maximizing problem in which a single seller wants to sell k different items to a single buyer, who has independently distributed values for the items with additive valuation. The case was completely resolved by Myerson’s classical work in 1981, whereas for larger k the problem has been the subject of much research efforts ever since. Recently, Hart and Nisan analyzed two simple mechanisms: selling the items separately, or selling them as a single bundle. They showed that selling separately guarantees at least a fraction of the optimal revenue; and for identically distributed items, bundling yields at least a fraction of the optimal revenue. In this paper, we prove that selling separately guarantees at least fraction of the optimal revenue, whereas for identically distributed items, bundling yields at least a constant fraction of the optimal revenue. These bounds are tight (up to a constant factor), settling the open questions raised by Hart and Nisan. The results are valid for arbitrary probability distributions without restrictions. Our results also have implications on other interesting issues, such as monotonicity and randomization of selling mechanisms.