Approximating the Revenue Maximization Problem with Sharp Demands

Abstract

Abstract. We consider the revenue maximization problem with sharp multi-demand, in which m indivisible items have to be sold to n potential buyers. Each buyer i is interested in getting exactly di items, and each item j gives a benefit vij to buyer i. We distinguish between unrelated and related valuations. In the former case, the benefit vij is completely arbitrary, while, in the latter, each item j has a quality qj , each buyer i has a value vi and the benefit vij is defined as the product viqj . The problem asks to determine a price for each item and an allocation of bundles of items to buyers with the aim of maximizing the total revenue, that is, the sum of the prices of all the sold items. The allocation must be envy-free, that is, each buyer must be happy with her assigned bundle and cannot improve her utility. We first prove that, for related valuations, the problem cannot be approximated to a factor O(m 1−ǫ ), for any ǫ > 0, unless P = NP and that such result is asymptotically tight. In fact we provide a simple m-approximation algorithm even for unrelated valuations. We then focus on an interesting subclass of "proper" instances, that do not contain buyers a priori known not being able to receive any item. For such instances, we design an interesting 2-approximation algorithm and show that no (2 − ǫ)-approximation is possible for any 0 < ǫ ≤ 1, unless P = NP. We observe that it is possible to efficiently check if an instance is proper, and if discarding useless buyers is allowed, an instance can be made proper in polynomial time, without worsening the value of its optimal solution.

Improving the search of optimal prices in envy-free perfect matchings

“…A wide range of envy-free pricing problem application proposals can be found in the literature, among which we can mention: plug-in eletric vehicles charging and display advertising [9], cloud computing [10], selling books online [11], sponsored search auctions and ad exchanges [12], allocation of projects to employees and allocating equipment time to scientists [13].…”

Pricing Problems with Buyer Preselection

“…Related hardness results are alsto studied in the literature (Briest, 2008;Chalermsook, Chuzhoy, Kannan, & Khanna, 2012;Chalermsook, Laekhanukit, & Nanongkai, 2013a, 2013bDemaine, Feige, Hajiaghayi, & Salavatipour, 2008). Further variants such as large markets (Anshelevich, Kar, & Sekar, 2015), sharp demands where each buyer asks for a fixed quantity of items (Bilò et al, 2017;Chen & Deng, 2010;Chen, Deng, Goldberg, & Zhang, 2016), market with metric substitutability among the items (Chen, Ghosh, & Vassilvitskii, 2011), and buyers with budgets (Colini-Baldeschi, Leonardi, Sankowski, & Zhang, 2014;Feldman, Fiat, Leonardi, & Sankowski, 2012) are also considered. Feldman et al (2016) propose an interesting relaxation of the notion of Walrasian Equilibrium, called Combinatorial Walrasian Equilibrium (CWE), obtained by grouping items into bundles so as to induce a "reduced market" to which, then, applying the notion of Walrasian Equilibrium.…”

“…From an economic point of view, envy-free pricing, by safeguarding the losers' interests, shelters the seller from possible future losses due to their dissatisfaction. However, Bilò et al (2017) show that, in certain markets, an intrinsic and unavoidable hurdle to the construction of a good quality envy-free solution may come from the presence of a set of "disturbing" customers, that is, a set of buyers such that at least one of them gets envious in any assignment of sufficiently high revenue. This observation naturally leads to the following intriguing question: "What happens if envy-freeness is restricted to apply only to the set of winners?…”

On Social Envy-Freeness in Multi-Unit Markets

“…Related hardness results were given by (Briest 2008;Chalermsook et al 2012;Chalermsook, Laekhanukit, and Nanongkai 2013b;2013a;Demaine et al 2008). Further variants were considered by (Chen and Deng 2010;Chen, Ghosh, and Vassilvitskii 2011;Feldman et al 2012;Anshelevich, Kar, and Sekar 2015;Bilò, Flammini, and Monaco 2017;Chen et al 2016).…”

“…Finally, (Bilò, Flammini, and Monaco 2017) considered the possibility of limiting the view of some buyers, not admitting them in the market. This might be seen as a related to a social envy-freeness setting in which such buyers are isolated in the social graph.…”