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Competitive Algorithms for Online Pricing
"... Abstract. Given a seller with m amount of items, a sequence of users {u1,u2,...} come one by one, the seller must set the unit price and assign some amount of items to each user on his/her arrival. Items can be sold fractionally. Each ui has his/her value function vi(·) such that vi(x) is the highes ..."
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Abstract. Given a seller with m amount of items, a sequence of users {u1,u2,...} come one by one, the seller must set the unit price and assign some amount of items to each user on his/her arrival. Items can be sold fractionally. Each ui has his/her value function vi(·) such that vi(x) is the highest unit price ui is willing to pay for x items. The objective is to maximize the revenue by setting the price and amount of items for each user. In this paper, we have the following contributions: if the highest value h among all vi(x) is known in advance, we first show the lower bound of the competitive ratio is O(log h), then give an online algorithm with competitive ratio O(log h); if h is not known in advance, we give an online algorithm with competitive ratio O(h 3log−1/2 h 1
Approximating the Revenue Maximization Problem with Sharp Demands
"... Abstract. We consider the revenue maximization problem with sharp multidemand, 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 val ..."
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Abstract. We consider the revenue maximization problem with sharp multidemand, 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 envyfree, 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−ɛ), for any ɛ> 0, unless P = NP and that such result is asymptotically tight. In fact we provide a simple mapproximation 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 2approximation algorithm and show that no (2 − ɛ)approximation is possible for any 0 < ɛ ≤ 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. 1
Online Pricing for Multi Type of Items
"... In this paper, we study the problem of online pricing for multitype items. Given a seller with k types of items where the amount of each type is m, a sequence of users {u1, u2,...} arrive one by one. Each user is singleminded, i.e., each user is only interested in a particular bundle of items. T ..."
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In this paper, we study the problem of online pricing for multitype items. Given a seller with k types of items where the amount of each type is m, a sequence of users {u1, u2,...} arrive one by one. Each user is singleminded, i.e., each user is only interested in a particular bundle of items. The seller must set the unit price and assign some amount of bundles to each user upon his/her arrival. Bundles can be sold fractionally. Each ui has his/her value function vi(·) such that vi(x) is the highest unit price ui is willing to pay for x bundles. The objective is to maximize the revenue of the seller by setting the price and amount of bundles for each user. In this paper, we first show that the lower bound of the competitive ratio for this problem is O(log h+log k), where h is the highest unit price to be paid among all users. We then give a deterministic online algorithm Pricing, whose competitive ratio is O ( √ k · log h log k). The lower and upper bounds match with the optimal result O(log h) asymptotically when k = 1.
Online pricing for bundles of multiple items
 J GLOB OPTIM
, 2013
"... Given a seller with k types of items, m of each, a sequence of users {u1, u2,...} arrive one by one. Each user is singleminded, i.e., each user is interested only in a particular bundle of items. The seller must set the price and assign some amount of bundles to each user upon his/her arrival. Bun ..."
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Given a seller with k types of items, m of each, a sequence of users {u1, u2,...} arrive one by one. Each user is singleminded, i.e., each user is interested only in a particular bundle of items. The seller must set the price and assign some amount of bundles to each user upon his/her arrival. Bundles can be sold fractionally. Each ui has his/her value function vi (·) such that vi (x) is the highest unit price ui is willing to pay for x bundles. The objective is to maximize the revenue of the seller by setting the price and amount of bundles for each user. In this paper, we first show that a lower bound of the competitive ratio for this problem is �(log h + log k), wherehis the highest unit price to be paid among all users. We then give a deterministic online algorithm, Pricing, whose competitive ratio is O ( √ k ·log h log k). When k = 1 the lower and upper bounds asymptotically match the optimal result O(log h).