Title Online pricing for multi-type of Items

In this paper, we study the problem of online pricing for multi-type 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 single-minded, 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.