Frequency capping in online advertising

We study the following online problem. There are n advertisers. Each advertiser a_i has a total demand d_i and a value v _i for each supply unit. Supply units arrive one by one in an online fashion, and must be allocated to an agent immediately. Each unit is associated with a user, and each advertiser a_i is willing to accept no more than f_i units associated with any single user (the value f_i is called the frequency cap of advertiser a_i). The goal is to design an online allocation algorithm maximizing the total value. We first show a deterministic 3/4 -competitiveness upper bound, which holds even when all frequency caps are 1, and all advertisers share identical values and demands. A competitive ratio approaching 1-1/e can be achieved via a reduction to a different model considered by Goel and Mehta (WINE '07: Workshop on Internet and Network, Economics: 335-340, 2007). Our main contribution is analyzing two 3/4 -competitive greedy algorithms for the cases of equal values, and arbitrary valuations with equal integral demand to frequency cap ratios. Finally, we give a primal-dual algorithm which may serve as a good starting point for improving upon the ratio of 1-1/e.