A study of compact reserve pricing languages

Abstract

Online advertising allows advertisers to implement fine-tuned targeting of users. While such precise targeting leads to more effective advertising, it introduces challenging multidimensional pricing and bidding problems for publishers and advertisers. In this context, advertisers and publishers need to deal with an exponential number of possibilities. As a result, designing efficient and compact multidimensional bidding and pricing systems and algorithms are practically important for online advertisement. Compact bidding languages have already been studied in the context of multiplicative bidding. In this paper, we study the compact pricing problem. More specifically, we first define the multiplicative reserve price optimization problem (MRPOP) and show that unlike the unrestricted reserve price system, it is NP-hard to find the best reserve price solution in this setting. Next, we present an efficient algorithm to compute a solution for MRPOP that achieves a logarithmic approximation of the optimum solution of the unrestricted setting, where we can set a reserve price for each individual impression type (i.e., one element in the Cartesian product of all features). We do so by characterizing the properties of an optimum solution. Furthermore, our empirical study confirms the effectiveness of multiplicative pricing in practice. In fact, the simulations show that our algorithm obtains 90-98% of the value of the best solution that sets the reserve prices for each auction individually (i.e., the optimum set of reserve prices). Finally, in order to establish the tightness of our results in the adversarial setting, we demonstrate that there is no compact pricing system (i.e., a pricing system using O(n1-ϵ) bits to set n reserve prices) that loses, in the worst case, less than a logarithmic factor compared to the optimum set of reserve prices. Notice that this hardness result is not restricted to the multiplicative setting and holds for any compact pricing system. In summary, not only does the multiplicative reserve price system show great promise in our empirical study, but it is also theoretically optimal up to a constant factor in the adversarial setting.