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In this paper, we propose a stochastic model to describe how search service providers charge client companies based on users' queries for the keywords related to these companies' ads by using certain advertisement assignment strategies. We formulate an optimization problem to maximize the long-term average revenue for the service provider under each client's long-term average budget constraint and design an online algorithm which captures the stochastic properties of users' queries and click-through behaviors. We solve the optimization problem by making connections to scheduling problems in wireless networks, queueing theory and stochastic networks. Unlike prior models, we do not assume that the number of query arrivals is known. Due to the stochastic nature of the arrival process considered here, either temporary “free” service, i.e., service above the specified budget (which we call “overdraft”) or under-utilization of the budget (which we call “underdraft”) is unavoidable. We prove that our online algorithm can achieve a revenue that is within O(ε) of the optimal revenue while ensuring that the overdraft or underdraft is O(1/ε) , where ε can be arbitrarily small. With a view towards practice, we can show that one can always operate strictly under the budget. In addition, we extend our results to a click-through rate maximization model and also show how our algorithm can be modified to handle non-stationary query arrival processes and clients with short-term contracts. Our algorithm also allows us to quantify the effect of errors in click-through rate estimation on the achieved revenue. We show that we lose at most Δ/1+Δ fraction of the revenue if Δ is the relative error in click-through rate estimation. We also show that in the long run, an expected overdraft level of Ω(log(1/ε)) is unavoidable (a universal lower bound) under any stat- onary ad assignment algorithm which achieves a long-term average revenue within O(ε) of the offline optimum.
Date of Publication: Nov. 2012