The Second-Price Knapsack Problem: Near-Optimal Real Time Bidding in Internet Advertisement
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In the past decade, Real Time Bidding (RTB) has become one of the most common purchase mechanisms of the online advertisement market. Under RTB a unique second-price auction between bidders is managed for every individual advertisement slot. We consider the bidder's problem of maximizing the value of the bundle of ads they purchase, subject to budget constraints, and assuming the value of each ad is known. We generalize the problem as a second-price knapsack problem with uncertain resource consumption: the bidder wins an auction when they bid the highest amount, but they pay an amount equal to the second-highest bid, unknown a priori. Surprisingly, because of the second-price mechanism, we prove that a linear form of bidding yields a near-optimal selection of ads. We extend to an implementable online one-shot learning algorithm. We prove that key points of the random permutation assumption hold in this setting, and that we can apply algorithms for the online-knapsack problem to this setting. Through this we recover a competitive ratio of 1-6ϵ, where ϵ is the training ratio and is small in general, yielding a strong theoretical result on an important variation of the standard online knapsack problem. Numerical results from the iPinYou dataset verify our results, recovering a bundle of ads with 99.5 optimal value.
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