The Second-Price Knapsack Problem: Near-Optimal Real Time Bidding in Internet Advertisement

Jonathan Amar, Nicholas Renegar, Haihao Lu

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. We study the online setting, where the random permutation assumption holds under 'stable' setting assumptions, and show general methods for adapting both primal and dual online knapsack algorithms to this setting, despite the prices no longer being known a priori. We give examples of these algorithms, and show that we can achieve 1-epsilon competitive ratios, where epsilon is very small in practice, and sublinear with respect to the number of ads. This stands in contrast to existing work on adaptive pacing, which offers less powerful guarantees, but does so under more general settings. Numerical results from the iPinYou dataset verify our results on a stable setting, and show that we can significantly outperform adaptive pacing for more selective bidders, recovering a bundle of ads with an average of 25.9% more value.

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