Real-time Bidding for Time Constrained Impression Contracts in First and Second Price Auctions -- Theory and Algorithms

We study the optimal behavior of a bidder in a real-time auction subject to the requirement that a specified collections of heterogeneous items be acquired within given time constraints. The problem facing this bidder is cast as a continuous time optimization problem which we show can, under certain weak assumptions, be reduced to a convex optimization problem. Focusing on the standard first and second price auction mechanisms, we first show, using convex duality, that the optimal (infinite dimensional) bidding policy can be represented by a single finite vector of so-called "pseudo-bids". Using this result we are able to show that, in contrast to the first price auction, the optimal solution in the second price case turns out to be a very simple piecewise constant function of time. Moreover, despite the fact that the optimal solution for the first price auction is genuinely dynamic, we show that there remains a close connection between the two cases and that, empirically, there is almost no difference between optimal behavior in either setting. Finally, we detail methods for implementing our bidding policies in practice with further numerical simulations illustrating the performance.