Optimal Real-Time Bidding Strategies

The ad-trading desks of media-buying agencies are increasingly relying on\ncomplex algorithms for purchasing advertising inventory. In particular,\nReal-Time Bidding (RTB) algorithms respond to many auctions -- usually Vickrey\nauctions -- throughout the day for buying ad-inventory with the aim of\nmaximizing one or several key performance indicators (KPI). The optimization\nproblems faced by companies building bidding strategies are new and interesting\nfor the community of applied mathematicians. In this article, we introduce a\nstochastic optimal control model that addresses the question of the optimal\nbidding strategy in various realistic contexts: the maximization of the\ninventory bought with a given amount of cash in the framework of audience\nstrategies, the maximization of the number of conversions/acquisitions with a\ngiven amount of cash, etc. In our model, the sequence of auctions is modeled by\na Poisson process and the \\textit{price to beat} for each auction is modeled by\na random variable following almost any probability distribution. We show that\nthe optimal bids are characterized by a Hamilton-Jacobi-Bellman equation, and\nthat almost-closed form solutions can be found by using a fluid limit.\nNumerical examples are also carried out.\n