## Research Papers on Real-Time Bidding

**Real-time bidding strategies with online learning, with J. Fernandez-Tapia and J.-M. Lasry.**

*Abstract: One critical issue in the control of Markov processes is that, in order to successfully apply dynamic programming tools, the knowledge of the statistical laws governing the system is required. When these laws are difficult to estimate beforehand using historical data, the estimation/calibration task requires to be performed at runtime (this is known as \textit{online learning}). Bayesian inference provides a useful way to address this problem by defining probability distributions for the model parameters and update them with the incoming information. It is particularly relevant in the case of most conjugate Bayesian priors as they preserve the Markovian properties of the model, thus making it possible to apply classical dynamic programming / stochastic control tools. In this study, we apply such a Bayesian approach for the control of a bidding algorithm participating in a high-frequency stream of (Vickrey) auctions, with the aim of maximizing an expected payoff depending on the state at the end of the period. This is of particular interest in real-time bidding (RTB) advertising.*

**On the pricing of programmatic ad-buying services, with J. Fernandez-Tapia and J.-M. Lasry.**

*Abstract: In this paper, we provide a mathematical framework for the rigorous pricing and risk management of performance-based programmatic ad-buying contracts. We mainly focus on the case of Real-Time Bidding (RTB) audience strategies, where ad inventory is purchased algorithmically through the participation to a huge number of Vickrey auctions. Our approach is based on stochastic optimal control techniques. It is a general approach in that it makes it possible to consider a broad range of practical situations. In addition to the pricing of ad-buying contracts, we obtain results on both the optimal bidding strategy and the risk associated with each contract, the latter being obtained thanks to Monte Carlo simulations. Besides the mathematical framework itself, our goal is to show that mathematical and numerical tools exist for giving a fair price to performance-based ad-buying contracts -- that are too rare in the industry, as of today -- and to assess and manage the associated risk.*

**Optimal Real-Time Bidding Strategies, with J. Fernandez-Tapia and J.-M. Lasry, Applied Mathematics Research Express 1-42**

*Abstrac*

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