## The 2020 Joseph L. Doob Prize will be awarded to the two-volume set Probabilistic Theory of Mean Field Games with Applications, I and II, written by René Carmona and François Delarue, and published in 2018 by Springer-Verlag in its series Stochastic Analysis and Applications.

Stochastic game theory is a vastly complicated subject that attempts to model how rational individuals interact within a random environment, while each pursues her/his own, usually conflicting, goals. Understanding even simple-looking model problems requires designing strategies and counterstrategies of often great complexity. But "mean field games" offer a resolution, looking for simpler strategic structures that sometimes appear in the limit as the number of identically motivated players goes to infinity. The analogy with statistical physics should be clear. The motivating insight is that each individual need not respond much to the precise decision of any other player, but rather to the empirical distribution of the cumulative effects of all the others' actions.

These two magisterial volumes by René Carmona and François Delarue provide an accessible, fully detailed introduction to the exciting prospects for mean field games and to their considerable mathematical subtleties.

Response of René Carmona and François Delarue: We are deeply honored and grateful to receive the 2020 Joseph L. Doob Prize, and especially humbled by Doob's towering figure in the history of probability and its foundations.

We began working together nearly a decade ago, driven by the common desire to develop stochastic analysis tools that could be used to solve applied problems in mathematical physics. A couple of published journal articles and two years of travel between Nice and Princeton later, we became excited by the potential of an otherwise nascent mathematical model for large anonymous games with mean field interactions. It quickly became clear to us that there was something really special and potentially far-reaching in these models that were introduced independently and simultaneously by a couple of French applied mathematicians (Jean-Michel Lasry and Pierre-Louis Lions) and a group of electrical engineers in Canada (Peter Caines, Minyi Huang, and Roland Malhamé). We are thankful for their groundbreaking innovation.

When we decided to write a book on the subject, we did not have a clue just how extensive this project would be, what the final product would look like, and of the toll it would take on our lives. We also did not expect this project to prompt such profound honor and satisfaction, epitomized by this award, which has been granted to so many prestigious and highly influential mathematicians before us.

Of particular note to us has been the wide range of scientific subfields beyond our immediate community that have already cited this book, and the extent to which new generations of researchers are both relying on our probabilistic approach and extending the scope of its applications.

Witnessing this, and the many opportunities, conversations, and collaborations that this book has made possible, has been deeply heartening.

We want to thank the committee for recognizing all of this, and above all else, for seeing this book’s potential to “provide an accessible, fully detailed introduction to the exciting prospects for mean field games and to their considerable mathematical subtleties" and to "contribute immensely to our understanding of mean field games."

**Biographical Sketch of René Carmona**: René Carmona has been a member of the faculty at Princeton University since 1995. He is a Fellow of the Institute of Mathematical Statistics and of the Society for Industrial and Applied Mathematics. Dr. Carmona is the founding chair of the SIAM Activity Group on Financial Mathematics and Engineering, a founding editor of the journal Electronic Communications in Probability, and the SIAM Journal on Financial Mathematics. Over the last decade he has tried to understand from a probabilistic point of view a class of mathematical models known as mean field games. Together with François Delarue, he developed a probabilistic approach to these models. This collaboration culminated in a two-volume book providing the state of the art on the subject.

**Biographical Sketch of François Delarue**: Since 2009, François Delarue has been a full professor at the research laboratory of mathematics, "Laboratoire Dieudonné," of the University of Nice Sophia Antipolis. In 2014, he was appointed a Junior Fellow of the Institut Universitaire de France and this year he became co-editor in chief, with Peter Friz, of the journal Annals of Applied Probability. His research is in stochastic analysis, including mean field particle systems and applications to partial differential equations. He met his co-author René Carmona in 2009 and invited him to spend one month in Nice in 2010 in order to initiate a collaboration. Since René’s visit, they have been working together on the probabilistic approach to mean field games.

About the Prize: The Joseph L. Doob Prize recognizes a single, relatively recent, outstanding research book that makes a seminal contribution to the research literature, reflects the highest standards of research exposition, and promises to have a deep and long-term impact in its area. The book must have been published within the six calendar years preceding the year in which it is nominated.

Books may be nominated by members of the Society, by members of the selection committee, by members of AMS editorial committees, or by publishers.

The 2020 prize will be awarded Thursday, January 16 during the Joint Prize Session at the 2020 Joint Mathematics Meetings in Denver.

Find out more about the prize and see previous recipients.

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