© 2020

A Course on Tug-of-War Games with Random Noise

Introduction and Basic Constructions


  • Provides a rigorous, self-contained introduction to the emerging field of tug-of-war games

  • Includes background and auxiliary material on probability and partial differential equations

  • Includes exercises with solutions


Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-ix
  2. Marta Lewicka
    Pages 1-7
  3. Back Matter
    Pages 163-254

About this book


This graduate textbook provides a detailed introduction to the probabilistic interpretation of nonlinear potential theory, relying on the recently introduced notion of tug-of-war games with noise.

The book explores both basic and more advanced constructions, carefully explaining the parallel between linear and nonlinear cases. The presentation is self-contained with many exercises, making the book suitable as a textbook for a graduate course, as well as for self-study. Extensive background and auxiliary material allow the tailoring of courses to individual student levels.


Tug-of-War games nonlinear potential theory game theory p-Laplacian discrete stochastic processes games with noise mean value properties regularity theory

Authors and affiliations

  1. 1.Department of MathematicsUniversity of PittsburghPittsburghUSA

About the authors

Marta Lewicka is a Polish mathematician at the University of Pittsburgh, specializing in Mathematical Analysis. Marta obtained her PhD in 2000 from Scuola Internazionale Superiore di Studi Avanzati with a thesis on Hyperbolic Systems of Conservation Laws. She has subsequently worked on Calculus of Variations, Partial Differential Equations, Continuum Mechanics, and Tug-of-War Games in relation to Nonlinear Potential Theory.

Bibliographic information

  • Book Title A Course on Tug-of-War Games with Random Noise
  • Book Subtitle Introduction and Basic Constructions
  • Authors Marta Lewicka
  • Series Title Universitext
  • Series Abbreviated Title Universitext
  • DOI
  • Copyright Information The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-030-46208-6
  • eBook ISBN 978-3-030-46209-3
  • Series ISSN 0172-5939
  • Series E-ISSN 2191-6675
  • Edition Number 1
  • Number of Pages IX, 254
  • Number of Illustrations 2 b/w illustrations, 24 illustrations in colour
  • Topics Potential Theory
    Probability Theory and Stochastic Processes
  • Buy this book on publisher's site


“The book is best suited for advanced graduate students and specialists in nonlinear potential theory. The author provides background appendices for probability, Brownian motion and PDEs.” (Bill Satzer, MAA Reviews, April 4, 2021)