#### Overview of this book

Python, one of the world's most popular programming languages, has a number of powerful packages to help you tackle complex mathematical problems in a simple and efficient way. These core capabilities help programmers pave the way for building exciting applications in various domains, such as machine learning and data science, using knowledge in the computational mathematics domain. The book teaches you how to solve problems faced in a wide variety of mathematical fields, including calculus, probability, statistics and data science, graph theory, optimization, and geometry. You'll start by developing core skills and learning about packages covered in Python’s scientific stack, including NumPy, SciPy, and Matplotlib. As you advance, you'll get to grips with more advanced topics of calculus, probability, and networks (graph theory). After you gain a solid understanding of these topics, you'll discover Python's applications in data science and statistics, forecasting, geometry, and optimization. The final chapters will take you through a collection of miscellaneous problems, including working with specific data formats and accelerating code. By the end of this book, you'll have an arsenal of practical coding solutions that can be used and modified to solve a wide range of practical problems in computational mathematics and data science.
Preface
Basic Packages, Functions, and Concepts
Free Chapter
Mathematical Plotting with Matplotlib
Working with Randomness and Probability
Geometric Problems
Finding Optimal Solutions
Miscellaneous Topics
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# Computing Nash equilibria

A Nash equilibrium is a two-player strategic game – similar to the one we saw in the Analyzing simple two-player games recipe – that represents a "steady state" in which every player sees the "best possible" outcome. However, this doesn't mean that the outcome linked to a Nash equilibrium is the best overall. Nash equilibria are more subtle than this. An informal definition of a Nash equilibrium is as follows: an action profile in which no individual player can improve their outcome, assuming that all other players adhere to the profile.

We will explore the notion of a Nash equilibrium with the classic game of rock-paper-scissors. The rules are as follows. Each player can choose one of the options: rock, paper, or scissors. Rock beats scissors, but loses to paper; paper beats rock, but loses to scissors; scissors beats paper, but loses to rock. Any game in which both players make the same choice is a draw...