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I am looking for the best book that contains a mathematically rigorous introduction to game theory.

I am a group theorist who has taken a recent interest in game theory, but I'm not sure of the best place to learn about game theory from first principles. Any suggestions? Thanks!

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    $\begingroup$ Are you interested in combinatorial game theory, non-cooperative game theory (Nash equilibria and the like), or any other particular part of the field? It's big, and the various parts of it are connected more by application than by mathematical content. $\endgroup$
    – Daniel Litt
    Commented Aug 2, 2011 at 2:51
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    $\begingroup$ Nobody seems to know what "game theory" covers. The term should be banned altogether in favor of more precise terms like "combinatorial game theory" or "von Neumann-Morgenstern-Nash game theory". $\endgroup$
    – Gro-Tsen
    Commented Feb 18, 2017 at 23:39

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As I note in my comment to the OP, game theory is a big field with several essentially disconnected areas, and one can't really hope for a comprehensive introduction from a single text. I'll recommend two, but this still shouldn't be thought of as a complete introduction.

I learned what I know of non-cooperative game theory from "Game Theory" by Fudenberg and Tirole. The book is well-written if terse, and covers a wide range of topics with a great deal of rigor. I would caution you that the book is written more as a reference than a gentle introduction, but it is certainly self-contained and I was able to read the book with no previous knowledge of the theory. It is a bit dry, however.

As for combinatorial game theory, I'd recommend Berlekamp, Conway, and Guy's "Winning Ways for Your Mathematical Plays," depending on your temperament. The book's style is pretty tongue-in-cheek, and some of the mathematics is non-rigorous (though the details are easy to fill in). But it's an absolutely beautiful book.

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    $\begingroup$ Winning Ways is by Berlekamp, Conway and Guy, not just by Conway. On Numbers and Games is by Conway alone; it is mathematically rigorous, and probably a better choice if Winning Ways seems too informal. $\endgroup$
    – Timothy Chow
    Commented Aug 2, 2011 at 3:55
  • $\begingroup$ @Timothy: Thanks, I've corrected the authors. Though I haven't read "On Numbers and Games" I've heard it's good. $\endgroup$
    – Daniel Litt
    Commented Aug 2, 2011 at 3:59
  • $\begingroup$ "tongue-in-cheek"? Gerhard "Ask Me About System Design" Paseman, 2011.08.02 $\endgroup$
    – Gerhard Paseman
    Commented Aug 2, 2011 at 18:23
  • $\begingroup$ @Gerhard: Yup, thanks! Must have been a little sleepy. $\endgroup$
    – Daniel Litt
    Commented Aug 2, 2011 at 19:04
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    $\begingroup$ I would disagree with the assertion that Fudenberg and Tirole has "a great deal of rigor". $\endgroup$
    – Michael Greinecker
    Commented Nov 5, 2013 at 15:30
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An excellent, rigorous introduction is provided by A Course in Game Theory by Martin Osborne and Ariel Rubinstein. They are very careful in setting up all conceptual machinnery. The book also contains some cooperative game theory at the end. The book can be downloaded freely and legally here.

A book that is useful for someone having already someo basic knowledge about game theory and what it is useful for, is Foundations of Non-Cooperative Game Theory by Klaus Ritzberger. The book is very conceptual and contains a lot of material that is usually not available in textbooks, such as normal form information sets, index theory of Nash components, and the structure theorem of Kohlberg-Mertens. The book is however somewhat short on motivation and is structured along conceptual lines, not pedagogy.

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An Introductory Course on Mathematical Game Theory by González-Díaz, García-Jurado and Fiestras-Janeiro

Reviews can be found here: 1, 2, 3.

Taken from the second review:

The book is self-contained and written very rigorously, but on the other hand, it is also very friendly to the reader, containing a lot of explanations and interpretations of game theory notions, as well as very many examples describing and analyzing various economic and other models with an application to game theory results. This makes it very suitable for graduate and advanced undergraduate courses on game theory.

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Daniel Litt's suggestion of Fudenberg and Tirole is a good one. Another standard text that takes a mathematically rigorous approach is Game Theory: Analysis of Conflict by Roger Myerson.

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For a rigorous introduction to combinatorial game theory, there are two other good references. (1) AN INTRODUCTION TO CONWAY’S GAMES AND NUMBERS by DIERK SCHLEICHER AND MICHAEL STOLL www.mathe2.uni-bayreuth.de/stoll/papers/games12.pdf which is succinct and might be the best choice given your background. (2) The book `Lessons in Play' by Albert, Nowakowski, Wolfe, is an introduction for undergraduate math majors.

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Game Theory: Mathematical Models of Conflict (Mathematics and Its Applications) [Hardcover] A.J. Jones

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I would look for introductions to combinatorial game theory by Elwyn Berlekamp alone. Not that the mathematics is any different from other treatments, but it probably stands clearer of the "recreational" background. I'm assuming here that you want to get past the notation for partisan games, Nim as reduction and so on, to issues to do with temperature and/or what you can do with the game tree. (As a go player I do have a bias, since Elwyn is interested in go while John Conway basically isn't beyond stealing the disjunctive game concept; but that's not why I'm commenting in this fashion.)

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  • $\begingroup$ Do you really think that Berlekamp "stands clear of the recreational background"? Berlekamp has written on Dots and Boxes, and on Go, which are both recreational. I can't think of any book by Berlekamp on combinatorial game theory that is not recreationally oriented. $\endgroup$
    – Timothy Chow
    Commented Aug 2, 2011 at 15:04
  • $\begingroup$ He has written introductory papers that aren't books, though. $\endgroup$
    – Charles Matthews
    Commented Aug 2, 2011 at 16:48
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This is probably not modern enough for you, but I'd suggest von Neumann and Morgenstern's Theory of Games and Economic Behavior. I'd be hard pressed to find anyone more rigorous than von Neumann. Plus, the text is available online.

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    $\begingroup$ A great deal has happened in game theory after this book, so it is not a good place to start. $\endgroup$
    – Jyotirmoy Bhattacharya
    Commented Mar 26, 2014 at 2:43
  • $\begingroup$ Indeed, I'd probably go for a more modern approach. $\endgroup$
    – Jon Bannon
    Commented Mar 26, 2014 at 11:50
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This question is old and since then a great book has come out: Michael Maschler, Eilon Solan, and Shmuel Zamir. Game Theory. 2nd Edition, 2020, Cambridge University Press.

It is extremely user friendly, but never sacrifices rigor. Its treatment of games with incomplete information is hands down the best I have seen.

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Besides Game Theory for Social Sciences (von Neumann, Morgenstern, Nash) and Nim-like Games (Berlekamp, Conway, Guy) there are at least three more flavors of game theory.

Differential Games or Continuous Pursuit-Evasion Games about which I know nothing.

Infinitely Long Discrete Games or "Polish games" introduced by Mazur, Banach, Ulam, Steinhaus, Mycielski, with applications to descriptive set theory and general topology. There may be some good books on the subject but they are over my head as they involve advanced set theory.

Tic-Tac-Toe-like Games, i.e., the other kind of combinatorial/recreational game theory. Here I can recommend a book (making this an answer rather than a comment): Combinatorial Games: Tic-Tac-Toe Theory by József Beck.

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Game Theory by Guillermo Owen.

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