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I am studying game theory, but I feel like I do not understand mixed-strategy equilibrium all that much. Yes, I know how to find them in 2x2 cases, and in some 3xn (n <= 2) cases as well where the normal form has been artificially constructed to make finding the mixed strategies very easily (e.g., one of the actions can conveniently be ignored because it's dominated).

But if I were to write down an nxn matrix right now, n >= 3, I wouldn't be able to solve for it's mixed strategies.

So I am looking for a really good really rigorous really thorough reference to understand mixed strategy equilibria. Any recommendations?

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    $\begingroup$ For any questions of this kind, the go-to reference is always Fudenberg & Tirole's "Game Theory". $\endgroup$
    – Bayesian
    Commented Jan 19, 2017 at 9:34
  • $\begingroup$ I want to second Bayesian's comment. $\endgroup$
    – 123
    Commented Jan 31, 2017 at 15:45

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The book by Gibbons that Tobias mentions is titled "A Primer in Game Theory" in Europe. It is excellent, especially in its coverage of more advanced games of imperfect/incomplete information. But it is quite concise in its treatment of games of complete information like the ones you are studying. The book by Osborne might therefore be a better choice. Both of these books are aimed at the introductory graduate level and are therefore quite rigorous.*

However, I feel compelled to say that it sounds like you are taking a first course in game theory, so I would be inclined to suggest also the book "Strategies and Games: Theory and Practice" bu Dutta. It is more verbose, but I think also provides a more gentle introduction to topics like Nash equilibrium and mixed strategies. Game theory, like much of economics is often best learnt by first obtaining a firm grasp of the intuition, and later supplementing it with sufficient rigor.

(*) There is also "Game Thoery" by Fudenberg and Tirole, which is less of a textbook and more of a reference. But it is, perhaps, even more rigorous as a result.

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I like Osborne "An Introduction to Game Theory" and Gibbons "Game Theory for Applied Economists" as textbooks. In any case, once you understand the concept, finding mixed equilibria boils down to solving a bunch of linear equations, it's just algebra.

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If your main focus is on equilibria in two-player strategic-form (i.e., bimatrix) games, then I would recommend the following:

  • B. von Stengel (2007), Equilibrium computation for two-player games in strategic and extensive form. Chapter 3, Algorithmic Game Theory, eds. N. Nisan, T. Roughgarden, E. Tardos, and V. Vazirani, Cambridge Univ. Press, Cambridge, 53-78.

  • B. von Stengel (2002), Computing equilibria for two-person games. Chapter 45, Handbook of Game Theory, Vol. 3, eds. R. J. Aumann and S. Hart, North-Holland, Amsterdam, 1723-1759.

  • D. Avis, G. Rosenberg, R. Savani, and B. von Stengel (2010), Enumeration of Nash equilibria for two-player games. Economic Theory 42, 9-37. doi:10.1007/s00199-009-0449-x

All available as PDFs from http://www.maths.lse.ac.uk/personal/stengel/bvs-publ.html

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I found this book to be very useful, especially understanding mixed-strategy equilibrium:

Peters, H. (2015). Game theory: a multi-leveled approach (2. Aufl.). Berlin ; Heidelberg [u.a.]: Springer.

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