big-listgame theorytextbook-recommendation
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!
Jürgen Elstrodt - Maß- und Integrationstheorie (only in German)
Fremlin - Measure Theory (freely available in the web space, contains pretty much every significant aspect of measure theory in appropriate depth)
Set is a card game that is very mathematical.
Set is played with a deck with $81$ cards. Each card corresponds to a point in affine $4$-space over $\mathbb Z/3$, with $3$ possible colors, shadings, shapes, and counts. The players must identify Sets, sets of $3$ cards corresponding to collinear points. Sets are also triples of cards which add up to the $0$-vector. The three cards pictured form a Set.
A natural question which arises during play is how many cards you can deal out without producing a Set. There can be $9$ cards in a codimension-$1$ subspace which do not contain a Set, corresponding to a nondegenerate conic in affine $3$-space such as $z=x^2+y^2$. There can be at most $20$ cards not containing a Set, corresponding to a nondegenerate conic in the projective $3$-space containing $10$ points.
Lest anyone think it is "just a game", this has led to quite a bit of research, including applications to computational complexity of matrix multiplication.
Best Answer
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.