CMPSCI 791JJ: Conway Game Theory

David Mix Barrington

Spring 2007

This is the home page for CMPSCI 791JJ. CMPSCI 791JJ is a one-credit graduate seminar in Conway's combinatorial game theory and the associated surreal number system.

Class meetings will be on Thursdays from 4:15 to 5:15 in room 243 of the Computer Science building. (The official start time is 4:00 but we will start late to allow underfed and/or undercaffeinated graduate students to take advantage of the Thursday department tea.)

The main goal of the course will be for the group to read and understand as much as possible of Conway's book On Numbers and Games, generally abbreviated "ONAG". (Hence, according to Conway, practitioners of the theory are known as onagers.) The book is available from the usual online sources for $39 plus shipping.

Two other relevant books are:

• Winning Ways by Berlekamp, Conway, and Guy, a treatment of the game theory with numerous examples. The book was originally in two volumes of which I own the first -- it has now been reissued in four volumes.
• Surreal Numbers by Donald Knuth, an explication of the number system in the form of a fictional narrative where two people on a desert island get the successive axioms of the theory as they wash ashore and derive the consequences.

The format will be very informal -- we will rotate the responsibility of leading the seminars with the assumption that the leader will prepare something to say about the agreed-upon section. I led the first class meeting and will lead the second. There will be no exams or written homework, unless I decide that we are slacking.

I will record our class activities and plans in the blog below.

Announcements (2 March):

• (2 Mar) Almost forgot -- Michael Sindelar will lead the next class on 8 March and probably the one after on 15 March. We can probably manage with eash person leading two classes.

• (2 Mar) We had our third class yesterday (1 March), and Robert Lychev led the discussion. He proved the Simplicity Theorem and went over some examples, particularly those from the end of Chapter 7. We discussed our first non-number game "*" or {0|0}, which is fuzzy because the first player always wins. We also talked about:
  • "Short" games (p. 97), where there is a bound on the number of possible positions and hence a bound on the number of possible moves. (There's a set theory fact called Konig's Lemma, which implies that any game where there are only finitely many options at each point is short. This is because KL says that any finitely-branching tree either has an infinite path or has only finitely many nodes. Infinite paths are ruled out by our definition of "game".) The "archimedian principle" (p. 98) says that any short game G, number or not, satisfies -n < G < n for some integer n. There's an easy game-theoretic proof of this -- let n be a bound on the length of G, then given the sum of n and G, Left makes all her moves in n, where Right cannot move, and Right eventually runs out of moves in G.)
  • The game "up" which is {0|*}. This game is positive because Left always wins -- if she moves first she moves to 0 and Right loses, if Right moves first he leaves Left in * which she wins. But up is less than any positive number (see the discussion on page 100).

• (21 Feb) I need to cancel tomorrow's class because of meetings with the AQAD visiting committee. Sorry for the late notice, I will also email the registered students.

• (21 Feb) In our second class on 15 February we mostly went over the motivation for the number and game definitions, and went through several more examples of finite games. One or more of Mike and Robert will lead the next class, which should include a proof of the Simplicity Theorem on page 23.

• (12 Feb) Our first class was on 8 February. We talked about the basic definition of Conway's class of games, and saw examples of games with each integer value. These were given both as abstract games using the {L|R} construction and as instances of the game of Dominoes. This involved material from Chapters 0 and 7 of ONAG. We defined positive, negative, zero, and fuzzy games. The exercise we left with was to show that if G is any game, then the sum of G and -G is a zero game.

  For the second class, we will go over the definitions of addition, comparison, and multiplication for the surreal numbers from Chapter 0, and try to convince ourselves that the surreal numbers form an ordered field.

Last modified 2 March 2007