Talk:Combinatorial game theory
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Having math on the same line as text doesn't render properly. I suggest that for simple expressions, math mode shouldn't be used and more complex ones should be on a separate line. -- Arvindn 12:31, 29 Feb 2004 (UTC)
Agree. But actually the problem with the page goes deeper - that's a severe case of dry textbook style, there.
Charles Matthews 14:09, 29 Feb 2004 (UTC)
Agree. It's a general problem with math articles (and this is a math topic). I'm not sure how one goes about solving it. You want the formalism to be there, for the people who are looking for it, and you also want a general discussion for people who are not mathematically literate -- a journalist, for example, should be able to get background here in case some mathematician makes news in CGT. How to arrange and intermix the two levels of presentation is a tricky conundrum.
There's another problem here, though, and that is that the article is very incomplete. The authors never get around to introducing the classic Berlekamp-Conway-Guy notation, and instead present the beginnings of the theory in their own notation, with the result that the beautiful heart of the theory, the mapping from games onto the real numbers, is completely missing.
I'd like to have more of a chat about this, to toss around ideas on how this material should be presented.
ACW 18:06, 11 Nov 2004 (UTC)
Well, the CGT people have a long tradition of not being able to explain themselves properly (IMHO). This page is too mathsy by half. I probably tend towards Berlekamp's attitude, and am less interested in ONAG. I have had weary discussions with Bill Spight, elsewhere, about the foundations. Charles Matthews 19:39, 11 Nov 2004 (UTC)
I think the ideal is a kind of two track description, so that the layman can read a fairly coherent explanation in plain English, while the mathematician can find the rigorous development as well. Maybe the two are interleaved, with an explanation up front that tells the lay reader than the stuff in small type (just brainstorming here) can be safely skipped. I can imagine a few ways of doing it.
Another thing that's troubling me is that the whole cluster of mathematical games articles could be better organized and more consistent ... more like they were articles from the same encyclopedia, if you get my meaning. I know the essence of True Wiki is little incremental changes without big Grand Plans, but I'm tempted to do a little Grand Planning here. You know, list the articles in the cluster, figure out what's missing, what should be moved, what are the expected reading paths, and so on. I'm too ambitious, right? ACW 02:15, 13 Nov 2004 (UTC)
Lists are good. A list of 'combinatorial games' topics would be a useful start, and a way to discriminate between CGT proper and its environs. Charles Matthews 10:06, 13 Nov 2004 (UTC)
I've just made some changes: (1) Most importantly, I've found and corrected what I think was an error (the author originally stated that C_fin is the smallest collection of games such that for every pair of H,K of elements there's a game G with L(G) = {H}, R(G) = {K}).
(2) I've explained what it means to 'play' these strange games.
(3) Using (2), I've added 'intuitive descriptions' of addition and negation of games.
(4) The author hadn't explained how to construct the set "P", so I worked out such a construction myself.
(Note that my only background in this theory came from actually reading the original page!)
I've modified your text a bit to speak of well-founded games, rather than finite non-loopy games. The distinction is a bit fine. For instance, consider the game G for which L(G)=R(G)=N, where N is the set of all nim-heaps. G is not finite, but it is well-founded. Conway calls these games "Enders".--Dan Hoey(because the signature feature seems to be malfunctioning)68.239.81.167 01:57, 24 October 2005 (UTC)
I see that what happened is I got logged out. Anyway, I've modified the definition of C_fin so that it covers all the well-founded collections of games. --Dan Hoey 02:18, 24 October 2005 (UTC)
I've fixed some problems with the definitions of well-founded games and C_fin. But what really needs to be done is to make this page more consistent with the Surreal numbers page. I've made a start by introducing the G = { L(G) | R(G) } notation. But the part about taking the quotient needs to be continued to talk about which games are numbers. Also something about the quotient being the greatest refinement that preserves outcomes of disjunctive sums. --Dan Hoey 15:22, 24 October 2005 (UTC)
This page lack of definiton of Combinatorial game theory (CGT). It is stated that is part of game theory, but not what is. In the formal definion it is not clear what a game is. Apart from mathemaical abstract aspect is not clear at all the correlation to real (normal) game AnyFile 13:31, 24 Jan 2005 (UTC)
Added a paragraph to the introduction to illustrate the purpose of CGT, and added a couple of good reference books.
The history section used the term disjoint sum which I corrected to disjunctive sum (ONAG, p. 74). This should be described more completely, since it is the fundamental insight that makes CGT an advance over the diffuse theory that preceded it.
I must say that I am quite uncertain about the History section. The mathematics of CGT were presented in ONAG six years earlier than WW, and the latter extended the theory in fairly minor ways. I don't imagine that Conway cares to argue priority; in fact, he acknowledges Berlekamp and Guy as his sources for much of the material in ONAG. But saying that WW "pioneered" CGT and was preceded by ONAG with some of its content seems an odd sort of horsecart. --Dan Hoey 23:34, 23 October 2005 (UTC)
- I have discussed this with Elwyn Berlekamp, and I know how Conway works (he used to be a colleague). The lags in writing-up are true enough to the historical record. Charles Matthews 15:39, 24 October 2005 (UTC)
[edit] Simple English
Please take a look at the Simple English version of this page. Despite some errors, it is much better than this one. Rcaetano 15:23, 28 May 2005 (UTC)
- I highly agree. Now that I've seen a "simple english" version of what this is, I've finally been able to figure out what the complex math notation is saying. I used the simple english version as a guide to add a "Simpler Definitions" section under formal definitions, which I hope will help people out. Needs some explanatory images of the tic-tac-toe boards I reference though... I'll see if I have the time to whip some up. Fieari 10:21, July 24, 2005 (UTC)
There are a number of ways in which this article could be improved. The article could be organized and the topics introduced in a way similar to the article on surreal numbers. Some game other than tic-tac-toe ought to be used for an example, since tic-tac-toe is not played according to the normal rule, and does not break into independent components, like most games studied in Combinatorial Game Theory. Some other game, like nim or domineering might be a better example.
The example ought to introduce some of the common operations and definitions in Combinatorial Game Theory, such as the sum operator, inverse operator, bracket notation, and comparisons, and equivalent games. Sums and inverses could be demonstrated much better with some game other than tic tac toe. Contrary to what the current article seems to imply, these concepts can be given intuitive definitions that the layman can understand.
Additionally, the formal definitions below don't seem to correspond to any standard approach to combinatorial game theory, and the notation seems very obscure. The surreal numbers article has better definitions than this one, and it would make sense to copy these definitions into this article.
This article also has a lot of room for expansion. Information on further Combinatorial Game Theory topics could be introduced, such as canonical normal forms, numbers, nimbers, stopping values, thermography, infinitesimal games, all-small games, ordinal sums, norton products, and atomic weights (as well as a lot of other stuff that I don't know about). Most of this stuff is in On Numbers and Games, and there is probably a lot more stuff in Winning Ways. —The preceding unsigned comment was added by 152.157.79.253 (talk • contribs) .
- Would you be able to suggest a textbook to reference? I haven't taken a class in this, I'm simply interested on a personal, almost hobby-like level, and to be honest, I can barely understand the current "Formal Definitions" at all. I simply took the "Simple English" version of this page and added it in... which was the only way I really understood what was going on. Fieari 18:15, 17 May 2006 (UTC)
[edit] Choice of Example Game
Using Tic-Tac-Toe as an example seems like a poor choice. This is not a typical sort of game that is analyzed in CGT, and it does not fit into the framework of the theory very well. For example, tic-tac-toe games can end in draws, although nonloopy games in CGT always end in one player winning. Also, it does not make much sense to add tic-tac-toe positions together, so the operation of addition cannot easily be demonstrated.
A better choice would be a game such as hackenbush, domineering, or nim, which are actually played under the Normal Rule. Positions in these games could be used to demonstrate some of the basic operations in CGT, such as addition and negation, in terms that the layman could understand.
[edit] Mathematical Presentation
The formal mathematical description in this article is vary confusing, even to someone who knows a lot about CGT, since it is completely unorthodox. The article on Surreal Numbers has a much better approach, that could be the model for this article, as the construction of Surreal Numbers is almost the same as the construction of the class of games. In general, this article should probably adopt the same notation and conventions as On Numbers and Games, or Winning Ways. Someone who has read these books should rewrite most of this article in a more orthodox notation.
