I am a geometer. but i have become interested in combinatoral game theory because of 2 things. 1)Go 2) proving mathematical theory's can be regarded as a 2 player game.

so i studied 100 pages of winning ways. and learnt about the {|} notation. at first i found it weird cause we always know who is going to make the first move. so why should i write it this way? then after learning several other weird things like {-2|+3} = 0 i kinda guessed that the whole theory is based on addition of games. we consider every players option cause we might choose a different game and then our opponent will have those options available to him or every game that won't change the result under addition is zero. so no surprise that the theory's examples usually come from games that can be broken to smaller games. even it's application to go is in the end game that the game is broken to pieces.

So the only way i know this notation helps is that if one can break a game to some sub games and understand those sub games well, then he can understand the whole game.

my question is: is there any other way this notation can help? for example is there an example that i break a game into sub games such that the product of those sub games is the whole game and this leads to understanding the whole game?


Essentially, no. For one thing, you can only take a product in a general way that works out at all if the game is cold (so that the game/summands act like numbers).

This is very contrived, but I suppose you could take a cold game that breaks up into sums, and use the definition of the exponential function in the Surreals to define the exponentiated game, which would break up as a product of the exponentials of the summands of the original. But if you're looking at some common game like Go, or even the contrived chess puzzles Noam Elkies applied CGT to, you're not going to have a shot at using "products" in any way.

  • $\begingroup$ thank you. are there anything else other than product that can help? if not then it seems one should be looking for a new notation if it is possible at all to find a notation to study go tsumego for example. $\endgroup$ – user76556 Jun 30 '13 at 14:27
  • $\begingroup$ Well there are some facts about areas of a Go board that are nearly independent (see if you can browse through a copy of Mathematical Go at a library and take a look at "gaskets", IIRC), but in general tsumego are going to be hard. In some sense, Conway wanted to exploit the near-independence of regions of a Go board towards the end of the game, and this allows one to do that. In a general tsumego problem without separate parts I doubt there's much to exploit. $\endgroup$ – Mark S. Jun 30 '13 at 17:57

This notation for combinatorial games generalizes surreal numbers, which in turn generalize real numbers to include infinitesimal and infinite numbers.

  • $\begingroup$ well yes i know that. but that is not exactly my question. i am talking about it's use in solving real games which are always finite. i am interested to know whether this notation will make solving games like chess and go easier or not? $\endgroup$ – user76556 Jun 13 '13 at 13:31
  • $\begingroup$ I believe there were two questions, and this answers the first of them "whether there is anything else rather than addition about this notation". With regards to the second one, chess and go are hard because they are large and complex, not because we don't have the right notation to understand them. $\endgroup$ – vadim123 Jun 13 '13 at 13:35
  • $\begingroup$ i see. i put the first question badly. i was interested in knowing how else it can help us study games rather than helping us adding games. for the second i think go endgame is difficult too. but the theory has been useful in it of course because the game is a sum of smaller games. $\endgroup$ – user76556 Jun 13 '13 at 13:39

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