[computer-go] From Surreal Numbers to Games

[Ray Tayek]

fyi:

http://scienceblogs.com/goodmath/2007/04/post_3.php


Today we're going to take our first baby-step into the land of surreal games.

A surreal number is a pair of sets {L|R} where 
every value in L is less than every value in R. 
If we follow the rules of surreal construction, 
so that the members of L sets are always strictly 
less than members of R sets, we end up with a 
totally ordered field (almost) - it gives us 
something essentially equivalent to a superset of 
the real numbers. (The reason for the almost is 
that technically, the surreals form a class not a 
set, and a field must be based on a set. But for 
our purposes, we can treat them as a field without much trouble.)

But what happens if we take away the restriction 
about the < relationship between the L and R 
sets? What we get is a set of things called 
games. A game is a pair of sets L and R, where 
each member of L and R is also a game. It should 
be obvious that every surreal number is also a 
game - but there are many more games than there 
are surreal numbers, and most games are not surreal numbers.

Games lose some of the nice properties of the 
surreal numbers. They are not a field. They are 
not totally ordered. In fact, they're not even 
all positive or negative. They're very strange things.

So why would we want to break the restriction on 
the surreals that gives us games? Naturally, 
because games have useful applications in 
modeling many things - in particular, games (in 
the non-mathematical sense - games like Go, Chess, Checkers, Poker, etc).

Let's take a bit more of a detailed look at games, and how they interact.

Game arithmetic is exactly the same as surreal 
arithmetic: addition, subtraction, 
multiplication, negation - even division (which 
we haven't looked at yet) are all defined in the 
same way of surreal numbers and games.

But: while surreal numbers are always either 
positive, negative, or zero, games can also be 
fuzzy. Remember, games are not fully ordered. 
That means that there are pairs of games (a,b) 
where ¬a b and ¬b a - that is, where the two 
games cannot meaningfully be compared. Fuzzy 
games are games that can't be compared to zero.

What does a fuzzy game look like? The simplest 
example is: {1|-1}. Try to use the definition of 
" " on that game with zero - it doesn't work.

Games also have some strange behaviors with 
respect to multiplication. If a, b, and c are 
games, then (as you would expect for numbers), if 
x×z=y×z then x=y. But, with games, x=y doesn't 
mean that x×z=y×z. Nasty, that, eh?

So what are these beasts useful for? Part of 
Conway's motivation was trying to analyze the 
game of Go (aka Wei-Chi). Go is one of the oldest 
strategic games in the world; it's been played 
for thousands of years in China, Japan, and 
Korea. Go is the Japanese name, which is 
generally used here in the US; Wei-Chi is the 
chinese name for it. It's a thoroughly fascinating game.

Go is a two-player game where the players have a 
17x17 grid. Each move, a player puts a piece of 
their own color on one of the intersections on 
the grid. The goal of the game is to surround 
territory using your pieces. Whoever has the most 
territory at the end wins. Mechanically, it's 
about as simple as a game can get. Strategically, 
it's unbelievably deep and complex. It's 
frequently compared to Chess in terms of depth 
and strategy. It's a wonderful game. ...


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