Welcome to week three of Games without Dice or Cards: Combinatorial Game Theory.
I'm Tom Morley. Today we want to talk about ordering
games. And these are our definitions.
They're like, games are like numbers, sort of, or at least sometimes, and ups-
numbers you can order. 1 is less than 2 is less than 3.
2 is equal to 2, minus 1 is less than 6. And here's what this means for games.
To say a game is negative means that right always wins, going first or going second.
To say a game is zero means the first player loses, whoever moves first loses,
and the second player wins all the time. To say a game is positive means left wins
always, going first or going second. And here's a new one for you that doesn't
come up in numbers. A game is fuzzy with zero, that's how
that's pronounced. If fuzzy, F U Z, Z Y.
It's fuzzy with zero if the first player wins.
So let's look at what these mean. And how they're used.
First of all, what, the way they're typically used is not comparing again with
0. But comparing one gain to another.
So, G is less than H.. Actually you just convert and, you know,
subtract G from both sides, and it's, this should be the same thing as saying that 0
is less than H minus G. That's what it is, the definition is
mathematically. What it means in terms of game play is
that H is better for left, even if it's part of a bigger gain.
And we'll have some examples as we, as the week progresses.
So, H is better for left than G, even if, they're both part of a much larger game.
If you're playing five games at once, and one of the pieces is G, replace it by H,
and the whole thing is better for left. Now you could also of course read this
backwards. This says G is better for right than H is.
Because to say that once something is less than something is the same thing as saying
that is bigger than the first thing. The interesting case perhaps is, is equal
to zero G is equal to H, and this is our definition, means H minus G is zero.
And what this turns out to mean is that G and H have the same outcome, the same
player wins, in best play, even if G and H are both part of some bigger game.
So you have a go game and towards the end you have this little piece in the corner
that's equal to G. Well replace it by that little piece in
the one corner by H and nothing is changed in terms of the outcome.
Now in real numbers you can say not only that one number is less than another
number. But you can also say one is less or equal
to two, or one is less or equal to one. And so you can combine these inequalities
and for instance the following way. G is less than or equal to H means just
like it would for real numbers, G is less than H or G is equal to H.
So let's see what this says. It says, to say that G is less than H
means that H is better for left than G is, and to say that G is equal to H means that
they're the same for left and right. So you put these together and less than or
equal to means H is at least as good, perhaps not better, but at least as good
for left as G. In other words, it says left with H minus
G going second. Okay, so that's a lot of stuff.
A bunch of definitions here. If you forget the definitions go back,
look them over. They're all right at the beginning of this
module. And we'll continue on in the next module
with some definitions with some examples.