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Welcome to week three of Games without
Dice or Cards: Combinatorial Game Theory.

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I'm Tom Morley.
Today we want to talk about ordering

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games.
And these are our definitions.

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They're like, games are like numbers, sort
of, or at least sometimes, and ups-

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numbers you can order.
1 is less than 2 is less than 3.

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2 is equal to 2, minus 1 is less than 6.
And here's what this means for games.

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To say a game is negative means that right
always wins, going first or going second.

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To say a game is zero means the first
player loses, whoever moves first loses,

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and the second player wins all the time.
To say a game is positive means left wins

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always, going first or going second.
And here's a new one for you that doesn't

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come up in numbers.
A game is fuzzy with zero, that's how

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that's pronounced.
If fuzzy, F U Z, Z Y.

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It's fuzzy with zero if the first player
wins.

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So let's look at what these mean.
And how they're used.

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First of all, what, the way they're
typically used is not comparing again with

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0.
But comparing one gain to another.

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So, G is less than H..
Actually you just convert and, you know,

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subtract G from both sides, and it's, this
should be the same thing as saying that 0

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is less than H minus G.
That's what it is, the definition is

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mathematically.
What it means in terms of game play is

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that H is better for left, even if it's
part of a bigger gain.

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And we'll have some examples as we, as the
week progresses.

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So, H is better for left than G, even if,
they're both part of a much larger game.

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If you're playing five games at once, and
one of the pieces is G, replace it by H,

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and the whole thing is better for left.
Now you could also of course read this

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backwards.
This says G is better for right than H is.

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Because to say that once something is less
than something is the same thing as saying

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that is bigger than the first thing.
The interesting case perhaps is, is equal

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to zero G is equal to H, and this is our
definition, means H minus G is zero.

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And what this turns out to mean is that G
and H have the same outcome, the same

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player wins, in best play, even if G and H
are both part of some bigger game.

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So you have a go game and towards the end
you have this little piece in the corner

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that's equal to G.
Well replace it by that little piece in

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the one corner by H and nothing is changed
in terms of the outcome.

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Now in real numbers you can say not only
that one number is less than another

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number.
But you can also say one is less or equal

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to two, or one is less or equal to one.
And so you can combine these inequalities

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and for instance the following way.
G is less than or equal to H means just

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like it would for real numbers, G is less
than H or G is equal to H.

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So let's see what this says.
It says, to say that G is less than H

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means that H is better for left than G is,
and to say that G is equal to H means that

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they're the same for left and right.
So you put these together and less than or

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equal to means H is at least as good,
perhaps not better, but at least as good

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for left as G.
In other words, it says left with H minus

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G going second.
Okay, so that's a lot of stuff.

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A bunch of definitions here.
If you forget the definitions go back,

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look them over.
They're all right at the beginning of this

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module.
And we'll continue on in the next module

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with some definitions with some examples.