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>> Games without chance.
Okay.

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I just dealt myself a card hand by chance.
And now look at what I got.

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You believe that?
Okay.

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But, not a chance mix.
This is the module.

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Simplicity.
It's all about simplicity.

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Okay, let's look.
What do I mean by simplicity?

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Remember last time we looked at things
like, well let's do minus 3, 1.

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Now, it turns out that this is some
number, and the number has to be between 1

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and minus 3.
Whenever you have a game like this that's

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a number, all the options over here are
less than, strictly less than all the

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options over here.
It turns out that unless that happens,

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that's not a number.
There are games, lots of games that aren't

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numbers.
But if it is a number, it has to be

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somewhere in between.
But, you see we have lots of choices here,

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you know, minus 2 is in between, minus
1.12345 is in between.

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All kinds of numbers are in between, and
if you actually believe in real numbers,

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which some people do.
I happen to believe in real numbers

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sometimes.
There are infinitly many numbers in

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between, and so, we have to decide, which
one.

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And the answer is, the simplest one.
Simplest one.

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So, This is some number.
It's got to be, if this is some number,

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it's got to be, be between minus 3 and 1,
and the question is which one.

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And the answer is the simplest number.
Now, 0 is the simplest number of all.

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And then you have 1, 2, 3, and also their
negatives.

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And then slightly more complicated than
these are 1 half minus a half, numbers

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that are, 3 and a half would be a,
actually, more complicated than 1 half,

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because it's bigger.
And then is the quarters, and then the

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8ths, and then the 16ths, etc., etc.
Okay.

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So we'd look at the game 1, and 1 and 3/8.
And ask, what's the simplest number that's

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in between?
Well, it turns out there's a number where

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the 4th.
Namely 1 and a quarter that's in between

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these 2.
Even though this is.

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Involves 8ths.
We don't have to go to 8th or 16ths to fit

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something in between.
We can actually get something strictly in

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between.
That's got a, that's got a 4 in the

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denominator.
And having a 4 in the denominator is

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simpler than having an 8th in the
denominator.

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So, so this is, 1 and a quarter which is
in between 1 and 1 and 3 8th.

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So, when there's a choice Of, of which one
to fit in between is the simplest one.

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Now let me actually explain something.
One more thing.

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Take a look at say one.
And 3 and 7 8ths.

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Okay, there's lots of numbers that fit in
between here and the, The value here X

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that is, this is equal to a number X.
The value of X is not the average of these

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two.
It's the simplest that's fixed in between

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There's an actually a whole number that
fits in between and so it's two.

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Okay, you can try one.
You can try this at home.

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Let's try 3 8th, 3 quarters.
What's the simplest number that's bigger

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than 3 8ths and less than 3 4ths?
Try it at home.

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I think you'll find that, that number is
halfway in between 0 and 1.

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Just to give away the answer.
So, and to prove that this number is equal

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to 1 half, you would have to show this
number, this game, minus this game, minus

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1 half, is a 0 game, that is, a loss for
the first player.

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So there we have it.