>> Welcome back.
We have, what do we have?
We have, we have a straight flush.
Small, small values.
But nonetheless, straight flush.
Not quite in order, but you can check it.
We're doing toads and frogs now.
You can, you can do this at home.
With people and it's fun play with people
but you can also do it with, with coins.
Here's toads and frogs.
Let's use nickels over here and Jefferson
nickels.
Let's use Lincoln pennies over here.
Here's toads and frogs.
The toads in this case are the nickels,
they move right and are controlled by
left.
The frogs are the Lincoln cents, they move
left and are controlled by right.
The toads can move one space to the right
if it's empty.
The frogs can move one space to the left
it it's empty.
But they also can jump over each other.
A toad can jump over a frog, like this.
Note that the, the coin when you jump over
isn't removed.
'Kay?
But you can jump over and then they, then
say the Lincoln cent can move over here
the, the nickel can do here.
The Lincoln cent can move over here My
concern is the frog.
And we see at this point that neither the
toads nor the frogs have a move so this
Lincoln cent, which was a frog, was right.
Right made the last move, it's now left's
turn, left doesn't have any moves, left
loses, Okay.
That's toads and frogs.
And there's lots of examples involving up,
down and integers and all that good stuff
involving with the small versions of toads
and frogs.
So, so you can try this at home.
You can either do this with, with, with
coins on a piece of paper or get your
friends and, that way you can jump over
one another and you, you draw these on the
floor, okay?
All right.
So but be careful.
Don't hurt yourself and don't try that at
home.
Probably do it with coins.
All right.
So lets look at its, at a, toads and frogs
position.
Now here we have toad, toad, they move
right, they're controlled by left.
Blank means an empty space.
Frog, frog, frogs are controlled by right.
They move left.
So, the only possible opening move for
left is to move to toad, is to move this
toad one to the right and that leaves toad
blank toad frog, frog.
And the only possible opening move for
right is to move this frog one left and
that's leaves the position, toad, toad,
frog, blank, frog.
Now, let's go over here and look at this
position.
If left moves in this position, then this
toad moves to the right.
And we're left with toad, toad, frog,
frog.
That's a zero position.
Okay?
Up and if right moves then right can jump
out take this frog.
The only possible case is for right to
move this frog jump over the toad and left
with toad, frog, toad blank frog.
Now let me remind you that when we're
analyzing games we have to consider like
two left moves in a row and right moves in
a row, because we want to analyze this
game in the context of a much larger game.
So we might have this toads and frogs over
here.
We might have a nim game someplace else,
cut cake someplace else, a game of chess
going on, a game of Go going on.
And then each player in turn chooses one
of these, one of these games and makes a
move in it.
This is how we add gain.
So and in doing so in adding this game to,
to others, left might make two, two moves
in a row in this, okay?
So, I'm going to leave out some, some
pretty long, I think, computation, but
this game down here.
Toad, frog, toad, blank, frog is actually
equal to star so we have this game up
here.
Toad, blank, toad, frog, frog.
Its left option, only left option is to
move to zero only left move is to move to
zero.
Only right move is to move to star and
therefore this game toad, blank, toad,
frog, frog is up.
Up, you remember, was left 0, right star.
Now, if you look at this game over here,
it's the same as the game over here, with
toads and frogs interchanged and left and
right interchanged.
Just, Just take the mirror image and
change all the toads to frogs and all the
togs, frogs to toads.
Therefore this gain on the right is the
negative of the gain over here.
So this gain on the right is minus up
which is sometimes written down.
So this whole gain is equal to up down.
And that's what I want to look at in this
module.
Okay.
So, there are no dominated moves.
Because there's only 1 move possible for
each player.
But down here.
If up is 0, star, then down is its
negative which is star, 0.
Remember that the negative of star is star
and the negative of 0 is 0.
And, so if right moves to down.
Left can move,[SOUND].
Left can move to star.
And my claim is, star is better.
For left than the original game.
That is to say, g is less than or equal to
star.
Which is the same thing as adding star to
both sides.
G plus star is less than or equal to 0.
Which says, that right wins this game
going second.
Okay, who will bit the[UNKNOWN] there?
So, but what this means is that if right
moved down, left moves immediately to up.
I'm sorry.
Right moves to down, left moves
immediately to star.
And now since star is 0, 0, the right
possibilities for to, to move From star
are only 0, so this reverses all the way
down to just 0 which means that if right
plays down left and mutely[UNKNOWN] plays
star and then rights only option is, is
down[SOUND].
So this says that g is actually equal to
up zero.
And this game simplifies a little bit.
But that's part of the exercises,
homework, quiz.
Whatever you want to call it, for this
week.
Alright.
Now, let me make some general comments
about this.
Simplifying games.
We, we came up with 2 ways of simplifying
games.
Dominated moves.
We can delete dominated moves.
If a move is better for left than another
move, then you can, you never have to do
the poor move.
If, if a move is better for right.
You never have to use the move that it's
better than.
And reversible move and that's a little
complicated but we have some examples and
that's probably technically the most
complicated thing in the whole course.
Now, the question is, is this enough?
That is are there other possible ways of
simplifying games?
Of course there are.
But what one can show is that this is
enough in the sense that if you keep doing
dominated moves and reversible moves and
keep doing them over and over again,
you'll reach a point where no moves are
dominated, no moves are reversible and
what's left is called the canonical form,
and if, that's unique and there's so, and
so in some sense some technical sense, in
fact that one can make very precise, the
simplest possible version of that game.
So all that you really need is this.
Now in terms, and that's how some computer
programs, and you can look up which ones
that are available open source up on the
web actually simplified games And, they
keep doing dominated moves and reversible
moves over and over again until there
aren't any.
And then when its left, they say, this is
as simple as possible, and that's what it
is.
Okay.
Let me just leave you with this which will
be posted in, in the assignments, et
cetera.
And the directions this week are simplify.
So, write down the simplest form of, of
each of these games.
Number 1 is numbers over here and numbers
over here.
2 and 3 are turn out to be numbers, I
think.
But, you try it yourself.
Number 4 was what we ended up with in the
exact last answer.
We took up down and simplified to this.
And now we want to simplify this further.
The last one is the toads and frogs
position.
Remember, frogs move right.
Toads move left.
Take this and see if you can come up with
the simplest possible description of that
and we'll see you all in a week or two.
Take care.
Next week I think is NIM.
We finally will, will analyze NIM in
general and answer the question, with the
arbitrary mid game how do you play and
when?
Sp, we'll see that all next time.
Take care.