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PROFESSOR: So let's get started.

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Today we're going to be
talking about preflop, which

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is the last thing
I definitely want

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to cover before the
[? Sakuna ?] tournament.

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As you know, this is going to
be one of my last lectures.

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I'm teaching one or to
over the next two weeks,

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but we're primarily
going to have

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guest speakers talking
about more the macro poker

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environment.

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So why are we doing preflop?

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So in tournaments,
most of your value

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is going to come from
what you do preflop--

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playing it close to optimally.

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And the reason is
because a lot of people

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don't do this good at all.

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Like they really,
really play preflop

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badly because it's
very counter intuitive.

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Especially live.

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Like people have a
much tougher time

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doing this because
either they're

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afraid of getting knocked
down on a bad hand,

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or their afraid of
showing down a bad hand,

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or live players are
just worse in general.

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So for whatever reason,
people screw this up online.

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A little bit on live a lot.

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In addition, one of
the reasons that we're

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spending an entire day on it is
it's relatively easier to solve

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from a mathematical standpoint.

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Like we get close to
a Nash equilibrium

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because there aren't
that many variables.

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Whereas postflop, there
are a million variables.

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It's more related to kind of
putting things into patterns

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that you might be close to.

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So that's why I'm doing this.

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And then let's start
with a scenario

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that we're going to be analyzing
for the rest of the class.

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There we go.

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So here's our scenario.

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So I'm-- how it works heads up
is the dealer button is a small

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blind.

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That's a slight
break in the rules

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because you assume I'm
button, he's small blind,

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I'm big blind, but they
change around a little bit.

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This way, the button isn't the
last one to act every round.

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So here, I'm first one to
act preflop and then last

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to act every round thereafter.

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If I were big blind, I'd
be less every single round.

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So it's a minor variation that
you guys should just know.

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So situation-- I'm
small blind for 125,

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he's big blind for
150, and there's

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a 25 ante, which is why
there's 50 in the pot.

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And the question is
what do we do here?

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OK, so we have 9-6
offsuit with 2 1/2 m.

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We're in the small
blind, and we're trying

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to figure out what do we do.

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So the answer here might
not be that intuitive,

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and I don't think I'd
give it to you right away.

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But how we figure it out is
just with a normal semi-bluffing

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equation.

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We take a look at-- OK,
so everything preflop

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is a semi-bluff because we
have some chance of winning,

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and we're virtually
never less than 30%.

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So we just use this
semi-bluffing formula

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that we've seen before
where our EV is just

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going to be the pot times
the chance he folds,

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plus the chance he doesn't fold
times our EV when he calls.

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We need to figure out some
sort of calling range.

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I just made up one
here that seemed

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like about at the pro level,
and this is pretty wide.

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Like him calling with like
Ace-2 or Jack-10 or 2-2.

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This is a wideish range.

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I think a lot of players
for the tournament life

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might not call this
wide, but it ends up

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being like 27.6% of
hands he's calling.

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So we're going to
use it as a baseline,

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and then later we'll show
why that doesn't matter.

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So our question is, what's
our equity versus range here?

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And what we can do there is
just plug into PokerTracker.

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So the idea is we
don't know what

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hand he has and we don't care.

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It's unrealistic to think
that we can get down

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to any sort of hand
he'll be calling with,

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but we can get down to a range.

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We can say he's
equally likely to have

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any assortment of the hands in
that range because if he calls,

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that means that he has
one of those hands.

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So we can compare our equity
versus any one of those hands

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to get our equity
versus range, and we

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can do that in PokerTracker.

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So what you're going to do
is you're going to open up

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the equity calculator, you're
going to put in 9-6 off,

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and then you're
going to put in this.

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I actually just
drew it in there,

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but that's the same
thing I showed before,

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which is pocket 2s are better.

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Ace-2, King-2 , Queen-- sorry.

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King-10, Queen-10,
Jack-10, and then the same

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for offsuit, which
is the representation

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of this entire thing.

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I just need to separate
offsuit or suited,

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and then I assumed by jack-10,
it meant that entire corner,

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although PokerTracker
likes specifying.

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Anyway, so this is
our equity versus.

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This is saying that
if we go in with 9-6

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and he calls with
anything in that range,

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we are 34% to win this hand.

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So I don't know if that's
about what you'd expect.

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It seems higher than I would
have thought initially,

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but that's our
equity verses range.

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If we go all in and
he calls, we can

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assume we're a 35-65
underdog without even

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knowing what he has.

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And then once we
find out what he has,

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we will find out whether we're
slightly better or worse,

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but that doesn't really matter.

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So we can calculate our EV
in this hand in the same way

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that we're used to
calculating semi-bluffing.

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The EV of the push
is going to be this.

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The chance of him
folding times 425.

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If he calls 27%
of the time, that

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means he folds whatever this
is like 73% of the time.

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So that's our value from him
folding-- the fold equity.

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And then 20% of the time
we're in a showdown situation

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where of that [? push ?]
time that he calls,

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we're 35 to win 1250 and
we're 65 to lose 950,

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resulting in a
total equity of 253.

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So that's a lot of
chips for what might

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seem like a very marginal move.

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By going all in
here, you're actually

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making that many chips.

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And in fact, a more
provocative way

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to describe it is if
you don't do this,

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you are losing 250 chips.

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You only need 1,000 to
win this whole tournament,

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and by folding what seems
like a very weak hand

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in this position, you are just
giving him 250 chips of value.

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So that's a lot, and that shows
how counterintuitive this is.

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That a situation where
you have really bad cards,

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you don't realize that
these are actually really,

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really good cards
in the situation.

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In fact, no matter
what, your cards

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are basically good
in this situation.

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Because this situation makes
any two cards good enough.

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So let's talk about-- so we
made one assumption, which

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is what his call range is.

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Did you have a question?

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Oh, OK.

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So we made some assumption about
what his call range is, and I

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said it didn't matter.

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Why?

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Because what we can do here is
make the call range a variable.

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So this fold percent
is a variable,

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and then our win percent
is related to the percent

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that he folds.

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Because if he calls a small
range of hands when we get

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called, he's going
to be crushing us.

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Whereas if he calls
like 90% of the time,

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we're probably
ahead of his range

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because he calls a
lot of worse hands.

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Anyway, so you can
take a graph and just

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look, OK, he calls somewhere
between 0% of the time

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in 100% of the time, and then we
can do this EV equation for all

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those hands by
calculating OK, so what's

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our equity versus calling range?

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This doesn't change.

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We still win or lose the
same amount every time.

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And say so if we were
always pushing here,

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if he calls with whatever
range, this is our EV.

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And then one thing
that's interesting

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about this graph is what?

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It's always
positive, which means

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that no matter what his calling
range is, this push is good.

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It might be more
intuitive this way.

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Like if he folds
100% of the time--

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if he only calls 0% of
the time, we trend up

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to a number around here.

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And what number's around here?

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That's just this.

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It's the pot before
we do anything.

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So if he folds 100% of the
time, we just win the pot.

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Like that's our EV every
hand, and it just trends

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based on this.

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Where the more he calls,
typically the better the worse

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our EVs going to be.

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Yes?

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00:08:12,910 --> 00:08:17,430

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[? Yes. ?] We'll get to that.

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And that's a good
question, although it's

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going to take us half
the class to answer that.

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Anyway, so that's
a cool thing here.

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So it's always
positive, which means

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I'm saying-- no matter what
his-- the villain does, always,

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always push all in with 9-6
offsuit in this position.

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So let's break it
down into components.

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So to explain
intuitively what drives

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this graph is your equity
there is split into two parts

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because it's a semi-bluff.

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It's your fold equity
and your showdown equity.

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So let's turn this into lines.

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So you see your fold equity
just decreases the more

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he calls in a linear fashion.

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So that should be
fairly intuitive.

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The proportion he calls
reduces your fold equity

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by that amount, and then
your showdown equity

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has a little curve
here where you

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get more value from the showdown
on average the more he calls.

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The only reason there's
a little bit of curvature

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is because it's multiplied
by the chances of him

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calling in the first plAce.

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So it's tilting
in some direction.

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But anyway, these are curved.

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Makes this a very interesting
optimization puzzle.

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And the reason we're positive
at the end is for this reason.

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We're basically-- so this
line, the total equity,

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is just the sum of these two.

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Where our showdown
value is going

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to be negative over
basically this entire range.

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Even when our fold
equity reaches zero,

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that's when our showdown
equity creeps just above zero.

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Regardless of what
his calling range is,

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00:10:01,260 --> 00:10:03,840
your average is like
175 chips of EV.

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00:10:03,840 --> 00:10:07,060
Like the 20% call--
or whatever we said.

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The 27% calling range
isn't necessarily optimal.

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He knows you have these cards.

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But on average, you're
getting-- whatever,

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like 100 and something chips.

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And just to show
how bad is folding,

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if you just open fold
this for some reason,

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that's as bad as
purposely calling it all

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in with 3-4 versus Ace-King.

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Like that's an equivalent
amount of EV loss

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as folding this hand.

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So now let's talk about how
hard this is going to be.

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So push/fold
decisions are really

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hard to kind of
intuitively figure out

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00:10:40,220 --> 00:10:44,050
just because of the curvature
and the steepness of those

247
00:10:44,050 --> 00:10:46,670
graphs are moving
in weird directions

248
00:10:46,670 --> 00:10:49,220
and they're both moving at
the same time with regard

249
00:10:49,220 --> 00:10:50,780
to all the other variables.

250
00:10:50,780 --> 00:10:53,720
So it makes it so that
it's very hard to come up

251
00:10:53,720 --> 00:10:55,950
with very quick rules.

252
00:10:55,950 --> 00:10:58,180
Some variables are going
to affect our decision,

253
00:10:58,180 --> 00:11:00,440
and the result is
either push or fold.

254
00:11:00,440 --> 00:11:02,350
Our result isn't
even as complicated

255
00:11:02,350 --> 00:11:04,670
as betting a certain
amount, but the variables

256
00:11:04,670 --> 00:11:08,540
you have to consider are our
cards, our position, our stack,

257
00:11:08,540 --> 00:11:11,100
and the villain's
call range, which

258
00:11:11,100 --> 00:11:13,570
results in this
five-dimensional ray that we

259
00:11:13,570 --> 00:11:18,500
have to slice and dice so that
we have our push/fold decision

260
00:11:18,500 --> 00:11:19,780
range.

261
00:11:19,780 --> 00:11:23,280
So what we want to do is
isolate for specific things.

262
00:11:23,280 --> 00:11:30,150
We want to say OK, let's take--
let's make this and this static

263
00:11:30,150 --> 00:11:31,850
so that we can
solve for these two,

264
00:11:31,850 --> 00:11:34,570
or let's make these two static
so we can solve for these two.

265
00:11:34,570 --> 00:11:36,740
So we can end up with a chart
that looks something like this.

266
00:11:36,740 --> 00:11:38,448
And this is a little
bit more manageable.

267
00:11:38,448 --> 00:11:41,570
We can just say this green
area is when you should push

268
00:11:41,570 --> 00:11:44,240
and this yellow area is
when you should fold.

269
00:11:44,240 --> 00:11:45,410
That's our goal here.

270
00:11:45,410 --> 00:11:47,880
To be able to develop
that sort of chart.

271
00:11:47,880 --> 00:11:51,770
And then figure out what's
the best thing to isolate.

272
00:11:51,770 --> 00:11:54,220
So let's get to it.

273
00:11:54,220 --> 00:11:55,739
So range is the
postset of hands.

274
00:11:55,739 --> 00:11:56,780
This is how you write it.

275
00:11:56,780 --> 00:11:59,262

276
00:11:59,262 --> 00:12:00,970
This is the assumption
that we're making.

277
00:12:00,970 --> 00:12:02,850
So we're doing it
for two reasons.

278
00:12:02,850 --> 00:12:04,680
Analyzing our opponent, which
we're not going to be doing.

279
00:12:04,680 --> 00:12:06,930
We're going to assume we
have no information about him

280
00:12:06,930 --> 00:12:10,430
because it's preflop, and a lot
of it-- one of our assumptions

281
00:12:10,430 --> 00:12:12,220
is that he hasn't
even acted yet, so we

282
00:12:12,220 --> 00:12:14,550
have no information at all.

283
00:12:14,550 --> 00:12:16,860
But we're going to be using
it to determine our plays.

284
00:12:16,860 --> 00:12:19,350
Basically we're saying we
have some sort of decision

285
00:12:19,350 --> 00:12:20,620
[INAUDIBLE] for our range.

286
00:12:20,620 --> 00:12:22,954
Like we certainly can't solve
this for every single hand

287
00:12:22,954 --> 00:12:25,411
or at least we can't remember
it, so what we're going to do

288
00:12:25,411 --> 00:12:27,630
is come up with a line
where we say every hand

289
00:12:27,630 --> 00:12:29,540
above that certain line, i.e.

290
00:12:29,540 --> 00:12:31,640
That range above
that line, is going

291
00:12:31,640 --> 00:12:34,790
to be good for our
specific decision.

292
00:12:34,790 --> 00:12:36,540
And we're bringing
these into percentiles,

293
00:12:36,540 --> 00:12:38,373
and that's how we're
going to describe them.

294
00:12:38,373 --> 00:12:40,620
And we're using
Sklansky-Carlson.

295
00:12:40,620 --> 00:12:44,880
His rankings for our
breakdown of percentiles.

296
00:12:44,880 --> 00:12:46,710
But here's his
original question,

297
00:12:46,710 --> 00:12:48,780
and this is just to explain
where these come from

298
00:12:48,780 --> 00:12:51,470
and why this is particularly
relevant to what we're doing.

299
00:12:51,470 --> 00:12:53,460
So his question is
you're the small blind

300
00:12:53,460 --> 00:12:55,520
and you have a choice
between either folding

301
00:12:55,520 --> 00:12:59,030
or open pushing a hand, or open
pushing here means you push

302
00:12:59,030 --> 00:13:01,540
and you show your hand
before he makes a call.

303
00:13:01,540 --> 00:13:05,210
And the question is how
many chips is this good for?

304
00:13:05,210 --> 00:13:07,820
If you have one chip, or
I guess we have two chips,

305
00:13:07,820 --> 00:13:11,320
you have since they're 1-2
blinds, if you have two chips

306
00:13:11,320 --> 00:13:14,950
and one is already in the small
blind, your pot odds are 25%.

307
00:13:14,950 --> 00:13:16,800
So even if he knows--
he's going to call

308
00:13:16,800 --> 00:13:18,940
if he's anything more
than a 50% favorite,

309
00:13:18,940 --> 00:13:23,350
so you have the odds to do that
with at least a pretty bad hand

310
00:13:23,350 --> 00:13:24,690
and a lot of good hands.

311
00:13:24,690 --> 00:13:27,570
So x here is how
big can your stack

312
00:13:27,570 --> 00:13:30,560
be before this starts
becoming unprofitable?

313
00:13:30,560 --> 00:13:34,590
And the answer for Ace-Ace
offsuit is about 70,

314
00:13:34,590 --> 00:13:36,502
and to explain how
we got to that,

315
00:13:36,502 --> 00:13:38,210
we're going to make
a couple assumptions.

316
00:13:38,210 --> 00:13:40,820
And you can see
the clear relation

317
00:13:40,820 --> 00:13:43,790
between this methodology
and what we're doing.

318
00:13:43,790 --> 00:13:46,620
So 1-2 blinds-- here a
small blind with whatever.

319
00:13:46,620 --> 00:13:47,834
Here are bets x all in.

320
00:13:47,834 --> 00:13:49,500
And we're going to
say big blind's going

321
00:13:49,500 --> 00:13:52,410
to call when EV's more than 50.

322
00:13:52,410 --> 00:13:55,082
Which he might do pot odds,
which we're just going

323
00:13:55,082 --> 00:13:56,290
to ignore now for simplicity.

324
00:13:56,290 --> 00:13:58,123
We're just going to say
big blind is calling

325
00:13:58,123 --> 00:14:00,230
when he's more than 50% to win.

326
00:14:00,230 --> 00:14:03,105
It's actually zero when
he has the pot odds, which

327
00:14:03,105 --> 00:14:05,547
is this equation, but we're
going to forget about that.

328
00:14:05,547 --> 00:14:07,130
Let's just say he's
going to only call

329
00:14:07,130 --> 00:14:08,880
with literal better hands.

330
00:14:08,880 --> 00:14:12,060
So here he's going
to call with Ace-8

331
00:14:12,060 --> 00:14:14,210
or pocket 2's because
we have Ace-8.

332
00:14:14,210 --> 00:14:16,460
He's going to call with a
pocket pair, which is always

333
00:14:16,460 --> 00:14:17,960
going to be better,
and Ace-8, which

334
00:14:17,960 --> 00:14:19,730
is going to be equal or better.

335
00:14:19,730 --> 00:14:22,290
So that's his calling
range 13% of the time.

336
00:14:22,290 --> 00:14:25,030
So we win 33% for
this range, which

337
00:14:25,030 --> 00:14:29,020
is you just plug-in
to the PokerTracker

338
00:14:29,020 --> 00:14:33,290
and it'll give you that
you're a 30% underdog, which

339
00:14:33,290 --> 00:14:34,650
is what I did here.

340
00:14:34,650 --> 00:14:35,540
That's how you do it.

341
00:14:35,540 --> 00:14:38,290

342
00:14:38,290 --> 00:14:40,580
And we get our EV
from the situation,

343
00:14:40,580 --> 00:14:42,160
and we have to solve for x.

344
00:14:42,160 --> 00:14:45,610
So we want EV to be 0 because
we're finding the marginal EV,

345
00:14:45,610 --> 00:14:47,820
and this is what we get.

346
00:14:47,820 --> 00:14:53,130
We get x, which is
all the chips we

347
00:14:53,130 --> 00:14:56,190
had before we paid
the big blind,

348
00:14:56,190 --> 00:15:00,080
and we lose x minus 1, which
is all the chips we have

349
00:15:00,080 --> 00:15:02,154
after paying the small blind.

350
00:15:02,154 --> 00:15:03,820
So a little bit of
nuance there and only

351
00:15:03,820 --> 00:15:07,030
changes our number by one, but
that gives us our break even.

352
00:15:07,030 --> 00:15:08,829
And our break even
here is like 62x.

353
00:15:08,829 --> 00:15:10,370
So the reason it's
a little bit lower

354
00:15:10,370 --> 00:15:13,390
is it doesn't factor in the
big blind doing pot odds,

355
00:15:13,390 --> 00:15:14,600
but that's the idea.

356
00:15:14,600 --> 00:15:17,162
So what we're going to do
is we're going to solve this

357
00:15:17,162 --> 00:15:18,370
and we're not going to do it.

358
00:15:18,370 --> 00:15:20,441
Someone else did it.

359
00:15:20,441 --> 00:15:22,440
We're going to solve this
for every single hand,

360
00:15:22,440 --> 00:15:23,960
and this is what Sklansky did.

361
00:15:23,960 --> 00:15:27,620
So he figured what your number
here is going to be two things.

362
00:15:27,620 --> 00:15:30,190
The number of hands you
are slightly better than,

363
00:15:30,190 --> 00:15:33,201
and why is this
factor important?

364
00:15:33,201 --> 00:15:35,700
What's he going to do it if you
have a better hand than him?

365
00:15:35,700 --> 00:15:38,910

366
00:15:38,910 --> 00:15:40,020
He's going to fold.

367
00:15:40,020 --> 00:15:43,050
So this determines
our fold equity,

368
00:15:43,050 --> 00:15:45,140
and then chance of
winning when you're behind

369
00:15:45,140 --> 00:15:47,370
determines our equity when
he actually does call.

370
00:15:47,370 --> 00:15:49,110
He just has perfect
information which

371
00:15:49,110 --> 00:15:51,600
is why it's a little
bit of a change

372
00:15:51,600 --> 00:15:57,310
and we're able to solve
it out perfectly here.

373
00:15:57,310 --> 00:15:59,380
So Sklansky didn't
know how to program,

374
00:15:59,380 --> 00:16:03,410
which is why Scott Carlson, or
Victor Chubukov a couple days

375
00:16:03,410 --> 00:16:06,440
later-- apparently
later the same day-

376
00:16:06,440 --> 00:16:07,440
came up with the answer.

377
00:16:07,440 --> 00:16:12,310
Where he said Aces have
infinity value here

378
00:16:12,310 --> 00:16:15,040
where you can have an
unlimited amount of chips

379
00:16:15,040 --> 00:16:18,430
and then show Aces and push
them, and it's plus EV.

380
00:16:18,430 --> 00:16:20,430
We're not saying it's the
most optimal decision,

381
00:16:20,430 --> 00:16:23,090
but you won't actually lose
chips on average even if you

382
00:16:23,090 --> 00:16:25,050
have an unlimited chip stack.

383
00:16:25,050 --> 00:16:28,880
Whereas King, it only
works for 1,290 chips.

384
00:16:28,880 --> 00:16:31,800
So we can solve it for
every hand to get down to,

385
00:16:31,800 --> 00:16:35,210
you could do this with 3-2
off with 1.8 chips and so on.

386
00:16:35,210 --> 00:16:38,620
So this is our primary
way to rank hands,

387
00:16:38,620 --> 00:16:41,740
and we're assuming that at
least in a heads-up situation,

388
00:16:41,740 --> 00:16:44,899
this is a pretty good
idea of what hands

389
00:16:44,899 --> 00:16:46,690
will have the most
value especially when it

390
00:16:46,690 --> 00:16:48,000
comes to going all in.

391
00:16:48,000 --> 00:16:52,100
One other method is going to
be equity versus three randoms,

392
00:16:52,100 --> 00:16:55,000
and it might be more
relevant for multi-way pots.

393
00:16:55,000 --> 00:16:57,150
You just rank them
by their expectation

394
00:16:57,150 --> 00:17:00,759
against three callers who
haven't looked at their hand.

395
00:17:00,759 --> 00:17:03,050
But we're not going to do
that especially because we're

396
00:17:03,050 --> 00:17:06,240
assuming generally we're going
to get one caller, especially

397
00:17:06,240 --> 00:17:07,819
heads-up where you
can only get one.

398
00:17:07,819 --> 00:17:11,419
But we're going to
expand it out later.

399
00:17:11,419 --> 00:17:12,960
So just an example
of what these are.

400
00:17:12,960 --> 00:17:16,560
Top 1% is Aces, top 5%
is 10s and Ace-Queens.

401
00:17:16,560 --> 00:17:18,109
30% is this.

402
00:17:18,109 --> 00:17:20,589
And sorry to break
it to you guys,

403
00:17:20,589 --> 00:17:22,990
but you're going to
have to memorize these.

404
00:17:22,990 --> 00:17:25,916
So 5% you just remember
their premium hands, just ten

405
00:17:25,916 --> 00:17:28,260
or Ace Queens, and I'm
giving you little mnemonics

406
00:17:28,260 --> 00:17:29,690
to help you remember it.

407
00:17:29,690 --> 00:17:32,796
Like Ace-10 or better
is 10%, and then

408
00:17:32,796 --> 00:17:35,420
like some sort of pocket pairs--
it's not even that big of deal

409
00:17:35,420 --> 00:17:37,690
if you go down to 2s here.

410
00:17:37,690 --> 00:17:41,000
20% is Ace-2 or pocket 2's.

411
00:17:41,000 --> 00:17:42,730
That's how you're
going to remember it.

412
00:17:42,730 --> 00:17:46,510
And then 30% is Broadway, so
Broadway is any two fAce cards.

413
00:17:46,510 --> 00:17:50,580
And an intuitive way to think
about it is based on this.

414
00:17:50,580 --> 00:17:56,260
Broadway is just this corner
here, which is about 1/3

415
00:17:56,260 --> 00:17:58,750
of the graph when you
count in pocket pairs.

416
00:17:58,750 --> 00:18:01,565
So that's how you just remember
30%-- it's that corner.

417
00:18:01,565 --> 00:18:02,440
So it's not that bad.

418
00:18:02,440 --> 00:18:05,870
Just remember top
20% is Ace-2 and 2s.

419
00:18:05,870 --> 00:18:10,990
30% is Broadway, and then
10% is Ace-10-- not that bad.

420
00:18:10,990 --> 00:18:14,330
So 50% is just going to be
the diagonal on the graph.

421
00:18:14,330 --> 00:18:18,560
It's any two cards
adding up to 15.

422
00:18:18,560 --> 00:18:20,060
This is what I mean
by the diagonal.

423
00:18:20,060 --> 00:18:23,930
Like this is your 13-by-13
graph where these are offsuit

424
00:18:23,930 --> 00:18:27,240
and these are suited, which
is why we double count them.

425
00:18:27,240 --> 00:18:30,300
And then this hand
adds up to 15.

426
00:18:30,300 --> 00:18:33,130
Ace-two and so does King-3
and so does Queen-4.

427
00:18:33,130 --> 00:18:35,160
So this is your 50th
percentile hand.

428
00:18:35,160 --> 00:18:37,946
So that's how you're
going to remember that.

429
00:18:37,946 --> 00:18:40,070
And then you don't really
need to remember anything

430
00:18:40,070 --> 00:18:40,810
more than that.

431
00:18:40,810 --> 00:18:44,190
So 100% is going to
be any two cards.

432
00:18:44,190 --> 00:18:46,430
It's going to be the
entire graph here.

433
00:18:46,430 --> 00:18:50,170
So we're going to be
talking about-- yeah?

434
00:18:50,170 --> 00:18:53,150
Percentage is the
percent of hands

435
00:18:53,150 --> 00:18:55,150
that that range represents.

436
00:18:55,150 --> 00:18:58,270
So like this is the
top 10% of hands.

437
00:18:58,270 --> 00:19:00,200
So if he calls--
if we assume he's

438
00:19:00,200 --> 00:19:02,440
going to call 10%
of the time, we

439
00:19:02,440 --> 00:19:04,410
are saying that he will
call with these cards

440
00:19:04,410 --> 00:19:05,710
and these cards only.

441
00:19:05,710 --> 00:19:08,120
And we're going to be
talking about ranges

442
00:19:08,120 --> 00:19:16,490
from now on because it helps us
understand what cards he has,

443
00:19:16,490 --> 00:19:18,980
to some extent, but
also lets us use it

444
00:19:18,980 --> 00:19:22,430
as an idea to get-- to figure
out exactly how often he's

445
00:19:22,430 --> 00:19:23,252
going to call.

446
00:19:23,252 --> 00:19:25,210
So we're just going to
be talking about ranges,

447
00:19:25,210 --> 00:19:27,190
and this is what I
mean when I say ranges.

448
00:19:27,190 --> 00:19:29,930
And to the extent that you're
just doing this at the table,

449
00:19:29,930 --> 00:19:32,962
you just have to memorize
these three numbers.

450
00:19:32,962 --> 00:19:34,420
Because this is
easy, this is easy,

451
00:19:34,420 --> 00:19:36,475
and then just remember
that tens are our 5%.

452
00:19:36,475 --> 00:19:41,670

453
00:19:41,670 --> 00:19:43,810
When we're talking
about ranges in general,

454
00:19:43,810 --> 00:19:47,460
a plus EV range--
a decision that's

455
00:19:47,460 --> 00:19:51,380
good for a range means that
on average, your decision is

456
00:19:51,380 --> 00:19:55,230
profitable for every hand
for that range in general.

457
00:19:55,230 --> 00:19:57,690
But an optimal range is
profitable for literally

458
00:19:57,690 --> 00:19:58,654
every hand.

459
00:19:58,654 --> 00:20:00,070
So you can do a
lot of things that

460
00:20:00,070 --> 00:20:01,970
are profitable
for any two cards,

461
00:20:01,970 --> 00:20:05,390
but 3-2 off might actually
not be profitable.

462
00:20:05,390 --> 00:20:06,740
So it's not necessarily optimal.

463
00:20:06,740 --> 00:20:09,320
You want to make sure that
every [INAUDIBLE] you can,

464
00:20:09,320 --> 00:20:12,090
every single card in
that range is optimal

465
00:20:12,090 --> 00:20:15,370
and is profitable, which
gives you the optimal range.

466
00:20:15,370 --> 00:20:20,100
It's the most plus EV range
for that type of decision.

467
00:20:20,100 --> 00:20:22,620
An example of this
is-- so say you're

468
00:20:22,620 --> 00:20:27,780
playing against Ace-Queen
and you call with this range.

469
00:20:27,780 --> 00:20:30,240
Pocket 5's and Ace-10 or better.

470
00:20:30,240 --> 00:20:33,630
So you are ahead of
this [INAUDIBLE] Queen

471
00:20:33,630 --> 00:20:36,856
because you're 53% to win
the hand with this range.

472
00:20:36,856 --> 00:20:38,230
However, we know
it's not optimal

473
00:20:38,230 --> 00:20:40,570
because we're calling
with two losing hands--

474
00:20:40,570 --> 00:20:42,000
Ace-10 and Ace-jack.

475
00:20:42,000 --> 00:20:44,370
So a more realistic,
more optimal

476
00:20:44,370 --> 00:20:49,529
range is going to be pocket
2s and Ace-King or better.

477
00:20:49,529 --> 00:20:51,320
So that's going to be
the difference there.

478
00:20:51,320 --> 00:20:54,200
So I want to make sure we're
not solving for something that's

479
00:20:54,200 --> 00:20:57,490
a slight favorite when actually
missing out on something

480
00:20:57,490 --> 00:20:59,450
a little bit more optimal.

481
00:20:59,450 --> 00:21:01,830
Here we go, so this
is a better range.

482
00:21:01,830 --> 00:21:04,420
5-5 or Ace-Queen,
actually 2s or Ace-Queen

483
00:21:04,420 --> 00:21:09,530
would be a little bit better,
and we win 60% of the time.

484
00:21:09,530 --> 00:21:11,720
And then if a range
is optimal, then

485
00:21:11,720 --> 00:21:15,420
we know for every
hand in that range,

486
00:21:15,420 --> 00:21:17,010
we have a plus EV decision.

487
00:21:17,010 --> 00:21:19,240
And that's why this
is the action we're

488
00:21:19,240 --> 00:21:24,550
going to use to prove that
if we solve for a range,

489
00:21:24,550 --> 00:21:27,520
then we know that if you
have any card in that range,

490
00:21:27,520 --> 00:21:28,982
our rule is still good for it.

491
00:21:28,982 --> 00:21:30,440
So that's why I'm
showing you that.

492
00:21:30,440 --> 00:21:31,273
Right, so that's it.

493
00:21:31,273 --> 00:21:33,430
So that's all we're
going to do in terms

494
00:21:33,430 --> 00:21:36,170
of defining what ranges are
and how we got those numbers.

495
00:21:36,170 --> 00:21:39,810
And from now, we're going to
talk about making decisions

496
00:21:39,810 --> 00:21:41,540
based on a range.

497
00:21:41,540 --> 00:21:42,750
So let's talk about preflops.

498
00:21:42,750 --> 00:21:45,527
Our assumption is hero
has M less than 10.

499
00:21:45,527 --> 00:21:47,110
We're in that period
of the tournament

500
00:21:47,110 --> 00:21:50,110
where your M is not
going to be that high.

501
00:21:50,110 --> 00:21:52,570
The villain is calling
some percentage

502
00:21:52,570 --> 00:21:56,260
of hands that are presumably
the top whatever of hands.

503
00:21:56,260 --> 00:21:58,800
We're guessing he's not
calling with a worse hand

504
00:21:58,800 --> 00:22:01,930
then-- and folding better hands.

505
00:22:01,930 --> 00:22:03,470
[INAUDIBLE]

506
00:22:03,470 --> 00:22:06,124
He might be a little
bit-- his view of what

507
00:22:06,124 --> 00:22:07,540
are good hands and
bad hands might

508
00:22:07,540 --> 00:22:09,490
be a little bit off of
ours, but we're just

509
00:22:09,490 --> 00:22:11,440
assuming everyone has the same.

510
00:22:11,440 --> 00:22:13,300
[? ICM ?] doesn't
matter, so we don't care

511
00:22:13,300 --> 00:22:15,420
about payout in tournaments.

512
00:22:15,420 --> 00:22:19,990
We're just trying to maximize
our chip-- our chip EV,

513
00:22:19,990 --> 00:22:22,462
and we only have two
decisions-- push or call.

514
00:22:22,462 --> 00:22:24,670
So the way that we're going
to come up with this rule

515
00:22:24,670 --> 00:22:27,540
here for heads-up is--
so what we're doing

516
00:22:27,540 --> 00:22:31,140
is we want to figure out
what our push/fold range is

517
00:22:31,140 --> 00:22:33,400
for heads-up in every scenario.

518
00:22:33,400 --> 00:22:37,290
And the way that we're
going to do this is first,

519
00:22:37,290 --> 00:22:40,970
we need an equation that tells
us range for range equities.

520
00:22:40,970 --> 00:22:46,640
So everyone can figure out Aces
versus whatever is like 80-20,

521
00:22:46,640 --> 00:22:50,960
and then 2 over cards with
a pocket pair is 50-50ish.

522
00:22:50,960 --> 00:22:53,540
But we want to know what's
a range versus a range.

523
00:22:53,540 --> 00:22:56,550
Like if we push 70%
of the time and he

524
00:22:56,550 --> 00:23:00,290
calls 30% of the time,
what does that translate to

525
00:23:00,290 --> 00:23:01,440
in terms of our equity?

526
00:23:01,440 --> 00:23:03,617
And it ends up being
like 60/40, but we

527
00:23:03,617 --> 00:23:05,950
want to get an idea of what
that trend is because that's

528
00:23:05,950 --> 00:23:10,640
going to materially change
our ability to come up

529
00:23:10,640 --> 00:23:13,200
with an optimal solution here.

530
00:23:13,200 --> 00:23:16,030
So we want to build a table
of range versus range,

531
00:23:16,030 --> 00:23:18,020
and then we come up
with a formula that

532
00:23:18,020 --> 00:23:21,440
will let us put in
two ranges and come up

533
00:23:21,440 --> 00:23:24,536
with an equity calculation
for both of those ranges.

534
00:23:24,536 --> 00:23:25,910
Then we're going
to develop an EV

535
00:23:25,910 --> 00:23:28,090
model for semi-bluffs,
which we already did,

536
00:23:28,090 --> 00:23:30,470
so that should be quick.

537
00:23:30,470 --> 00:23:34,600
For each m, find Nash
equilibrium if one exists.

538
00:23:34,600 --> 00:23:37,490
So a lot of people-- if you
Google "Nash equilibrium"

539
00:23:37,490 --> 00:23:40,200
for a preflop, you'll find
something, it's wrong,

540
00:23:40,200 --> 00:23:42,240
and later I'll show you why.

541
00:23:42,240 --> 00:23:44,760
And therefore,
unstable equilibriums--

542
00:23:44,760 --> 00:23:46,380
it pretty much always is.

543
00:23:46,380 --> 00:23:48,400
Find a reasonable
range and figure out

544
00:23:48,400 --> 00:23:50,170
what kind of assumptions
we need to make

545
00:23:50,170 --> 00:23:54,980
to make our push/fold
decisions correct.

546
00:23:54,980 --> 00:23:56,610
So let's do this first thing.

547
00:23:56,610 --> 00:23:59,590
So if we just-- for
each-- say that we

548
00:23:59,590 --> 00:24:01,330
use a hero's range of 50%.

549
00:24:01,330 --> 00:24:04,340
So in PokerTracker, we
put top 50% of hands.

550
00:24:04,340 --> 00:24:07,010
And then for each range
of hands for the villain,

551
00:24:07,010 --> 00:24:09,530
we calculate the
equity for that.

552
00:24:09,530 --> 00:24:13,590
And we do that for whatever I
did, like 15 different ranges.

553
00:24:13,590 --> 00:24:18,740
So if we push 50% of the time
and he calls 50% of the time,

554
00:24:18,740 --> 00:24:19,560
what's our equity?

555
00:24:19,560 --> 00:24:22,520
What's our chance of winning?

556
00:24:22,520 --> 00:24:23,400
50, right.

557
00:24:23,400 --> 00:24:25,160
So we have an even
equity because we

558
00:24:25,160 --> 00:24:28,160
are pushing and calling the
same types of hands on average.

559
00:24:28,160 --> 00:24:30,260
Whereas if he calls
100% of the time,

560
00:24:30,260 --> 00:24:32,480
our equity's actually
closer to 60.

561
00:24:32,480 --> 00:24:36,020
We are the favorite-- we are
a 60/40 favorite on that hand.

562
00:24:36,020 --> 00:24:39,960
Whereas if he calls with just
Aces, then we are like 17%

563
00:24:39,960 --> 00:24:40,820
to win the hand.

564
00:24:40,820 --> 00:24:41,810
That's the idea.

565
00:24:41,810 --> 00:24:46,240
So our goal here is to come
up with some sort of equation

566
00:24:46,240 --> 00:24:50,255
that we can just plug-in
to-- we can do math on.

567
00:24:50,255 --> 00:24:52,130
Like we want to be able
to do calculus on it,

568
00:24:52,130 --> 00:24:54,250
so we need an equation.

569
00:24:54,250 --> 00:24:55,960
So what type of
function does this look

570
00:24:55,960 --> 00:25:01,500
like because we want to
try to fit a curve to it.

571
00:25:01,500 --> 00:25:02,090
Exactly.

572
00:25:02,090 --> 00:25:03,720
It seems like a
logarithmic function

573
00:25:03,720 --> 00:25:06,470
and within r squared
of 99, it says

574
00:25:06,470 --> 00:25:08,670
that that's the logarithmic
equation for it.

575
00:25:08,670 --> 00:25:10,930
So that works when
the range is 50%,

576
00:25:10,930 --> 00:25:13,540
but let's take a look at when
we change the range to 30%.

577
00:25:13,540 --> 00:25:17,280
So what this is saying is we
are pushing 30% of the time

578
00:25:17,280 --> 00:25:19,900
and he is calling x
percent of the time.

579
00:25:19,900 --> 00:25:24,710
And what this line is our chance
of winning when he does that,

580
00:25:24,710 --> 00:25:26,450
and this is also logarithmic.

581
00:25:26,450 --> 00:25:28,200
So we're seeing a
little bit of a pattern,

582
00:25:28,200 --> 00:25:31,150
and then it also has a
really good r squared.

583
00:25:31,150 --> 00:25:33,350
And then if we push
10% of the time,

584
00:25:33,350 --> 00:25:37,200
it still like, OK,
it's 98 and change

585
00:25:37,200 --> 00:25:39,900
when we look at our equities
versus his calling range.

586
00:25:39,900 --> 00:25:42,060
But then it starts to get bad.

587
00:25:42,060 --> 00:25:44,170
If he calls 5% of
the time, our r

588
00:25:44,170 --> 00:25:48,110
squared when we fit a
log normal function,

589
00:25:48,110 --> 00:25:51,140
or a logarithmic
function, is only 95.

590
00:25:51,140 --> 00:25:55,100
And if he only calls 3% of
the time, our r squared is 89.

591
00:25:55,100 --> 00:25:58,280
Like this is not
really lot logarithmic

592
00:25:58,280 --> 00:26:00,420
when we start getting
in very tight ranges.

593
00:26:00,420 --> 00:26:01,930
And in 1%, it sucks.

594
00:26:01,930 --> 00:26:03,710
It's not even close.

595
00:26:03,710 --> 00:26:06,480
So what we're trying to
do is develop an equation

596
00:26:06,480 --> 00:26:08,180
for figuring out
range versus ranges,

597
00:26:08,180 --> 00:26:11,450
and we see that logarithmic
works some of the time.

598
00:26:11,450 --> 00:26:14,140
So the reason it
doesn't work, just

599
00:26:14,140 --> 00:26:17,320
to give you kind of an
idea why it doesn't work,

600
00:26:17,320 --> 00:26:20,940
is because the top 1%
of hands is three hands.

601
00:26:20,940 --> 00:26:23,260
It's Aces, Ace-King, and Kings.

602
00:26:23,260 --> 00:26:28,190
So it's materially changing
based on whether he's-- if we

603
00:26:28,190 --> 00:26:33,790
push Ace-King, whether he calls
with only Aces or Kings or also

604
00:26:33,790 --> 00:26:36,780
Ace-King jumps us
between here and here.

605
00:26:36,780 --> 00:26:39,180
So we have huge
gaps when it comes

606
00:26:39,180 --> 00:26:41,280
to very tight
percentages, so that's why

607
00:26:41,280 --> 00:26:44,410
we break a little bit up here.

608
00:26:44,410 --> 00:26:47,540
And when we turn it around, we
have the same type of thing.

609
00:26:47,540 --> 00:26:49,780
This is looking at OK,
say that we know he's

610
00:26:49,780 --> 00:26:52,460
going to call 50% of the time.

611
00:26:52,460 --> 00:26:54,430
If we push x
percent of the time,

612
00:26:54,430 --> 00:26:55,890
what is our chance of winning?

613
00:26:55,890 --> 00:27:01,200
So if we 100% of the time
and he calls 40% of the time,

614
00:27:01,200 --> 00:27:04,420
then we are 40% to win.

615
00:27:04,420 --> 00:27:07,539
So that's what this
chart is telling us,

616
00:27:07,539 --> 00:27:09,080
and you see the same
type of pattern.

617
00:27:09,080 --> 00:27:13,260
Where 50% range versus
this kind of range is good.

618
00:27:13,260 --> 00:27:16,130
If we push whatever and
he calls 30% of the time,

619
00:27:16,130 --> 00:27:17,300
it's still good.

620
00:27:17,300 --> 00:27:20,059
But then if he calls 2% of the
time, it starts becoming bad.

621
00:27:20,059 --> 00:27:21,600
So what are we taking
away from this?

622
00:27:21,600 --> 00:27:23,580
And the whole plan is we want
to come up with an equation that

623
00:27:23,580 --> 00:27:24,950
just gives us these numbers.

624
00:27:24,950 --> 00:27:27,440
I only got these from pushing
them into PokerTracker

625
00:27:27,440 --> 00:27:31,080
and were trying to fit a
certain equation onto it.

626
00:27:31,080 --> 00:27:34,840
So our takeaways here
is that like range

627
00:27:34,840 --> 00:27:37,800
versus range relationship
is probably logarithmic.

628
00:27:37,800 --> 00:27:39,380
That seems to be
a good estimate,

629
00:27:39,380 --> 00:27:43,490
but it's definitely
not good in the top 5%.

630
00:27:43,490 --> 00:27:45,250
So with regard to
our model, we're

631
00:27:45,250 --> 00:27:48,480
just going to say
this is probably not

632
00:27:48,480 --> 00:27:51,310
that good when you're
talking about ranges in 5%.

633
00:27:51,310 --> 00:27:54,300
But realistically,
when m is less than 10,

634
00:27:54,300 --> 00:27:56,400
no one is do anything
in the 5% range,

635
00:27:56,400 --> 00:27:59,620
so it's not that big of
a deal just to ignore it.

636
00:27:59,620 --> 00:28:04,090
So what we did is we
populated this table just now.

637
00:28:04,090 --> 00:28:06,435
So we took a look at what's
the villain's range up here

638
00:28:06,435 --> 00:28:07,310
and what's our range?

639
00:28:07,310 --> 00:28:11,790
So when we push 100% of the time
and he calls 30% of the time,

640
00:28:11,790 --> 00:28:13,940
we win 39% of the time.

641
00:28:13,940 --> 00:28:17,450
Where red means that we are
not likely to win and green

642
00:28:17,450 --> 00:28:19,150
means that we are likely to win.

643
00:28:19,150 --> 00:28:22,140
Like when we push 5% and
he calls with anything,

644
00:28:22,140 --> 00:28:24,870
we're 73% to win.

645
00:28:24,870 --> 00:28:27,860
So that's what this
table is telling us,

646
00:28:27,860 --> 00:28:31,380
and we want to do is we want to
come up with an equation that

647
00:28:31,380 --> 00:28:34,360
lets us populate this table
without actually having

648
00:28:34,360 --> 00:28:38,700
to do the range versus
range calculations by hand.

649
00:28:38,700 --> 00:28:40,950
So what I did was I
just ran a regression

650
00:28:40,950 --> 00:28:44,000
based on logarithmic variables,
and I found something

651
00:28:44,000 --> 00:28:47,460
that I found to be
really, really cool.

652
00:28:47,460 --> 00:28:50,890
So these seem to
be the coefficients

653
00:28:50,890 --> 00:28:53,930
with an r squared of 98.

654
00:28:53,930 --> 00:28:56,430
So this equation seems
to basically nail

655
00:28:56,430 --> 00:29:00,780
these range versus range
equities, which is, I think,

656
00:29:00,780 --> 00:29:01,880
really cool.

657
00:29:01,880 --> 00:29:06,330
So our guess is probably a
reason that it starts at 50%,

658
00:29:06,330 --> 00:29:08,670
and it seems to be symmetrical.

659
00:29:08,670 --> 00:29:11,490
Which is-- I don't
know if there's

660
00:29:11,490 --> 00:29:14,110
a statistical reason
for that, but that's

661
00:29:14,110 --> 00:29:15,660
extremely fascinating.

662
00:29:15,660 --> 00:29:18,150
And it could make
sense intuitively

663
00:29:18,150 --> 00:29:21,060
that-- so this is your
chance of winning.

664
00:29:21,060 --> 00:29:24,680
So would it make sense
that the wider he calls--

665
00:29:24,680 --> 00:29:28,760
when we take the natural log of
the percentage of him calling,

666
00:29:28,760 --> 00:29:30,542
our percent win goes up.

667
00:29:30,542 --> 00:29:32,000
So I think it does
because it means

668
00:29:32,000 --> 00:29:34,210
he's calling with a worse hand.

669
00:29:34,210 --> 00:29:36,764
That's why it goes up
when he calls greater,

670
00:29:36,764 --> 00:29:39,180
but then goes down when we
push greater because this means

671
00:29:39,180 --> 00:29:40,920
that we're pushing worse hands.

672
00:29:40,920 --> 00:29:42,670
So that's the equation
there, and that's

673
00:29:42,670 --> 00:29:44,550
giving us something that we
can actually differentiate when

674
00:29:44,550 --> 00:29:45,990
we're trying to solve this.

675
00:29:45,990 --> 00:29:49,400
And just in terms of
errors, it's not that bad.

676
00:29:49,400 --> 00:29:51,260
We have an r squared of 98.

677
00:29:51,260 --> 00:29:52,920
So I think this is
good enough to use.

678
00:29:52,920 --> 00:29:55,220
We can actually use
this to try to optimize

679
00:29:55,220 --> 00:29:57,420
our decision making preflop.

680
00:29:57,420 --> 00:29:59,710
So we solved-- we came
up with an equation

681
00:29:59,710 --> 00:30:02,540
that we can use for determining
our chance of winning

682
00:30:02,540 --> 00:30:06,281
a hand compared to our pushing
range and his calling range.

683
00:30:06,281 --> 00:30:08,280
So let's go back to our
EV model for semi-bluff.

684
00:30:08,280 --> 00:30:11,020

685
00:30:11,020 --> 00:30:14,825
So we already did that in
the fold equity portion,

686
00:30:14,825 --> 00:30:16,200
but we're going
to build this out

687
00:30:16,200 --> 00:30:20,450
to be relevant to our
situation in particular.

688
00:30:20,450 --> 00:30:23,070
So we're assuming we're
talking about in terms of M,

689
00:30:23,070 --> 00:30:26,040
so what's our-- so our blinds
are going to be equal to 1.

690
00:30:26,040 --> 00:30:30,440
Like this means this is 1 M and
our EV is going to be in terms

691
00:30:30,440 --> 00:30:31,150
of M's.

692
00:30:31,150 --> 00:30:34,690
We're going to-- if our
EV ends up being 0.35,

693
00:30:34,690 --> 00:30:37,630
it means 0.35 times all
the blinds combined just

694
00:30:37,630 --> 00:30:38,950
to get rid of blinds here.

695
00:30:38,950 --> 00:30:41,450
So all the blinds
combined equals

696
00:30:41,450 --> 00:30:43,770
M. Our showdown
value is just going

697
00:30:43,770 --> 00:30:45,570
to be-- so our fold
equity is going

698
00:30:45,570 --> 00:30:48,820
to be the pot times the
chance of him folding, i.e.

699
00:30:48,820 --> 00:30:50,100
1 minus calling.

700
00:30:50,100 --> 00:30:52,280
The chance of him--
our showdown value

701
00:30:52,280 --> 00:30:56,650
is chance of calling times win
amount times win percentage

702
00:30:56,650 --> 00:30:59,770
minus lose amount
times lose percentage.

703
00:30:59,770 --> 00:31:02,700
The win amount is going
to be stacked plus 2/3

704
00:31:02,700 --> 00:31:05,850
because if the blinds are
1-2 or something where

705
00:31:05,850 --> 00:31:07,890
the big blind is
double the small blind,

706
00:31:07,890 --> 00:31:11,710
it means that you
win his big blind

707
00:31:11,710 --> 00:31:14,840
and then you lose
your whole stack.

708
00:31:14,840 --> 00:31:16,900
Depending on when
we mark our stack,

709
00:31:16,900 --> 00:31:19,120
it changes it a
little bit with regard

710
00:31:19,120 --> 00:31:21,880
to whether we count it as before
we pay the blind or after.

711
00:31:21,880 --> 00:31:23,750
But it ends up not
making a huge difference.

712
00:31:23,750 --> 00:31:27,190
It just impacts whether
this is plus 2/3 or plus

713
00:31:27,190 --> 00:31:29,060
1/3 or just your stack.

714
00:31:29,060 --> 00:31:31,880
But anyway, we also
have this equation,

715
00:31:31,880 --> 00:31:33,700
which we just solve for.

716
00:31:33,700 --> 00:31:35,210
Where the chance
of us winning is

717
00:31:35,210 --> 00:31:38,530
related to the
chance to him calling

718
00:31:38,530 --> 00:31:41,970
and the range that
we're pushing.

719
00:31:41,970 --> 00:31:44,130
So what you might
see here is all

720
00:31:44,130 --> 00:31:47,970
of these are related to the
same three variables, where it's

721
00:31:47,970 --> 00:31:49,810
either the call
percentage, the push

722
00:31:49,810 --> 00:31:51,545
percentage, or a stack size.

723
00:31:51,545 --> 00:31:54,170
So we combine this to one giant
equation, which obviously we're

724
00:31:54,170 --> 00:31:57,510
not going to remember but we
can use to start solving this

725
00:31:57,510 --> 00:31:58,530
mathematically.

726
00:31:58,530 --> 00:32:01,280
So this is what we
would end up with.

727
00:32:01,280 --> 00:32:06,560
So when M is 1, we end up
with this sort of graph.

728
00:32:06,560 --> 00:32:09,450
When the hero's push
range goes up to 100%

729
00:32:09,450 --> 00:32:14,530
here and the villain's call
range goes up to 100% here.

730
00:32:14,530 --> 00:32:17,214
This is our equity,
which I just color-coded

731
00:32:17,214 --> 00:32:18,880
so we don't need to
look at the numbers.

732
00:32:18,880 --> 00:32:21,950
Where green means it's in
the hero's favor and yellow

733
00:32:21,950 --> 00:32:23,960
means it's close to
zero and red means

734
00:32:23,960 --> 00:32:26,270
it's in the villain's favor.

735
00:32:26,270 --> 00:32:30,720
So what we see here is
it's really green over here

736
00:32:30,720 --> 00:32:32,220
and really yellow over there.

737
00:32:32,220 --> 00:32:34,010
And this is telling
us this is already

738
00:32:34,010 --> 00:32:37,410
factoring in fold equity and our
chance of winning if he calls.

739
00:32:37,410 --> 00:32:41,750
This is saying that if
we call 100% of the time,

740
00:32:41,750 --> 00:32:44,500
he should call 100%
of the time because he

741
00:32:44,500 --> 00:32:48,340
wants the yellowest area and
we want the greenest area.

742
00:32:48,340 --> 00:32:51,220
So that's with 1 M, so it
shouldn't be surprising

743
00:32:51,220 --> 00:32:53,900
that our equilibrium
at 1 M is going

744
00:32:53,900 --> 00:32:57,580
to be everyone like getting
all in 100% of the time.

745
00:32:57,580 --> 00:33:00,720
We see when we switch to 10
M, it's pretty different.

746
00:33:00,720 --> 00:33:04,290
In fact, this 100 is one
of the yellowest areas,

747
00:33:04,290 --> 00:33:09,305
whereas our green area is
either up here or down here.

748
00:33:09,305 --> 00:33:10,680
So we're going to
use this to get

749
00:33:10,680 --> 00:33:14,200
an idea of how our
value changes based

750
00:33:14,200 --> 00:33:15,930
on these variables
changing to figure out

751
00:33:15,930 --> 00:33:21,030
if we can isolate it to a corner
and make that our kind of rule.

752
00:33:21,030 --> 00:33:24,010
So first, let's find out when
we can get a Nash equilibrium.

753
00:33:24,010 --> 00:33:27,730
So the villain
gets to pick this.

754
00:33:27,730 --> 00:33:29,980
The villain gets to
pick his call range

755
00:33:29,980 --> 00:33:32,390
and we get to pick
our push range.

756
00:33:32,390 --> 00:33:37,440
That's the flexibility
each person has.

757
00:33:37,440 --> 00:33:41,490
So if we push 100% of the time
and he calls 100% of the time,

758
00:33:41,490 --> 00:33:43,760
this is a Nash
equilibrium for 1 M

759
00:33:43,760 --> 00:33:46,430
because he can't do
any better by going up

760
00:33:46,430 --> 00:33:49,400
here because a higher
number is worse for him

761
00:33:49,400 --> 00:33:52,040
because this is our
equity in the hand.

762
00:33:52,040 --> 00:33:54,370
And the equity comes
directly from him

763
00:33:54,370 --> 00:33:56,990
because it's a heads-up
situation whereas we can't

764
00:33:56,990 --> 00:33:58,590
do better by pushing left.

765
00:33:58,590 --> 00:34:00,440
Like when we go down
here, the next number

766
00:34:00,440 --> 00:34:03,580
is 0.32, meaning that
if we push any less,

767
00:34:03,580 --> 00:34:05,360
we're actually losing
value because we

768
00:34:05,360 --> 00:34:06,700
want the number to be higher.

769
00:34:06,700 --> 00:34:09,620
And when it comes to 2 M,
100% isn't that much off.

770
00:34:09,620 --> 00:34:12,320
We're talking about 0.01
M of a difference in EV,

771
00:34:12,320 --> 00:34:15,275
but we do actually reach a
Nash equilibrium here-- 2 M--

772
00:34:15,275 --> 00:34:17,790
where we can't do anything
any better by moving

773
00:34:17,790 --> 00:34:19,830
and he can't do any
better by moving.

774
00:34:19,830 --> 00:34:22,130
So when M is 3, we
have an unstable Nash,

775
00:34:22,130 --> 00:34:25,260
and this is when stuff
is getting interesting.

776
00:34:25,260 --> 00:34:27,469
If we push 80, he
should call 100.

777
00:34:27,469 --> 00:34:30,860
If he calls 100, we
should push 65, and so on.

778
00:34:30,860 --> 00:34:32,310
So we end up in a
bit of a circle.

779
00:34:32,310 --> 00:34:35,280
It's a rock, paper,
scissors situation.

780
00:34:35,280 --> 00:34:36,340
What do we have?

781
00:34:36,340 --> 00:34:37,431
M equals 3.

782
00:34:37,431 --> 00:34:39,639
So this lets us know that
this whole Nash equilibrium

783
00:34:39,639 --> 00:34:41,170
thing is not going to work out.

784
00:34:41,170 --> 00:34:43,080
Like it only really
works for M of 2,

785
00:34:43,080 --> 00:34:46,675
and that's not the most
important situation

786
00:34:46,675 --> 00:34:48,050
to figure out
because you're know

787
00:34:48,050 --> 00:34:50,750
that you should be pushing
a very, very wide range when

788
00:34:50,750 --> 00:34:53,340
M is 2.

789
00:34:53,340 --> 00:34:55,370
So we need to figure out
some sort of pattern.

790
00:34:55,370 --> 00:35:00,340
How do these colors move
when your M is more than 2?

791
00:35:00,340 --> 00:35:02,999
So let's make-- let's
go to M equals 5

792
00:35:02,999 --> 00:35:04,790
and make the gradient
a little bit steeper.

793
00:35:04,790 --> 00:35:07,610
So these are the same numbers
except I just made it--

794
00:35:07,610 --> 00:35:11,290
we're talking about very
slight changes in EV, where

795
00:35:11,290 --> 00:35:16,350
the difference between yellow
here and green here is 0.1 M,

796
00:35:16,350 --> 00:35:19,330
and this will help us
understand where this value is

797
00:35:19,330 --> 00:35:20,610
going to come from.

798
00:35:20,610 --> 00:35:22,870
So let's see how this
changes over time,

799
00:35:22,870 --> 00:35:25,110
because our goal here
is-- the villain's goal

800
00:35:25,110 --> 00:35:27,550
is to cause this to
be in a yellow area,

801
00:35:27,550 --> 00:35:31,950
and the hero's goal is to cause
this to be in a green area.

802
00:35:31,950 --> 00:35:33,880
And you can see
why we inherently

803
00:35:33,880 --> 00:35:37,670
have this need to figure out
what they're going to do.

804
00:35:37,670 --> 00:35:40,890
Because say that
we're the villain.

805
00:35:40,890 --> 00:35:43,730
If we know the hero
pushes 100, we call 100.

806
00:35:43,730 --> 00:35:48,280
Whereas if we think the hero
pushes 30%, we should call 5%.

807
00:35:48,280 --> 00:35:50,650
So let's see how this
changes over time.

808
00:35:50,650 --> 00:35:52,840
So this is when M
is 2, and then this

809
00:35:52,840 --> 00:35:57,290
is us increasing as M
is 3, 4, 5, 6, 7, 8, 9.

810
00:35:57,290 --> 00:35:58,680
We're trying to
pick the box that

811
00:35:58,680 --> 00:36:01,090
is kind of the yellowest area.

812
00:36:01,090 --> 00:36:03,490
Where can we
realistically come up

813
00:36:03,490 --> 00:36:06,740
with a rule for ending up there?

814
00:36:06,740 --> 00:36:09,210
Based on what this
x-coordinate is, what

815
00:36:09,210 --> 00:36:11,100
should be our y-coordinate?

816
00:36:11,100 --> 00:36:13,200
What should be our call?

817
00:36:13,200 --> 00:36:17,450
Do we see any way to
figure this thing out?

818
00:36:17,450 --> 00:36:20,460
So what I think we should do
is just draw a line right here

819
00:36:20,460 --> 00:36:23,380
and say let's just figure out
what the equation of that line

820
00:36:23,380 --> 00:36:26,280
is, and then come up
with an estimate of what

821
00:36:26,280 --> 00:36:32,330
his pushing range is, and then
throw it into whatever function

822
00:36:32,330 --> 00:36:33,560
we used to get that line.

823
00:36:33,560 --> 00:36:36,750
Which can't be very complicated
in order to make sure,

824
00:36:36,750 --> 00:36:39,630
no matter what his pushing
range is that we read him for,

825
00:36:39,630 --> 00:36:41,860
we end up in this yellow area.

826
00:36:41,860 --> 00:36:45,540
So what I did was for each M, I
highlighted what the lowest EV

827
00:36:45,540 --> 00:36:48,670
decision is for the hero,
meaning the best decision

828
00:36:48,670 --> 00:36:49,880
for the villain.

829
00:36:49,880 --> 00:36:52,120
And for 10 M, it's
this diagonal.

830
00:36:52,120 --> 00:36:54,320
But for 1 M, it's all
the way down here.

831
00:36:54,320 --> 00:36:56,910
Like best you can
do is over here.

832
00:36:56,910 --> 00:37:02,670
So are our function is we call
10 times his pushing range.

833
00:37:02,670 --> 00:37:06,380
So if we have 1 M and he
pushes 5% of the time,

834
00:37:06,380 --> 00:37:08,560
we call about 50% of the time.

835
00:37:08,560 --> 00:37:11,160
If he pushes 20% of the
time, we're calling always,

836
00:37:11,160 --> 00:37:13,870
and anything more than
20%, we're always calling.

837
00:37:13,870 --> 00:37:15,230
That's where M equals 1.

838
00:37:15,230 --> 00:37:18,720
In general, the rule is going
to be if M is 1, always call,

839
00:37:18,720 --> 00:37:21,520
and if M is 2, you call
twice his pushing range.

840
00:37:21,520 --> 00:37:25,880
So if we think he's pushing 50%
or more, we're always calling.

841
00:37:25,880 --> 00:37:30,550
If we think he's pushing
20%, we call 40% of the time.

842
00:37:30,550 --> 00:37:32,740
Like that's the
optimal calling range

843
00:37:32,740 --> 00:37:34,800
in this heads-up situation.

844
00:37:34,800 --> 00:37:37,930
So when M is 4-- so I
skipped one, but when M is 4

845
00:37:37,930 --> 00:37:40,040
we have basically this
straight diagonal.

846
00:37:40,040 --> 00:37:42,250
We try to match
his pushing range.

847
00:37:42,250 --> 00:37:44,180
So when he pushes
60, we call 60.

848
00:37:44,180 --> 00:37:46,470
When he pushes 100, we call 100.

849
00:37:46,470 --> 00:37:50,770
That's the best we can do
to dominate his range there.

850
00:37:50,770 --> 00:37:53,710
When M is 6, we
call 2/3 his range.

851
00:37:53,710 --> 00:37:56,307
And when M is 9, we
call half is range.

852
00:37:56,307 --> 00:37:57,890
So those are the
rules that we're just

853
00:37:57,890 --> 00:37:58,973
going to have to remember.

854
00:37:58,973 --> 00:38:00,920
When you're in a
heads-up situation,

855
00:38:00,920 --> 00:38:04,960
if the stack is basically 1 or
2, you're always calling him.

856
00:38:04,960 --> 00:38:07,530
At 4, you're calling
him an even amount.

857
00:38:07,530 --> 00:38:11,880
At 6 and 9, you're calling him
like half of what he pushes.

858
00:38:11,880 --> 00:38:13,830
And that's the optimal move.

859
00:38:13,830 --> 00:38:17,670
That is how you will-- if you're
in a heads-up situation, which

860
00:38:17,670 --> 00:38:20,100
you will be at the end of
every tournament to the extent

861
00:38:20,100 --> 00:38:25,380
that you get there, you will
dominate his playing style

862
00:38:25,380 --> 00:38:28,540
based on what you read
as his pushing amount.

863
00:38:28,540 --> 00:38:31,930
And you can just
count how many hands

864
00:38:31,930 --> 00:38:33,840
did he push versus
how many hands

865
00:38:33,840 --> 00:38:36,490
does he fold to get an idea
of what his range is here.

866
00:38:36,490 --> 00:38:38,150
And this also works
when it's folded

867
00:38:38,150 --> 00:38:39,810
to you in the small blind.

868
00:38:39,810 --> 00:38:41,969
Like even if you're
10 handed, by the time

869
00:38:41,969 --> 00:38:44,010
you get to the small blind,
you're heads up again

870
00:38:44,010 --> 00:38:47,100
and these rules apply.

871
00:38:47,100 --> 00:38:49,040
So to the extent
you can memorize

872
00:38:49,040 --> 00:38:52,420
this thing will give you the
right move in that scenario

873
00:38:52,420 --> 00:38:55,110
for Ms up to 10.

874
00:38:55,110 --> 00:38:59,040
So when you're the hero-- when
you're the small but, rather,

875
00:38:59,040 --> 00:39:01,780
and you're pushing, it
requires you to estimate

876
00:39:01,780 --> 00:39:03,180
what his calling range is.

877
00:39:03,180 --> 00:39:09,430
Because you end up all
over the plAce based

878
00:39:09,430 --> 00:39:12,120
on making different estimates
of what you can call with,

879
00:39:12,120 --> 00:39:14,530
and you really don't
have any information.

880
00:39:14,530 --> 00:39:17,210
But I'm going to
graph it out here

881
00:39:17,210 --> 00:39:19,780
and then we can see what's
going to be a good estimate

882
00:39:19,780 --> 00:39:21,460
in basically any scenario.

883
00:39:21,460 --> 00:39:24,260
So there are a couple different
ways I can think about this.

884
00:39:24,260 --> 00:39:26,690
So we're going to come up
with a bunch of numbers

885
00:39:26,690 --> 00:39:30,050
based on the villains'
call range here.

886
00:39:30,050 --> 00:39:33,830
And we can either-- so
we're targeting a column

887
00:39:33,830 --> 00:39:36,860
and then like the
row is going to be

888
00:39:36,860 --> 00:39:38,550
information we don't have.

889
00:39:38,550 --> 00:39:41,870
So the question is do
we pick the column that

890
00:39:41,870 --> 00:39:46,520
has the highest average
EV, the highest minimum EV,

891
00:39:46,520 --> 00:39:49,880
or the highest EV versus
like a particular bad player

892
00:39:49,880 --> 00:39:51,390
that we're going
to be targeting?

893
00:39:51,390 --> 00:39:57,080
And the blue here is what
column maximizes your EV

894
00:39:57,080 --> 00:39:58,510
for that scenario?

895
00:39:58,510 --> 00:40:00,220
And your guess is
as good as mine

896
00:40:00,220 --> 00:40:03,090
when it comes to what's the
best way to strategize it.

897
00:40:03,090 --> 00:40:09,400
Are we looking for-- maximizing
the min will help us make sure

898
00:40:09,400 --> 00:40:11,410
that we're not dominated
by someone who's really

899
00:40:11,410 --> 00:40:13,860
got our number, whereas
maximizing the average

900
00:40:13,860 --> 00:40:15,960
might be better when
we're trying to figure out

901
00:40:15,960 --> 00:40:19,650
against a player we know nothing
about what would be better.

902
00:40:19,650 --> 00:40:22,400
And maximizing versus
tight-- or loose--

903
00:40:22,400 --> 00:40:25,740
will help us figure out some
sort of-- how to capitalize

904
00:40:25,740 --> 00:40:27,340
on the reads that we're making.

905
00:40:27,340 --> 00:40:30,760
So when M is 1, he's over there.

906
00:40:30,760 --> 00:40:32,780
No matter what the
scenario, you should

907
00:40:32,780 --> 00:40:34,469
be pushing 100% of the time.

908
00:40:34,469 --> 00:40:35,760
That's what this is telling us.

909
00:40:35,760 --> 00:40:38,790
It shouldn't be a surprise based
on all the stuff I told you

910
00:40:38,790 --> 00:40:40,482
about 1 M situations.

911
00:40:40,482 --> 00:40:42,440
No matter what kind of
assumption we're making,

912
00:40:42,440 --> 00:40:43,990
push 100%.

913
00:40:43,990 --> 00:40:45,700
So let me pick up to M is 10.

914
00:40:45,700 --> 00:40:48,630

915
00:40:48,630 --> 00:40:49,630
So what's going on here?

916
00:40:49,630 --> 00:40:53,430
So as M increases, they all
kind of move at the same time,

917
00:40:53,430 --> 00:40:54,970
except what?

918
00:40:54,970 --> 00:40:58,240
If we're talking loose or
we're targeting best/worst case

919
00:40:58,240 --> 00:41:00,890
scenario or best
average, it starts

920
00:41:00,890 --> 00:41:03,670
to trickle down to like 50%.

921
00:41:03,670 --> 00:41:06,780
But it's certainly-- they're
relatively near each other,

922
00:41:06,780 --> 00:41:08,620
and what's good about
this is that means

923
00:41:08,620 --> 00:41:12,490
that if we target any of these,
we're basically in the ballpark

924
00:41:12,490 --> 00:41:16,030
where the difference between
this column and this column

925
00:41:16,030 --> 00:41:19,450
for any of M is going
to be not that material.

926
00:41:19,450 --> 00:41:20,990
If you were just
in that ballpark,

927
00:41:20,990 --> 00:41:25,810
you're fine, but which one of
them is completely different.

928
00:41:25,810 --> 00:41:27,350
Yeah, against a tight player.

929
00:41:27,350 --> 00:41:29,030
And it should make
sense intuitively

930
00:41:29,030 --> 00:41:32,410
why if you read him as tight,
as someone who only calls

931
00:41:32,410 --> 00:41:36,220
15% of the time,
even with 9 and 10 M,

932
00:41:36,220 --> 00:41:38,040
you should push
100% of the time.

933
00:41:38,040 --> 00:41:39,470
Why?

934
00:41:39,470 --> 00:41:42,840
It's because 85% of the time,
he's just going to fold,

935
00:41:42,840 --> 00:41:45,240
and even when you're
called, the amount

936
00:41:45,240 --> 00:41:48,730
of value you get from him
folding most of the time

937
00:41:48,730 --> 00:41:50,230
just crushes him.

938
00:41:50,230 --> 00:41:54,360
So you should-- to the
extent that you can encourage

939
00:41:54,360 --> 00:41:56,280
him to be tight, do it.

940
00:41:56,280 --> 00:41:59,710
But absolutely, if he's
tight, push every single hand.

941
00:41:59,710 --> 00:42:03,190
You're never in the scenario
where pushing less than top 50%

942
00:42:03,190 --> 00:42:05,950
is good.

943
00:42:05,950 --> 00:42:07,310
So that's it for heads up.

944
00:42:07,310 --> 00:42:09,410
So now let's talk
about other positions,

945
00:42:09,410 --> 00:42:15,300
and we end up in a lot of
complicated situations, which

946
00:42:15,300 --> 00:42:17,899
we have to just
assume away here.

947
00:42:17,899 --> 00:42:19,440
So we can lose in
two different ways.

948
00:42:19,440 --> 00:42:21,490
We can lose if we
call and he beats us,

949
00:42:21,490 --> 00:42:24,260
or we could also lose if we
call and then someone behind us

950
00:42:24,260 --> 00:42:24,770
calls.

951
00:42:24,770 --> 00:42:26,520
So we're estimating
that someone behind us

952
00:42:26,520 --> 00:42:29,351
is only going to call if he has
a premium hand because if we're

953
00:42:29,351 --> 00:42:31,225
in a short stack situation
and someone pushes

954
00:42:31,225 --> 00:42:33,930
and then we call, someone is
only calling behind us when

955
00:42:33,930 --> 00:42:35,162
they have a really good hand.

956
00:42:35,162 --> 00:42:36,870
And let's just assume
we're going to lose

957
00:42:36,870 --> 00:42:37,960
if we get another caller.

958
00:42:37,960 --> 00:42:40,376
Like we are almost certainly
going to be dominated-- let's

959
00:42:40,376 --> 00:42:41,710
say we have 0 EV there.

960
00:42:41,710 --> 00:42:44,020
And because of our
equation here, we

961
00:42:44,020 --> 00:42:45,370
can actually solve that.

962
00:42:45,370 --> 00:42:47,790
We can actually figure
out what range is going

963
00:42:47,790 --> 00:42:49,780
to be 60% versus another range.

964
00:42:49,780 --> 00:42:52,750
So this is resulting in a really
cool rule of thumb here, which

965
00:42:52,750 --> 00:42:54,924
is when you're in a
10 M or less situation

966
00:42:54,924 --> 00:42:57,340
and you're trying to decide
whether to call an all-in, you

967
00:42:57,340 --> 00:42:59,930
just ask yourself,
are you calling such

968
00:42:59,930 --> 00:43:04,660
that his range is three
times more than your range?

969
00:43:04,660 --> 00:43:06,760
And some questions you
might ask yourself is

970
00:43:06,760 --> 00:43:10,490
say that you have Ace-10, like
you have a 10% range here.

971
00:43:10,490 --> 00:43:13,790
Would he push all in into you
with King-Jack-- something

972
00:43:13,790 --> 00:43:15,520
in a 30%?

973
00:43:15,520 --> 00:43:17,820
If you have King-Queen,
which is like 30%,

974
00:43:17,820 --> 00:43:22,800
would he push all in with 8-5,
which is 50% range, and so on.

975
00:43:22,800 --> 00:43:24,825
So when you're in a
[? full ring ?] situation

976
00:43:24,825 --> 00:43:26,990
and you're trying to
decide whether to call,

977
00:43:26,990 --> 00:43:29,010
figure out in general
what do you think

978
00:43:29,010 --> 00:43:30,540
his pushing range is there.

979
00:43:30,540 --> 00:43:32,920
And it's a good call
if your calling range

980
00:43:32,920 --> 00:43:35,787
is 1/3 of his pushing range.

981
00:43:35,787 --> 00:43:36,620
Let's call it a day.

982
00:43:36,620 --> 00:43:38,340
Thanks, guys.

983
00:43:38,340 --> 00:43:39,257