MIT 15.

S50 LECTURE 4
Wednesday, January 23rd, 2013
REVIEW OF EXPECTATION AND EQUITY:
DO YOU GO ALL-IN OR FOLD?
OBVIOUSLY IT DEPENDS ON THE
PLAYERS…

 Is the BB a gambly player? Is it likely that he or

she will call with a wide range of hands?
 How have I been playing? Do I look like a crazy
all-in monkey, or will my all-in get the credibility
of a very good hand?

 (Suppose it is far from the final table.)

ANALYSIS
 Let’s suppose the BB calls a “reasonable” range:

 Your equity against this range:

DO THE MATH
 75% of the time, he folds, and we win 2.5BB’s
 25% of the time, he calls,
 36% of the time, we win the all-in and win 16.5BB’s
 64% of the time, we lose the all-in and lose 14.5BB’s

 0.75(+2.5) + 0.25( 0.36(+16.5) + 0.64(-14.5) )

 =0.75(+2.5) + 0.25(-3.34)

 Clearly positive.

 =1.04BB

 You’re making an entire big blind by shoving

instead of folding! We really are maximizing our
expected # of chips by shoving!
IF WE HAD THE WORST HAND POSSIBLE...
DO THE MATH AGAIN
 Now we have 28% equity, instead of 36% equity,
against his calling range
 Expectation when he calls:
 0.28(+16.5) + 0.72(-14.5) = -5.82
 Overall expectation:
 0.75(+2.5) + 0.25(-5.82)
 Still postive! This shove is an EXCELLENT play
that earns you 0.42BB’s, not a “crazy bad” play.
We are still earning positive chips/money!
THE POINT?
 If you’re only calling this 25% from the Big Blind
in this situation, then you’re allowing the Small
Blind to shove any 2 cards profitably.
 Let’s suppose we consider adding QJo to our
calling range. We need to call 14BB to win a
total pot of 31BB, so we need 45% equity.
 Nonetheless, since most Small Blinds don’t shove
enough, we might not actually have this!
MORE MATH
 Let’s assume that the Small Blind only shoves
the top 25% of hands (instead of 100%, which
includes 32o).
 Your equity with QJo:

 You only have 42% equity. Not enough to call the

all-in!
LET’S TAKE A LOOK AT THE NASH RANGES
 SB 66.8%, 22+ Kx+ Q2s+ Q5o+ J2s+ J7o+ T2s+ T7o+
93s+ 96o+ 84s+ 86o+ 73s+ 76o 63s+ 65o 53s+ 43s
 BB 38.5%, 22+ Ax+ K2s+ K5o+ Q6s+ Q8o+ J8s+ JTo
T9s
 This is a potential formula to follow, when figuring
out which hands to go all-in with from the SB, and
which hands to call with from the BB.
 However, if the BB is calling way fewer than 38.5% of
hands, then as SB you can go all-in with a lot more
than 66.8% of hands.
 If the SB is going all-in with way fewer than 66.8% of
hands, then as BB it is a huge mistake to call with as
much as 38.5% of hands.
BUT IT’S AN EQUILIBRIUM…
 SB 66.8%, 22+ Kx+ Q2s+ Q5o+ J2s+ J7o+ T2s+
T7o+ 93s+ 96o+ 84s+ 86o+ 73s+ 76o 63s+ 65o
53s+ 43s
 BB 38.5%, 22+ Ax+ K2s+ K5o+ Q6s+ Q8o+ J8s+
JTo T9s
 If the SB is going all-in with 66.8% of hands, then
the optimal strategy for the BB is go call with
38.5% of hands.
 If the BB would call an all-in with 38.5% of
hands, then the optimal strategy for the SB is to
go all-in with 66.8% of hands.
A FEW MORE NASH RANGES
 BTN, 10BB
 43.9%, 22+ Ax+ K2s+ K6o+ Q5s+ Q9o+ J7s+ J9o+
T6s+ T9o 96s+ 98o 85s+ 75s+ 65s 54s
 CO, 7BB
 38.8%, 22+ Ax+ K2s+ K9o+ Q5s+ QTo+ J7s+ J9o+
T7s+ T9o 96s+ 86s+ 75s+ 65s 54s
 LJ, 10BB
 23.4%, 22+ A2s+ A8o+ K7s+ KJo+ Q8s+ QJo J8s+
JTo T8s+ 98s
 UTG (9 handed), 3BB
 24.1%, 33+ A2s+ A8o+ K6s+ KTo+ Q8s+ QTo+ J9s+
JTo T9s
SUPPOSE I AM TOO SCARED TO GO ALL-IN…
CAN I RAISE TO 2BB AS AN ALTERNATIVE?
LET’S PUT OURSELF IN BB’S SHOES…
 If we go all-in, BU will need to call 6 to win a pot
of 17.5, so he will call so long as he thinks he has
at least 34% equity against our range.
 So he will pretty much always call. (“When given 2-
to-1 to call a pre-flop all-in, you can pretty much
always call unless you are guaranteed to be a 70-30
or worse.”)
 When we go all-in, assuming BU calls, we risked
7 to win a pot of 17.5, so we need 40% equity for
this to be profitable.
ANALYSIS AS THE BB
 If our hand has 40% equity vs. BU’s range, then
going all-in is +EV.
 If our hand doesn’t have 40% equity vs. BU’s
range, calling could still be +EV, since we are
given 4.5-to-1 to see the flop.
 Even if our hand has 40% equity vs. BU’s range,
calling could be more +EV than going all-in.
STRATEGY FOR BB
 If our hand has more than 50% equity vs. BU’s
range, then it is most likely in our interest to get
all-in instead of just calling, since it is in our
favor to put money in 1-to-1.
 If our hand has less than 50% equity vs. BU’s
range, then it is most likely in our interest to just
call, even though we have Reverse Implied Odds
postflop from being out of position.
 Even if our hand has less than 40% equity vs.
BU’s range, as long as it has more than ~30%
equity, it is probably +EV to call given 4.5-to-1
odds to see the flop.
BIG LESSON
 None of these opportunities for the BB would’ve
arisen had the BU just went all-in in the first
place!
 The BB would’ve just had to call the hands with
40% equity, and fold the rest.
 But now, we’re allowing the BB to still get the
money in when he has 50%+ equity, play any
hand with ~30% equity, etc.
WHAT ABOUT RAISING TO 4X?
ANALYSIS
 If the BB were to fold, then raising 4x is
equivalent to going all-in.
 If the BB were to re-raise all-in himself, then
raising 4x is equivalent to going all-in (since we
are going to call his re-raise).
 Are there situations where the BB is better off
calling, and folding some flops? Probably not,
since the pot will be 1880 on the flop where each
player has 800 left. So the BB calling is
equivalent to the BB re-raising all-in, since no
player is ever folding postflop.
SUCKER RAISE
 However, the analysis for the last bullet is hard.
Maybe there are some opportunities for the BB to
squeeze extra EV with a few of the hands in his
range, but just calling preflop.
 Going all-in gives the BB strictly fewer options,
so it is the “right” play.
 For raising to 2x or 3x or 4x to be a better play
than going all-in, the BB has to be SO STUPID
that having more options will somehow cause
him to make a worse decision. This is the same
concept as what Bill Chen called a “sucker bet” in
his lecture.
CHECKPOINT
 We just:
 Reviewed how to do an Expectation / Equity
calculation.
 Explained why going all-in is so profitable when both
 Everyone in front of you has folded
 You are risking less than 15BB

 Explained why raising to 2.25BB as a cowardly

alternative, when the above two factors are present,
is nowhere near as profitable
 Give the other plays more options
 Allow them to play a larger % of hands, instead of having to
fold everything
AN IN-DEPTH COMBINATORIAL HAND
ANALYSIS
CALLING FOR IMPLIED ODDS IN POSITION
BB ALSO CALLS.
CO CONTINUATION BETS
WE CALL WITH OUR MEDIUM STRENGTH
HAND
ANALYSIS
 Definitely not folding, and raising is suicidal
when we have a medium-strength hand that
could play well in position on a lot of turns. We
have a back door flush draw, and our straight
draw is very legitimate when there is no flush
draw out there.
 We must be wary of a Big Blind check-raise,
although this is less likely when a flush draw
isn’t out.
BB FOLDS
HE BETS 30 ON TURN
ANALYSIS
 Benefits of Calling:
 See the river for sure, instead of having to fold to his turn
3-bet all-in. We really want to see the river because we
have plenty of opportunities to improve, and already have
a pair to start with, so will win the hand by checking
sometimes.
 Benefits of Raising:
 Can get him to fold better hands, say JJ.
 Can bet the river and win a bigger pot when hit a club or
an 8. (It is hard to do this when we hit a 9 or 7, since in
both cases there is 4-to-a-higher-straight.)
 I like calling in this situation. With a draw that
cannot win showdown like 95cc, I would raise. Or
with a draw that is good enough to call his turn 3-bet
all-in, like AJcc, I would raise.
WE CALL.
HE CHECKS RIVER.
WE BLUFF THE RIVER!
ANALYSIS
 Yes, we do have some chance of winning with our
pair of 8’s, so there may be no need to bluff. But:
 There are a lot of higher cards on the board now. The
pair of 8’s that looked pretty good on the flop is a lot
worse after a Q turn and A river.
 The pot is big; money went in on both the flop and
turn. If our opponent has a Ten or Queen or even
weak Ace, he will fold to our bet.
 The Ace is always a scary card.
 The advantage of being in position is that we already
know our opponent checked the river. So he is
unlikely to have a great hand, and more likely to fold
to our bluff.
CO FOLDS.
EXACT ANALYSIS
 What range of hands do we put our opponent on
exactly?
 Obviously this is impossible to answer exactly,
the entire purpose of poker is to approximate this
as well as possible.
 To do this, we must put ourselves in our
opponent’s shoes.
WHAT IS OUR OPPONENT’S RANGE HERE?
ANALYSIS
 Usually, trying to put him on exact hands preflop
is difficult, especially when he opens as late as
the Cutoff.
 Estimating that he opens somewhere around 33%
is probably reasonable. This includes pair, any
suited ace, any two broadway cards, but not
unsuited hands, and stuff like T8o.
WHAT IS OUR OPPONENT’S RANGE HERE?
ANALYSIS
 Our opponent continuation bet.
 What do we put him on? Let’s look at what he
would and wouldn’t continuation bet.
 Let’s review the factors of when a player would
continuation bet. We talked about this when we
introduced continuation betting, and also when
we talked about playing draws postflop.
RULES FOR CONTINUATION BETTING
 Incentives for Continuation Betting:
 Your hand is good enough that it beats even most of his
calling hands.
 Your showdown value is poor, but you have some equity (or
back door equity).
 You are out of position and cannot see a free turn by
checking.
 Few opponents (1 or 2).
 Incentives against Continuation Betting:
 Your hand is so dominant that you need to give him a turn
card to hope that he improves.
 You have decent showdown value.
 You have zero equity because your hand is just so terrible.
 You are in position and can see a free turn by checking.
 There are too many people in the pot to get them all to fold.
THE SITUATION AGAIN…
ANALYSIS
 Hands that he checks to trap or check-raise:
 97, TT, maybe 88 (unlikely because we have an 8)
 Hands that he value bets:
 AA-JJ, 66, T8, 86, T6, AT, KT, etc.
 Hands of medium-strength that he check-calls:
 JT, T9, 99, 77, T7, 76, etc.
 Hands that he “bluffs”:
 A6, K8, AK, AJ, Q9, QJ, J7, etc.
 Hands that he check-folds because they have zero
equity:
 Q5, 33, A2ss, etc.
“BAYESIAN UPDATING”, OR MORE
“BAYESIAN ELIMINATING”
 Hands that he checks to trap or check-raise:
 97, TT, maybe 88 (unlikely because we have an 8)
 Hands that he value bets:
 AA-JJ, 66, T8, 86, T6, AT, KT, etc.
 Hands of medium-strength that he check-calls:
 JT, T9, 99, 77, T7, 76, etc.
 Hands that he “bluffs”:
 A6, K8, AK, AJ, Q9, QJ, J7, etc.
 Hands that he check-folds because they have zero
equity:
 Q5, 33, A2ss, etc.
THE SITUATION ON THE TURN
HIS INCENTIVES FOR BETTING AGAIN ON
THE TURN

 You bet a good hand for value on the flop, and

your hand is still good on the turn. You’re
betting for more value.
 You bluffed a speculative hand on the flop, and
now either hit the turn or improved your draw.
 An overcard to the board came, so you thought
bluffing the turn could be good.
THE SITUATION AGAIN…
RIVER SITUATION
HE CHECKS RIVER…
 It seems improbable that our opponent would be
trying to trap us with his check:
 He’s already been showing plenty of aggression, so
it’s not like he can pretend he has a bad hand by
checking.
 With the pot already so big, he can get a large % of
the remaining stacks by just betting (instead of
check-raising).
 The Ace is not a good card for our range, so not a card
we’re likely to bet.
 So it’s unlikely that he has a nutty hand.
 However, he could very easily still have a hand
that beats us and calls our potential bluff.
ANALYSIS ON BOARD…
 We conclude that his range on the river will consist of
 AKs, AJs, KQs (that had backdoor flush draw on flop)
 QJ, Q9s, KK
 A9, T8s, 86s

 Assuming that he will call with exactly top pair or

better.
 We are risking 70 to win 100.
 Since he calls about 50% of the time, the bluff is
profitable.
 Note that we are only winning once in a blue moon by
checking, in the rare case he has K9 or something
weird like J7s, 74s and decided to not bluff the river.