Avoiding getting freerolled on the river
Abdul Jalib
A fairly common situation in a heads up river situation in Hold'em
(and other poker games) is to feel like it's a toss up whether you
beat your opponent or not. From the action, you're sure that if
you bet, your opponent will just call (or if he raises, you will be
sure you are beat and will then fold.) If you check, your hand is
good enough that you will call your opponent if he bets. This type of
situation arises often with overpair versus overpair or top pair
versus top pair. First to act, should you check or bet?
There are two diametrically opposed considerations here: you wish
to induce a bluff (or value bet) from weaker hands and you wish
to not be freerolled. You get freerolled if your opponent will tend
to check hands weaker than yours and bet hands better than yours.
Obviously, if your opponent is very aggressive, you'll tend more
towards checking, while if your opponent is conservative, or if you
think he thinks you have about the hand you do, then you should bet
to avoid being freerolled. I'm going to argue that very often you
want to be betting here to avoid that freeroll, even if you are an
underdog to win the pot if called! (But in slightly different
circumstances, you might lean towards inducing a bluff by checking.)
We're making some assumptions here - that neither you nor your
opponent is going to fold or try any tricky raises. Given this,
the math is relatively simple. The pot size is irrelevant(!) Suppose
p is the probability that you have the best hand. Then your expected
value for betting one big bet is:
p-(1-p)=2p-1
Now if you check and your opponent checks, then that's zero e.v.
If you check and your opponent bets, then your chance of winning
is p|(opponent bet) or let's just call it p' (p-prime.) Now your
expected value is:
p'-(1-p')=2p'-1
If your opponent bets with probability b, then your overall
expected value if you check is:
(1-b)0+b(2p'-1)
So the point you are indifferent about checking or betting is:
2p-1=b(2p'-1)
Let's play with this with some values for p,b, and p'. If your
opponent is likely to bluff or bet for value overaggressively,
then you might have p=.5, b=.9, p'=.6. Here betting has an
expected value of zero, whereas checking has an expected
value of 0.18 big bets. On the other hand, if your opponent is
more conservative or knows you are going to call, then you might
have p=.5, b=.2, p'=.2. Here betting has an expected value of
zero again, whereas checking has an expected value of -.12. By
betting you avoid that freeroll where your opponent bets mostly
hands that beat you and checks mostly the hands you beat.
Note that if b=.2 and p'=.2, you'd want to bet even when you're
almost a 2p-1=-.12 => p=.44 => 8% underdog to win the pot if you
bet.
So when you get into that situation on the river where you and your
opponent know that you each have top pair or overpairs and you are
a little unsure as to who has the other one notched, you might
wish to go ahead and bet on the river in first position, even if
you feel like your opponent is a bit more likely to have the better
hand.
The logic in a nutshell is:
If your opponent knows you're going to call, it makes no
sense for him to bluff or overaggressively value bet, as he has
no chance to win the pot with the worst hand... so don't expect him
to bluff or aggressively bet there... and so you should likely bet
to protect yourself from getting freerolled.
If the second player might raise with a worse hand, that opens up a
whole other can of worms, so be careful.
I don't know if I did the math correctly or the best way, but
I'm confident of the logic.
Comments?
--
Abdul Jalib | Help in the fight against Doug Grant spam from
| dgrant1tm/oghobo/cass...@aol.com and others, see
abd...@earthlink.net | http://home.earthlink.net/~abdulj/dg-index.html
Erik Reuter
> We're making some assumptions here - that neither you nor your
> opponent is going to fold or try any tricky raises.
...
> Let's play with this with some values for p,b, and p'. If your
> opponent is likely to bluff or bet for value overaggressively,
> then you might have p=.5, b=.9, p'=.6. Here betting has an
> expected value of zero, whereas checking has an expected
> value of 0.18 big bets.
So when you bet you have the best hand 50% of the time, when you check and
your opponent bets (90% of you checks) you have the best hand 60% of the
time, hence when you check and your opponent checks behind you (10% of
your checks) then you have the best hand -40% of the time? Seems to be a
problem here. Even if the numbers were modified for consistency, I can't
really understand this 90% betting opponent with p' > p . Suppose you win
52% of the times he bets, then you must only win 32% of the time he checks
behind you. In other words he is intentionally betting his bad hands and
not betting his good hands. Sounds like a great opponent to play against!
> On the other hand, if your opponent is
> more conservative or knows you are going to call, then you might
> have p=.5, b=.2, p'=.2. Here betting has an expected value of
> zero again, whereas checking has an expected value of -.12. By
> betting you avoid that freeroll where your opponent bets mostly
> hands that beat you and checks mostly the hands you beat.
At first I was bothered by the fact that when your opponent bets, you call
even though you are a big underdog, but if the pot is giving you 4:1 odds
you have to call anyway, so this actually makes sense. What still bothers
me is that your opponent seems to know 20% of the time that he has a
pretty good hand, since he bets and wins 80%. Is your opponent more
knowledgeable than you, or do you also have an idea of your top 20% hands?
If you know this, then you should bet those 20% and check the other 80% to
obtain an overall EV of at least 0 (0 if he always bets when you check,
more than 0 if he sometimes checks after you check). If you don't know,
then it seems this strategy is useful against an opponent who can read
hands better than you, but not against an equal or inferior opponent.
--
Erik Reuter, e-re...@uiuc.edu
Abdul Jalib
Erik Reuter wrote:
> [Abdul wrote:]
> > Let's play with this with some values for p,b, and p'. If your
> > opponent is likely to bluff or bet for value overaggressively,
> > then you might have p=.5, b=.9, p'=.6. Here betting has an
> > expected value of zero, whereas checking has an expected
> > value of 0.18 big bets.
>
> So when you bet you have the best hand 50% of the time, when you check and
> your opponent bets (90% of you checks) you have the best hand 60% of the
> time, hence when you check and your opponent checks behind you (10% of
> your checks) then you have the best hand -40% of the time? Seems to be a
> problem here.
No, it was intentional. The type of opponent I'm modelling here is the
overaggressive one who winds up on the river with nothing or a very weak
hand and who will make one final stab to win the pot with nothing (while
checking down a lot of his better hands.)
> Even if the numbers were modified for consistency, I can't
> really understand this 90% betting opponent with p' > p . Suppose you win
> 52% of the times he bets, then you must only win 32% of the time he checks
> behind you. In other words he is intentionally betting his bad hands and
> not betting his good hands.
Don't you do this to some extent? Bluff with missed hands, check your
made weak hands? But in this particular case, this type of play makes no
sense, at least if your opponent is sure you are going to call.
>Sounds like a great opponent to play against!
Yes, it is, too good, which is why I'm suggesting that in this circumstance
you may need to bet to avoid being freerolled, rather than checking to
induce a bluff where your opponent is not likely to bluff.
> > On the other hand, if your opponent is
> > more conservative or knows you are going to call, then you might
> > have p=.5, b=.2, p'=.2. Here betting has an expected value of
> > zero again, whereas checking has an expected value of -.12. By
> > betting you avoid that freeroll where your opponent bets mostly
> > hands that beat you and checks mostly the hands you beat.
>
> At first I was bothered by the fact that when your opponent bets, you call
> even though you are a big underdog, but if the pot is giving you 4:1 odds
> you have to call anyway, so this actually makes sense. What still bothers
> me is that your opponent seems to know 20% of the time that he has a
> pretty good hand, since he bets and wins 80%. Is your opponent more
> knowledgeable than you, or do you also have an idea of your top 20% hands?
> If you know this, then you should bet those 20% and check the other 80% to
> obtain an overall EV of at least 0 (0 if he always bets when you check,
> more than 0 if he sometimes checks after you check). If you don't know,
> then it seems this strategy is useful against an opponent who can read
> hands better than you, but not against an equal or inferior opponent.
My idea, and perhaps it is flawed, is that you give information to your
opponent if you check. If you check, your opponent can be fairly sure
that you do not have a very strong hand, and hence he is free to bet more
of his hands. Also, I was modelling a very conservative opponent here,
one that would only bet if he was fairly confident he had you beat.
Let's do a full scenario here.
You raise under the gun with QQ. A tight player in middle position
who respects your early raises reraises you. You know he would only do
this with AA, KK, QQ, JJ, TT, or AKs. The flop comes 752 offsuit. You bet,
he raises, you call. The turn is an offsuit 4. You bet, he raises, you call.
The river is a 5. Given this action, board, and knowledge of your opponent,
you can be nearly sure that he has AA, KK, QQ, JJ, or TT, and he probably
puts you on the same group of hands since he respects your play. If you
check, he's not going to bet his JJ or TT, but he will bet his AA, KK,
or QQ. If you bet, he's just going to call with KK, QQ, JJ, or TT. With
AA, he'll either raise or call, and if you feel comfortable about folding
to his raise, then his raise is no problem. So, in this scenario, betting
out seems to save you a lot of money (half a big bet), and you'd still do
it even if you thought TT was a bit less likely than those other hands.
On the other hand, if your opponent is the type who would three bet you
with KQs, raise the flop with two overcards, raise the turn with two
overcards, and make a final stab at the pot if you check to him on the
river, then obviously you'd be inclined to let him make a foolish bluff.
But I find that even fish are rarely this stupid. You've shown strength
preflop, on the flop, and on the turn, so by this point almost any
opponent will give you credit for having a hand that you'll call with.
Thanks for the comments. I'm not particularly argueing with them. My
article was somewhat crippled by using bogus arbitrary numbers, and you're
right to question them. Perhaps a game theoretic solution would be less
arbitrary and more insightful.
--
Abdul Jalib |
| A chip and a terminal...
abd...@earthlink.net |
Erik Reuter
In article <33AFF5...@earthlink.net>, abd...@earthlink.net wrote:
> Erik Reuter wrote:
> > Even if the numbers were modified for consistency, I can't
> > really understand this 90% betting opponent with p' > p . Suppose you win
> > 52% of the times he bets, then you must only win 32% of the time he checks
> > behind you. In other words he is intentionally betting his bad hands and
> > not betting his good hands.
>
> Don't you do this to some extent? Bluff with missed hands, check your
> made weak hands?
Not in the situation you described.
> But in this particular case, this type of play makes no
> sense, at least if your opponent is sure you are going to call.
Even if your opponent isn't sure you are going to call, his strategy is
irrational. Maybe he is crazy enough to bet the bottom 50% of his hands,
as an impossible bluff, after you check. But why would he check his
stronger hands after you check? The only reason to do this would be fear
of a check-raise, but that was ruled out by your initial assumptions of no
raising and no folding. A key is idea is that the opponent MUST have some
idea of the strength of his hand since he checks with a stronger set of
hands than he bets with.
> Yes, it is, too good, which is why I'm suggesting that in this circumstance
> you may need to bet to avoid being freerolled, rather than checking to
> induce a bluff where your opponent is not likely to bluff.
Do you mean the example of the conservative player? The irrational
opponent discussed above is much better to check to, EV wise. He is polite
enough to bet your hand for you when you are favored, and check for you
when you're the underdog.
> You raise under the gun with QQ. A tight player in middle position
> who respects your early raises reraises you. You know he would only do
> this with AA, KK, QQ, JJ, TT, or AKs. The flop comes 752 offsuit. You bet,
> he raises, you call. The turn is an offsuit 4. You bet, he raises, you call.
> The river is a 5. Given this action, board, and knowledge of your opponent,
> you can be nearly sure that he has AA, KK, QQ, JJ, or TT, and he probably
> puts you on the same group of hands since he respects your play. If you
> check, he's not going to bet his JJ or TT, but he will bet his AA, KK,
> or QQ. If you bet, he's just going to call with KK, QQ, JJ, or TT. With
> AA, he'll either raise or call, and if you feel comfortable about folding
> to his raise, then his raise is no problem. So, in this scenario, betting
> out seems to save you a lot of money (half a big bet), and you'd still do
> it even if you thought TT was a bit less likely than those other hands.
This is a great example. I'm going to have to think about it some more,
but look at your EV when first to act with various hands against the given
opponent:
Your EV EV
Hand you bet you check
AA +0.96 +0.52
KK +0.48 0
QQ 0 -0.48
JJ -0.48 -0.72
TT -0.96 -0.72
You should bet your best hand, AA, and check your worst hand, TT. But
surprisingly, you should bet all the others, even with JJ. I think this is
the point you were trying to make: you should bet a below-median hand like
JJ even though you are an underdog and know your opponent will call. This
is because if you check, your opponent will bet when he is ahead and check
when he is behind. If you bet, you still lose when your opponent is ahead,
but those times when your JJ beats his TT you make a little EV consolation
prize.
This situation seems familiar, did Sklansky deal with anything like this
in Theory of Poker?
One thing that bothers me: if you have JJ or TT, and you know your
opponent is conservative (only betting QQ, KK, AA), you should check and
fold to obtain a 0 EV (your JJ or TT have precisely 0 chance of winning
when he bets, so the pot is irrelevant). Maybe to make your model more
realistic, you need to allow for your opponent to be somewhere between the
crazy one and the conservative one: betting good hands after you check but
occasionally betting a bad one so you can't fold. To take this much
further, it will be necessary to use game theory, as you say.
--
Erik Reuter, e-re...@uiuc.edu
Erik Reuter
In article <e-reuter-250...@yoda.ccsm.uiuc.edu>,
e-re...@uiuc.edu (Erik Reuter) wrote:
> This situation seems familiar, did Sklansky deal with anything like this
> in Theory of Poker?
I checked _ToP_ and there is a long and complex treatment of this
situation, over 3 pages. The subsection is about first to act with an
underdog hand (another subsection for the favorite is there, too) in the
chapter about "head's up on the end".
The discussion, while thought provoking and worth reading, seemed tedious.
There were no equations. I think I can summarize 3 pages of prose in a few
equations (and some definition of terms):
You're first to act. No raising. Choose the action with highest EV:
p = pot size before you act, in big bets
x = probability your opponent calls your bet
y = probability your opponent bets after you check
a = probability you win given that you both check
b = probability you win given that your opponent called your bet
c = probability you win given that you checked and your opponent bet
EV_bet = x (2 b - 1) + p (x (b - 1) + 1)
EV_check = y (2 c - 1) + p (y (c - a) + a)
EV_fold = 0
This certainly isn't easy to apply on the fly in a game. But the equations
should be useful for analyzing situations away from the table. If you do
the math for the common situations now, you can then be prepared when they
come up in the game.
--
Erik Reuter, e-re...@uiuc.edu
HitTheFlop
e-re...@uiuc.edu (Erik Reuter) writes:
>This certainly isn't easy to apply on the fly in a game. But the
equations
>should be useful for analyzing situations away from the table. If you do
>the math for the common situations now, you can then be prepared when
they
>come up in the game.
>
>
There are many problems that lend themselves to detailed analysis
away from the table. My problem is that any equation with 3 variables
or more is likely to be of little value in actual play. I don't mean to
sound
like a Philistine, after all I enjoy a good math buzz as much as the next
guy, but poker is such a varied and inexact game that most complex
mathematical analysis is wasted. The time needed to probe a situation
deeply on the fly just isn't part of the game as I know it. I'll admit
that
there are a very few occasions where Baye's Theorem has led me to
the correct conclusion on the fly but even this simple tool is of value
only rarely. I don't believe I'll ever use game theory during play but
will
use it in a general sense to adjust my overall frequencies. Does any
member of this NG have a good first hand example of complex analysis
during the play of a hand leading to success? Please keep in mind that
my comments here are specific to limit poker only. I have not outlet
for regular ring play at pot limit or no limit.
Best Luck
Ed
Erik Reuter
In article <19970626031...@ladder01.news.aol.com>,
hitth...@aol.com (HitTheFlop) wrote:
> The time needed to probe a situation
> deeply on the fly just isn't part of the game as I know it.
Time is relative! Probe deeply before takeoff! Fly fast and shallow! Trust
your feelings! Use the force!
> I enjoy a good math buzz as much as the next guy
Here's a chaser for the last one, same taste and less filling:
p = pot size before you act
u = the amount by which the fraction of hands with which opponent calls and
loses exceeds the fraction opponent calls and wins
v = the amount by which the fraction of hands with which opponent bets and
loses exceeds the fraction opponent bets and wins
w = fraction of hands opponent loses if you bet
z = fraction of hands opponent loses if you check
EV_bet = u + w p
EV_check = v + z p
EV_fold = 0
--
Erik Reuter, e-re...@uiuc.edu
Erik Reuter
Here's a correction:
p = pot size before you act
u = the amount by which the fraction of hands with which opponent calls and
loses exceeds the fraction opponent calls and wins
v = the amount by which the fraction of hands with which opponent bets and
loses exceeds the fraction opponent bets and wins
w = fraction of hands opponent loses if you bet
z = fraction of hands opponent loses if you check
a = fraction of hands opponent will check and lose
EV_bet = u + w p
EV_check_and_call = v + z p
EV_check_and_fold = a p
--
Erik Reuter, e-re...@uiuc.edu
Zachariah Love
Erik Reuter wrote:
>
> In article <e-reuter-250...@yoda.ccsm.uiuc.edu>,
> e-re...@uiuc.edu (Erik Reuter) wrote:
> > This situation seems familiar, did Sklansky deal with anything like this
> > in Theory of Poker?
>
> I checked _ToP_ and there is a long and complex treatment of this
> situation, over 3 pages. The subsection is about first to act with an
> underdog hand (another subsection for the favorite is there, too) in the
> chapter about "head's up on the end".
>
> The discussion, while thought provoking and worth reading, seemed tedious.
> There were no equations. I think I can summarize 3 pages of prose in a few
> equations (and some definition of terms):
>
> You're first to act. No raising. Choose the action with highest EV:
>
> p = pot size before you act, in big bets
> x = probability your opponent calls your bet
> y = probability your opponent bets after you check
> a = probability you win given that you both check
> b = probability you win given that your opponent called your bet
> c = probability you win given that you checked and your opponent bet
>
> EV_bet = x (2 b - 1) + p (x (b - 1) + 1)
>
> EV_check = y (2 c - 1) + p (y (c - a) + a)
>
> EV_fold = 0
>
> should be useful for analyzing situations away from the table. If you do
> the math for the common situations now, you can then be prepared when they
> come up in the game.
>
> Erik Reuter, e-re...@uiuc.edu
My rule of thumbs is that I am always luckey with my 5's.
--
Zachariah Love, Commissioner
The Craig Kilborn Memorial Rotisserie League
and The Lee Atwater Invitational Dead Pool
http://home.earthlink.net/~ghicks/