Sklansky's hand revisited
Howard Lederer
limit Bellagio player, and he came up with a formula to determine player A's
bluffing frequency. To review the situation, assume there are N bets in the
pot and the bet size on the river is 1. Player A has the best hand going in
and Player B will draw to beat Player A's current hand 50% of the time.
But, Player A will re-draw and lock Player B out 50% of the time. This
makes Player A a 75% favorite to end up with the best hand. The cards have
been exposed, but the river will be dealt down. So both player's know the
current situation.
In Sklansky's problem, N was very large, and I feel that understanding the
proper strategies in this situation can be applied to many other real world
poker situations. To review the earlier discussion on this problem, Google
search my thread titled, "Sklansky asked me this question." By using
Warren's formula it is now easy to determine the proper strategy given pot
size N where N is a more normal number.
Warren's formula is this: 1/(4N+2). This will give you the total fraction
of the time Player A should bluff (betting after not improving) given all
outcomes. Once you get the bluff frequency, you can then construct a total
strategy that maximizes your equity in the pot. Remember your overall
strategy should have you checking after missing exactly twice as often as
you check with the intention of check-raising. I will choose three
different pot sizes and give you a chance to solve for the total final round
betting strategies. You should determine how often you bluff, you bet after
improving, you check after missing, and you check after improving. The
first two should offer some insight into common poker situations, the third
is just for fun.
Try to solve for pot sizes 9.5, 1, and 0. Remember the final betting round
is 1.
Howard Lederer
tadperry
news:T9tN9.38196$6k.20...@news1.west.cox.net...
> I discussed Sklansky's theoretical poker situation with Jimmy Warren, a
high
> limit Bellagio player, and he came up with a formula to determine player
A's
> bluffing frequency. To review the situation, assume there are N bets in
the
> pot and the bet size on the river is 1. Player A has the best hand going
in
> and Player B will draw to beat Player A's current hand 50% of the time.
> But, Player A will re-draw and lock Player B out 50% of the time. This
> makes Player A a 75% favorite to end up with the best hand. The cards
have
> been exposed, but the river will be dealt down. So both player's know the
> current situation.
>
> In Sklansky's problem, N was very large, and I feel that understanding the
> proper strategies in this situation can be applied to many other real
world
> poker situations. To review the earlier discussion on this problem,
> search my thread titled, "Sklansky asked me this question." By using
> Warren's formula it is now easy to determine the proper strategy given pot
> size N where N is a more normal number.
>
> Warren's formula is this: 1/(4N+2).
Howard,
Can you give the derivation of this formula?
I'm having trouble seeing why this would be the number.
Regards,
Tad Perry
Howard Lederer
1/(4(N+.5)). This might help you figure it out.
Howard Lederer
"tadperry" <tadp...@attbi.com> wrote in message
news:u8DN9.312720$GR5.1...@rwcrnsc51.ops.asp.att.net...
tadperry
> He did not tell how he arrived at the formula. But he gave it to me as:
> 1/(4(N+.5)). This might help you figure it out.
Do any of the math people reading happen to know the derivation of this
formula?
If so, please be so kind as to step forward and explain it for those of us
who find it mystifying.
Thanks in advance.
tvp
Howard Lederer
family in town for the holidays.
Question #1 asks for the proper betting strategy for a 9.5 bet pot:
Using the formula, you get the bluffing frequency at 1/(4(9.5)+4) = 1/40
We can now convert all our frequencies to x/40. Thus we make our hand
20/40th of the time and miss 20/40th of the time. We will miss and bet 1/40
times. Therefore we will miss and check 19/40 times. We also know that we
need to check with the intention of check-raising half as often as we check
after missing. Thus we will check-raise 9.5/40 times. This leaves us
betting after making our hand 10.5/40 times. To review:
1/40 Bluff
19/40 Check after missing
9.5/40 Check-raise
10.5/40 Bet after making
-------
40/40
We end up with the best hand 75% of the time. So, if there was no betting,
our equity in the pot would be 7.125 bets. Let's see what using this
strategy does to our equity. Player B could use a number of strategies,
though they should not effect Player A's equity. We can easily look at two
of his simple strategies.
Strategy 1 would have Player B call every time we bet and he improves, fold
when we bet and he doesn't improve, and check when we check. Remember
Player A has committed to his strategy so he can't take advantage of Player
B's simple plan. Using strategy 1 Player B will still win exactly 25% of
the pots, so pre-river betting equity stays the same. But, Player A will
bet after improving .2625 of the time and get called half of those times.
He will also bluff .025 of the time and get caught half of those times.
This means Player A will reap a positive EV of (.2625/2)-(.025/2) = .119
bets on the river. We add this to our pre-river betting equity and we get
7.244 bets out of the pot.
Strategy 2 calls for Player B to fold whenever Player A bets and check
whenever Player A checks. Using this strategy Player B will lose 20/40
times when Player A improves. He will lose 1/80 times that player A bluffs
and Player B folds the best hand. He will lose 1/80 times that Player A
bluffs and Player B does not improve. He will also loose 9.5/40 times when
Player A checks and Player B misses. He will win 9.5/40 times when Player A
misses and checks and Player B improves. This means he will win only 9.5/40
times, but he will not lose any money on the river betting.
Let's see what this does to Player A's pre-river betting equity. He now
wins the 9.5 bets 30.5/40 times which makes his equity in the pot 7.244
bets. It is no surprise that this equals Player A's equity if Player B
calls all bets. He just gets his value by winning more often, instead of
earning on the river betting.
I find it instructive to think about this simple hand. I am often surprised
to see how seldom you should bluff in certain situations. In a real life
hand like this, I would find it hard to imagine a player actually bluffing
only 1/40 times. This tells me that folding on the end in limit, when you
have chance to win, is a losing proposition. I am sure that in general
poker players bluff way too often in limit. But, conversely players fold
too much, thus justifying the high bluffing frequencies. Tackling a problem
like this gives me renewed conviction that my stubborn ways on the river are
correct.
Will spending the time to solve this problem prepare me for when this exact
same hand comes up in the future? No, but it will make it more likely that
my on-the-fly decisions at the table will be correct when a similar hand
does come up.
Howard Lederer
"Howard Lederer" <how...@lvcm.com> wrote in message
news:T9tN9.38196$6k.20...@news1.west.cox.net...
Scott N
Very interesting stuff, definetly enlightening and worth more
scrutiny/debate. Seems that this can only apply to certain types of
games (tight/high-limit?) though?
But, keeping track of where you are in the trials is unfeasible. How is
someone supposed to know where he is at in regards to what he should do?
Meaning, in your example you gave 40 hands as the trial. Saying Bluff
once, check after missing 19 times, etc. So in order to fully utilize
the advantages, you'd have to remember what you have done so far? Like
"Ok, this situation has come up for me 8 times already and I already
used my Bluff and missed twice and checked, etc. So lets throw a bet in
here this time and use one of my 10.5".
Or is all this geared to what you pointed out in the end of your post?
To give you a feel of proper strategy at the moment in accords to the
situation curently at hand? Fine-tuning your "on-the-fly" decision
making process.
Scott N
Howard Lederer
1/((4*1) + 2) = 1/6. So, we will bet after missing 1/6 times, and check
after missing 2/6 times. We need to check raise half as often as we check
after missing, so we check raise 1/6 times and bet after making 2/6 times.
To review:
1/6 Bluff
2/6 Check after missing
1/6 Check raise
2/6 Bet after making
---
6/6
Please also notice that we bet with the best hand 2 times and bluff once.
Since we are laying Player B 2-1 when we bet, this makes his decision
irrelevant when we bet.
If Player B uses strategy 1 from the previous post, he will win the pot 25%
of the time. This makes Player A's pot equity .75 bets. But Player A will
bet with the best hand 1/3rd of the time and get called 1/6th of the time.
He will also bluff 1/6th of the time and get caught 1/12th of the time.
This makes his river betting equity (.333/2)-(.167/2) = .083. And it makes
his total equity .750 + .083 = .833.
If player B uses strategy 2, he will only win half the time when Player A
checks after missing. But he won't lose anything during the river betting.
This means that Player A will win 5/6ths of the time. Which makes his
equity .833.
You might ask why this hand is relevant. Certainly a pot with 1 bet in it
with a betting limit of 1 never comes up in limit poker. True, but think
about pot-limit or no-limit. In pot-limit and no-limit, betting the pot is
a standard bet on the end. If there is $200 in the pot and you bet $200 on
the river, we now have the 1-1 pot size to river bet ratio we find in this
example.
We can learn a lot by examining the proper strategy in this hand. This hand
teaches us that we should be bluffing about 1/3rd of time we bet the pot in
pot-limit. Also, you should notice that having the dominant hand in
pot-limit is worth a lot more than in limit. Player A's river action
increases his equity by more than 10%, from .75 to .833. Where as in the
previous 9.5 bet example, he increases is equity by only about 1.5%, from
7.125 to 7.244.
Unlike limit, I don't think players are bluffing enough in situations like
this. Also, if you were playing pot-limit, you would not need to
check-raise as often, as the punishment when you successfully check-raise in
much greater in pot-limit than in limit.
I have found David's problem to be quite ingenious. It informs us on so
many levels, many of which, I'm sure, were unintended by David himself.
Howard Lederer
"Howard Lederer" <how...@lvcm.com> wrote in message
news:T9tN9.38196$6k.20...@news1.west.cox.net...
Howard Lederer
actions. Let's say you miss. You now have the number generator pick a
number from 1-40. If it picks 1, you bluff, otherwise you check. Now in
real life this is not feasible. But, you can use the cards to give you a
bluff frequency. Let's say you are playing lo-ball, and you will make the
winning hand 33% of the time. You might want to bluff 4% of the time. You
could then tell yourself that you will bet if you make your hand, and bluff
if you hit a red king. This would give about the right bluffing frequency.
Of course, you should always modify your strategy for any given situation.
If your opponent is easy to bluff, you should increase your bluff frequency.
If he is a calling station, bluff less often. If you are losing and have
bad table image, you should definitely slow down on the bluffing. Figuring
out these thing away from the table only gives you guidance on how to play
the hand. You must still play poker, where the psychology of that
particular hand will dictate your play more than the numbers.
Howard Lederer
"Scott N" <bigs...@linuxmail.org> wrote in message
news:Xns92EEF362251Dbi...@68.6.19.6...
Howard Lederer
> 1/40 Bluff
> 19/40 Check after missing
> 9.5/40 Check-raise
> 10.5/40 Bet after making
> -------
> 40/40
You should also notice that we are only bluffing 1 time for every 10.5 times
we bet after improving. And, since Player B is getting 10.5/1 after our
bet, he has no defense to the bluff.
Howard Lederer
Howard Lederer
> In theory, you could use a random number generator to produce your proper
> actions. Let's say you miss. You now have the number generator pick a
> number from 1-40. If it picks 1, you bluff, otherwise you check.
Oops! That should read "a number from 1 to 20" as you are bluffing 1 out
every 20 times you miss.
Howard Lederer
Scott N
> In theory, you could use a random number generator to produce your
> proper actions. Let's say you miss. You now have the number
> generator pick a number from 1-40. If it picks 1, you bluff,
> otherwise you check. Now in real life this is not feasible. But, you
> can use the cards to give you a bluff frequency. Let's say you are
> playing lo-ball, and you will make the winning hand 33% of the time.
> You might want to bluff 4% of the time. You could then tell yourself
> that you will bet if you make your hand, and bluff if you hit a red
> king. This would give about the right bluffing frequency.
>
This is Game Theory.
Someone able to perfectly implement Game Theory into Poker is
unbeatable.
(hit upon lightly by Sklansky in some of his material and a few posts
here in the past)
> Of course, you should always modify your strategy for any given
> situation. If your opponent is easy to bluff, you should increase your
> bluff frequency. If he is a calling station, bluff less often. If you
> are losing and have bad table image, you should definitely slow down
> on the bluffing. Figuring out these thing away from the table only
> gives you guidance on how to play the hand. You must still play
> poker, where the psychology of that particular hand will dictate your
> play more than the numbers.
Well Said.
You are on the path of new discovery I think. The one where most high
level thinkers playing the game of poker come upon. Game Theory applied
to Poker.
I like your explanations though and your ideas posted in the parent
post. But the real breakthroughs I think are being able to put this all
into play integrated together to produce strategy way advanced of 99.9%
of other players. Just imaging being able to tie all this together in a
system in your mind. The Game Theory stuff, the reading players, the
psychology, the card math, etc. A system that would allow you to explain
and give a reason exactly for any course of action. Possible? Maybe.
(But can Poker ever be "solved" you think to a point of killing it?
Hmm..)
Howard Lederer
Wait, something must be wrong. Jimmy's formula tells us to bet every time
we miss! And since we never check after missing, we never have to
check-raise. So a proper betting strategy would be to bet 100% of the time!
Sadly, since there is no money in the pot, Player B can safely fold 100% of
the time, and not worry about whether or not he got bluffed. ;-). What does
this example teach us? Almost nothing. But, it does show that Jimmy's
formula does work, even when the pot size going in is zero.
Howard Lederer
"Howard Lederer" <how...@lvcm.com> wrote in message
news:T9tN9.38196$6k.20...@news1.west.cox.net...
Scott N
Noticed but forgiven.
Just kidding. Some very poor late night humor...
Howard Lederer
> Well Said.
> You are on the path of new discovery I think. The one where most high
> level thinkers playing the game of poker come upon. Game Theory applied
> to Poker.
>
> I like your explanations though and your ideas posted in the parent
> post. But the real breakthroughs I think are being able to put this all
> into play integrated together to produce strategy way advanced of 99.9%
> of other players. Just imaging being able to tie all this together in a
> system in your mind. The Game Theory stuff, the reading players, the
> psychology, the card math, etc. A system that would allow you to explain
> and give a reason exactly for any course of action. Possible? Maybe.
I have been trying to do this for many years now. It is why I will never
get bored playing this game. Trying to perfectly put ALL of these things
together is probably beyond human capability. I recently had an interesting
discussion with Gus Hansen (the winner of the $10,000 Bellagio tournament).
He firmly believes that poker is in its infancy. He thinks that the play,
even in the biggest games, isn't even close to perfect. I know that he is a
deep thinker, and he is going to try to revolutionize the game in the
process.
> (But can Poker ever be "solved" you think to a point of killing it?
> Hmm..)
Gus does think that it is possible to program a computer that could trounce
any human at heads-up hold-em. I tend to agree, but I argued that the
complexity would spiral out of control as soon as you moved to ring game
poker. Even if we could program the perfect poker computer, I would argue
that poker, unlike chess and backgammon, would lose its basic elements of
emotion and tells when played by computers. I don't think that poker
without tells is truly poker.
Howard Lederer
Lou Krieger
>> "Howard Lederer" <how...@lvcm.com> wrote in message
news:fKXN9.155958$6k.26...@news1.west.cox.net...
Gus does think that it is possible to program a computer that could trounce
any human at heads-up hold-em. I tend to agree, but I argued that the
complexity would spiral out of control as soon as you moved to ring game
poker. Even if we could program the perfect poker computer, I would argue
that poker, unlike chess and backgammon, would lose its basic elements of
emotion and tells when played by computers. I don't think that poker
without tells is truly poker.
Howard Lederer <<
I agree completely. Against a single player a computer could play optimally
using game theory to make decisions such as those pointed out in Howard's
earlier posts. But the moment a suboptimal player is introduced into the
game, everything changes. For example, if a third player who is a real
calling station enters the game, then the computer's ability to bluff is
defeated, and if a maniac with an unlimited bankroll joins the game as the
fourth player, the computer must make other adjustments. It is posible to
have a mix of players in the game such that the best strategy is simply to
"win more with your best hands, lose less with your bad ones."
So how is that achieved? In many cases it comes down to reading opponents
and gathering tells along the way. While a computer could learn to read
opponents through statistical analysis, this would be a formative process
and probably take some time to fine tune to the point that the computer was
dialed in on each player at the table. And even something so simple as a
seat change -- for example, moving from the maniac's right to his left --
would change the mover's playing proclivities and the computer would have to
zero in on that player again. As for reading tells through verbal and
visual cues? I think we're a long, long way from a computer being able to
handle that.
Thanks to Howard for a series of thought-provoking posts. This is an example
of RGP at its best, and why, even with the current signal to noise ratio on
this NG, it's still worth reading.
Paul L. Schwartz
I am more a less a lurker here but I like math problems so here you
go:
Let:
x = overall checking frequency (missed hand)
y = overall check raise frequency (made hand)
z = overall value bet frequency (made hand)
w = overall bluff frequency (missed hand)
N = pot size
Note: The term overall is used above to indicate that the respective
frequencies are over all hands. That is w is the fraction of all
hands that are bluffed, not the fraction of bets that are bluffs.
The goal is to solve for w in terms of N.
The odds offered on a bet are N+1:1
The correct bluffing frequency is such that the odds against a bet
being a bluff are equal to the pot odds.
The odds against a bet being a bluff is simply the ratio of value bets
to bluffs or (z/w).
Therefore:
N+1 = z/w (eqn 1)
We also know that 50% of the time we make the hand and 50% of the time
we miss the hand so:
w+x=0.5 (eqn 2) and y+z=0.5 (eqn 3)
Additionally, we know that we should check twice as often as we
check-raise (this was discussed elsewhere) so:
x=2y (eqn 4)
Substituting for x in eqn 2 above we get:
w+2y=0.5 (eqn 5)
Equations 3 and 5 can be combined to eliminate y and solve for z to
give:
z=(w+0.5)/2 (eqn 6)
Finally equation 6 can be substituted for z in equation 1 and then
solved for w to give:
w=1/(4N+2)
Paul
Dsklansky
will usually not win as much as the best players with those same opponents and
the same cards.
Barbara Yoon
> With one bet in the pot, using Jimmy's formula we get a bluff frequency
> of 1/((4*1) + 2) = 1/6. So, we will bet after missing 1/6 times, and check
> after missing 2/6 times. We need to check raise half as often as we check
> after missing, so we check raise 1/6 times and bet after making 2/6 times.
> To review:
>
> 1/6 Bluff
> 2/6 Check after missing
> 1/6 Check raise
> 2/6 Bet after making
> ---
> 6/6
Howard.....unless I missed something previously, given that your strategy
here, for purposes of theoretical discussion, is made known to your
opponent (right?!), then because you "Check raise" only with a winning
hand, your opponent will know that, and will never call any "Check raise"
-- right?! I mean, then couldn't you just as well call that category, "Check,
hoping to sucker opponent into betting"...?!
tadperry
news:50541c61.02122...@posting.google.com...
Perfect!
Thanks.
tvp
Jerrod Ankenman
> A computer using game theory can guarantee that it has the best of it. But it
> will usually not win as much as the best players with those same opponents and
> the same cards.
Heads up only, I'm sure David meant to say.
Jerrod Ankenman
Howard Lederer
for example, that Player A announces his strategy with regards to the
initial bet first. Player B now realizes that Player A will check with
intention of check raising half as often as he checks after missing. Now
Player B decides to only call or fold. I have made no statement about
whether or not Player A will check raise bluff, as he will never face the
decision.
I made this simplification for the N = Small Number analysis. I wanted to
brake the problem down to its simplest case before opening the can of worms
that Player B betting will produce.
If you would like to analyze the proper strategies after Player B bets, I
would welcome your or anyone's input. I will take a look at this over the
next few days. I am particularly interested if Player B betting can result
in +EV and if so, does it change the proper betting strategy on Player A's
initial action.
Howard Lederer
"Barbara Yoon" <by...@erols.com> wrote in message
news:aub089$rgt$1...@bob.news.rcn.net...
Barbara Yoon
>>> With one bet in the pot, using Jimmy's formula we get a bluff
>>> frequency of 1/((4*1) + 2) = 1/6. So, we will bet after missing
>>> 1/6 times, and check after missing 2/6 times. We need to
>>> check raise half as often as we check after missing, so we
>>> check raise 1/6 times and bet after making 2/6 times.
>>> To review:
>>>
>>> 1/6 Bluff
>>> 2/6 Check after missing
>>> 1/6 Check raise
>>> 2/6 Bet after making
>>> ---
>>> 6/6
>>>
me:
>> Howard.....unless I missed something previously, given that your
>> strategy here, for purposes of theoretical discussion, is made known
>> to your opponent (right?!), then because you "Check raise" only with
>> a winning hand, your opponent will know that, and will never call
>> any "Check raise" -- right?! I mean, then couldn't you just as well
>> call that category, "Check, hoping to sucker opponent into betting"...?!
>> [and on any such bet by him, you would reap exactly the same by
>> just CALLING.]
Howard Lederer:
> In the post on the 9.5 bet hand, I do not allow Player B to bet.
Ohh.....I didn't see that (as I've focused mainly on this 'one-bet' case)...
Howard Lederer:
> I made this simplification for the N = Small Number analysis. I wanted
> to brake the problem down to its simplest case before opening the
> can of worms that Player B betting will produce.
Not to mention the '55-gal. barrell of worms' from Player A check-raise
BLUFFING...!!
Howard Lederer:
Barbara Yoon
> I do not allow Player B to bet. I made this simplification for the
> N = Small Number analysis. I wanted to brake the problem down
> to its simplest case before opening the can of worms that Player B
> bets, I would welcome your or anyone's input. I am particularly
> interested if Player B betting can result in +EV and if so, does it
> change the proper betting strategy on Player A's initial action.
Howard.....I'm not sure if I can readily prove it, but I think this 'game'
(the equivalent of each player tossing a coin that only he can see
before the betting, with 'heads' beating 'tails,' and Player A winning
all tie coin tosses.....right?!) might be just too heavily in Player A's
favor, making any betting at all by Player B 'suicidal'.....and thus,
we might really be at a game theory 'dead-end' with regard to any
further interest in this question as stated.....what do you think...?!
Tom Weideman
Perhaps, but even then the statement is living on the edge. I think David
is used to LV poker, where most or all of a player's ev comes from "bunny
bashing", i.e. taking money from truly terrible poker players. Only one or
two shockingly poor tourists are all that is required to make a full game
profitable for the pros.
If an optimal-playing computer program played heads-up against one of these
clueless players, it would certainly not perform as well as a very good
human player would in the same spot. But if the play of the opponent is
reasonably competent (say a typical break-even 30-60 player at the
Bellagio), then the computer program would fare better than even the best
human player. If I had a choice of taking an optimal heads-up program or
the human who I think is the best in the world at a game against a random
player who plays 60-120 or higher, I definitely take the computer.
Tom Weideman
Vince lepore
Sounds like you and Sklansky believe it is possible to develop an
unbeatable heads-up Holdem program. I don't. Of course Game Theory
hasn't gone to my head.
Vince
Jerrod Ankenman
>
> Vince
Contender for post of the year.
Jerrod Ankenman
'not only that, we can PROVE it.'
KrisppyKreme
<how...@lvcm.com> writes:
You people have way too much book learnin for me.
Krisppy Kreme