Game Theory

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Ken Ent

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Jun 18, 1999, 3:00:00 AM6/18/99
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How do you use the "Game Theory" technique, as mentioned by David Sklansky,
in Holdem, when
you are trying to decide whether to bluff or not. I could not seem to apply
the theory?

Foolproof2

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Jun 18, 1999, 3:00:00 AM6/18/99
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First you determine the odds on when you opponent is bluffing. Then you
randomize to call/raise when the odds are in your favor that he is bluffing.

The problem is how to determine the odds that your opponent is bluffing, which
Sklansky never explains. For example, in HFAP in several places he says
things like *...raise if you think you will have the best hand 55 percent of
the time...*

I have never figured out how you are supposed to determine 55 percent of the
time. You would have to observe the same opponent in the same situation at
least 20 times to measure if he did one thing (e.g. bluff on the end with
nothing) 11 times and the other (really have a hand) 9 times. Furthermore, 20
observations is too small a number of trials to make an accurate prediction.

To get an accurate sample, you would have to observe more like 100 situations
to even come close to being accurate at predicting a 5 percent probability.
Furthermore, you can hardly ever observe the successful bluffing situations,
because they are seldom shown down.

Does anyone have an insight into how Sklansky puts this theory into practice?

David, am I the only person who has trouble with this aspect of your otherwise
excellent book?
FoolProof2

xMarc

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Jun 18, 1999, 3:00:00 AM6/18/99
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It is not possible to make these estimations in an exact matter. It is
purely a matter of experience and empirical intuition. The point of the
specific percentages offered in S&M is to provide a theoretical foundation.
You should be thinking along these lines even if you can't nail down exact
numbers (nobody can).

In practice, the best poker players are those whose decisions most closely
reflect the course of action suggested by these percentages. Keen
observation and lots of hours at the table will develop your ability to
behave in accordance with the probabilities.

Nobody said it was gonna be easy...

-Marc

Foolproof2 wrote in message
<19990618122446...@ng-fn1.aol.com>...

Doug

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Jun 18, 1999, 3:00:00 AM6/18/99
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In article <19990618122446...@ng-fn1.aol.com>,

Foolproof2 <foolp...@aol.com> wrote:
>
>The problem is how to determine the odds that your opponent is bluffing, which
>Sklansky never explains. For example, in HFAP in several places he says
>things like *...raise if you think you will have the best hand 55 percent of
>the time...*
>
>I have never figured out how you are supposed to determine 55 percent of the
>time. You would have to observe the same opponent in the same situation at
>least 20 times to measure if he did one thing (e.g. bluff on the end with
>nothing) 11 times and the other (really have a hand) 9 times. Furthermore, 20
>observations is too small a number of trials to make an accurate prediction.
>
> To get an accurate sample, you would have to observe more like 100 situations
>to even come close to being accurate at predicting a 5 percent probability.
>Furthermore, you can hardly ever observe the successful bluffing situations,
>because they are seldom shown down.

As a beginning poker player, I have to admit that this is one of the
things that I am having the hardest time with. I think that in some part
this is at the heart of the math vs personal intuition/experience
debate. The problem is that many of the formulas that are supposed to
apply to poker require at least one, if not several inputs whose value
cannot be accurately measured and therefore can only be arrived at
through an educated guess.

This is a classic case of that old engineering joke "measure with
calipers, cut with an axe". Does it make sense to apply such an accurate
method for determining the best action (mathematical formulas) if your
input data may be wildly inaccurate? Put another way, if the numbers you
plug into your formula are guesses, can you claim that the course of
action recommended is anything but a guess multiplied?

It seems to me that a serious danger in using mathematical formulas in
poker is that you may use them to convince yourself that some bad course
of action is favorable to you, despite common sense, expert advice or
other means of decision making. Since I'm not sure that this is all very
clear, here's a hypothetical example to illustrate.

Suppose that you have been killing the 3/6 game and want to move up to
10/20. You know that many poker greats recommend that you need a bankroll
of $5000-$10000 to play this game, but you only have $3000. You might
decide to use some calculations from Mason Malmuth or others to
determine if your bankroll is large enough to play the 10/20. These
formulas will generally rely on your estimation of your EV (win rate)
as well as your standard deviation for the game that you will be
playing. Quite simply you have no way of accurately measuring these
values until you have played hundreds of hours in this game. The best
you can do is take your current win rate and standard deviation in the
3/6 and hope that it applies to the 10/20. It almost surely will not.
So you decide to take 3/4 your win rate since you expect the 10/20 game
to be tougher. There is a lot more raising in the 10/20 game (assuming
the 3/6 is very loose passive) so you double your standard deviation as
well. What are these numbers that you are plugging into your formula?
Little more than guesses. Now, what if, based on your numbers you
determine that you have an adequate bankroll to play the 10/20, when in
actuality you do not? You will have convinced yourself to ignore the
experts' advice based on the strength of flawed mathematical
calculations. I believe that this is a real danger.

I assert that the strength of the mathematical formulas are not in
determining the correct course of action but in pointing out what
factors are most important in making your decision and knowing which of
those factors greatly affect your outcome and which have a minor role. In
the bankroll example above, you should study the 10/20 game and think
about your possible EV and standard deviation in that game. Unless both
of these factors seemed very positive to you, I would follow the experts
advice and wait. If one factor is positive and one negative (say there
are some real calling stations that would greatly increase your EV but
there is also a lot of preflop raising that would increase your standard
deviation) I would still wait. In this way, the formula has helped me
because it has convinced me to consider a factor (standard deviation)
that I might not have considered if I were just chasing the fish.

In the example you give "*...raise if you think you will have the best
hand 55 percent of the time...*", here is a story that happened to me.

I was in late position with 77. The flop comes J86. I prepare to fold to
a bet (the pot was not nearly big enough to call) but it is checked
around. "Hmm," I think "is it possible that there is no J or 8 out
there?" The turn brings another 8. It is checked to the player to my
right who bets. Now I must consider the possibilities. Maybe he was
sand-bagging with a J (that's just silly). Maybe he has an 8 (pretty good
probability). Maybe he has a 6 or is on a stone cold bluff. It all boils
down to what are the chances that he is bluffing vs what are the chances
that he has an 8? Well, based on what I know to be good play, I would
have tried a bluff here. In fact I was planning to. I think that it's a
pretty good probability that the rest of the table would fold to a bet
(if there really isn't an 8 or a J out). Would he bluff more than
half of the time in his position? I say yes, so I raise limiting the pot
to me and him. I don't need to assign an accurate number to the
probability that he is bluffing, the formula merely dictates that I
consider what he might have, whether or not he is more likely to have a
better or worse hand than me. When it comes down to it, you are really
just making an educated guess. I don't think that you can be so
accurate as to say that the chance is 50% that he is on a bluff in one
case and 70% in another. I certainly wasn't very accurate in this
situation. He showed me the 8 :-)

tha...@nmia.com

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Jun 18, 1999, 3:00:00 AM6/18/99
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A minor nit. The actual percentage is 54.93628 + or - Sqrt(2)/2.

In article <19990618122446...@ng-fn1.aol.com>, foolp...@aol.com
says...


>
>First you determine the odds on when you opponent is bluffing. Then you
>randomize to call/raise when the odds are in your favor that he is bluffing.
>

>The problem is how to determine the odds that your opponent is bluffing, which
>Sklansky never explains. For example, in HFAP in several places he says
>things like *...raise if you think you will have the best hand 55 percent of
>the time...*
>
>I have never figured out how you are supposed to determine 55 percent of the
>time. You would have to observe the same opponent in the same situation at
>least 20 times to measure if he did one thing (e.g. bluff on the end with
>nothing) 11 times and the other (really have a hand) 9 times. Furthermore, 20
>observations is too small a number of trials to make an accurate prediction.
>
> To get an accurate sample, you would have to observe more like 100 situations
>to even come close to being accurate at predicting a 5 percent probability.
>Furthermore, you can hardly ever observe the successful bluffing situations,
>because they are seldom shown down.
>

Heldar

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Jun 18, 1999, 3:00:00 AM6/18/99
to
Doug, I agree with you completely. It is very easy to unconsciously fudge
your probability estimates in order to justify doing what you wanted to do
anyway, be it call or raise or fold.

Do "expert" poker players really think in terms of percentages, and their
computations are more accurate than those of us regular folks, or are they
just going on instinct, but their instincts are more accurate than ours?
Inquiring minds want to know, you experts out there.

Jazbo

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Jun 19, 1999, 3:00:00 AM6/19/99
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Foolproof2 wrote:
> The problem is how to determine the odds that your opponent is bluffing, which
> Sklansky never explains.

No, that's psychology, not game theory. The game theoretic bluffing
frequency
is purely mathematical (at least in certain cases). Let's do a
(simplified) example.
Before the river in seven card stud, player Ann has a flush draw and
player Bob has
a big pair (let's assume this is known to both). Ann will always bet
her
flush if it gets there. Let's say making the flush has probability A,
Ann bluffs
with probability B, and Bob calls with probability C. There are P bets
in the pot.

Ann wins an extra 1*A*C bets from Bob's calls of her flush bets, and an
extra
P*B*(1-C) when he folds to her bluffs. This costs her 1*B*C when Bob
calls
her bluffs.

Bob wins (P+1)*B*C for catching Ann's bluffs and loses 1*A*C when he
pays her off.

Game theory designs to play optimally against an optimal opponent. If
Ann thinks
Bob is an expert, she should bluff with a frequency B=A/(P+1). This
will guarantee
her a profit of A*C + P*A*(1-C)/(P+1) - A*C/(P+1) = P*/(P+1), regardless
of what Bob
does.

To make these numbers concrete, assume Ann will make her flush 32% of
the time
and there are 7 bets in the pot. Then Ann should bluff an extra 4%
(1/8th as
often as she makes her flush). This will give her an extra profit of
7*.32/8
= 0.28 bets, whatever Bob does.

Notice that when Ann is playing optimally, Bob's calling frequency
doesn't matter.
However, suppose Ann bluffs a little too much (say 5% instead of 4%).
Then
Bob should *always* call. The higher Ann's bluffing frequency, the more
Bob's
profit. Similarly, if Ann bluffs anything less than 4%, Bob should
*never*
call.

Generally, when the final bet is made, if you are confident you are
either facing a bluff or a hand you can't beat, calling or folding
depends not
on a precise evalutation of your opponent's bluffing frequency, but just
whether
they tend to bluff above or below optimal.

Of course, if you always call or always fold and your opponent is able
to pick this up, the situation may change. But that's poker.

Note: In the absence of information of your opponent's bluffing
tendancies,
calling randomly P/(P+1) of the time is the game theoretic optimal.
That is,
you should almost always call on the end if the pot is large and there
is
a chance your opponent is bluffing.

--jazbo <ja...@jazbo.com>
Video Poker and Poker Information
http://www.jazbo.com

Abdul Jalib

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Jun 19, 1999, 3:00:00 AM6/19/99
to
Doug <Thats_hersh_foll...@tc.umn.edu> writes:

>
> In article <19990618122446...@ng-fn1.aol.com>,
> Foolproof2 <foolp...@aol.com> wrote:
> >
> >The problem is how to determine the odds that your opponent is bluffing,
> >which Sklansky never explains. For example, in HFAP in several places
> >he says things like *...raise if you think you will have the best hand
> >55 percent of the time...*
> >
> >I have never figured out how you are supposed to determine 55 percent of the
> >time.
>

> As a beginning poker player, I have to admit that this is one of the
> things that I am having the hardest time with. I think that in some part
> this is at the heart of the math vs personal intuition/experience
> debate. The problem is that many of the formulas that are supposed to
> apply to poker require at least one, if not several inputs whose value
> cannot be accurately measured and therefore can only be arrived at
> through an educated guess.

> In the example you give "*...raise if you think you will have the best


> hand 55 percent of the time...*", here is a story that happened to me.
>
> I was in late position with 77. The flop comes J86. I prepare to fold to
> a bet (the pot was not nearly big enough to call) but it is checked
> around. "Hmm," I think "is it possible that there is no J or 8 out
> there?" The turn brings another 8. It is checked to the player to my
> right who bets. Now I must consider the possibilities. Maybe he was
> sand-bagging with a J (that's just silly). Maybe he has an 8 (pretty good
> probability). Maybe he has a 6 or is on a stone cold bluff. It all boils
> down to what are the chances that he is bluffing vs what are the chances
> that he has an 8? Well, based on what I know to be good play, I would
> have tried a bluff here. In fact I was planning to. I think that it's a
> pretty good probability that the rest of the table would fold to a bet
> (if there really isn't an 8 or a J out). Would he bluff more than
> half of the time in his position? I say yes, so I raise limiting the pot
> to me and him. I don't need to assign an accurate number to the
> probability that he is bluffing, the formula merely dictates that I
> consider what he might have, whether or not he is more likely to have a
> better or worse hand than me. When it comes down to it, you are really
> just making an educated guess. I don't think that you can be so
> accurate as to say that the chance is 50% that he is on a bluff in one
> case and 70% in another. I certainly wasn't very accurate in this
> situation. He showed me the 8 :-)

Wrong thinking is punishable. Your raise probability here is not
a matter of whether you have a 50%+ (or 55%+) chance of having the
best hand. The actual formula would be pretty complicated, but
let's move the action to the river, and assume the players behind you
will fold to your raise, your opponent will 3-bet you only if he has
you beat, and your opponent will not call your raise with a worse hand.
Then your expected value is:

B*P+(1-B)*(-2)

where B is the chance your opponent is bluffing
and P is the pot size after your opponent bets

Folding has an expected value of zero, so raising would be better when:

B*P+(1-B)*(-2) > 0
B*P-2+2B > 0
B(2+P) > 2
B > 2/(2+P)

So, if there are 4 big bets in the pot, then raising would be better
than folding if there were more than a 1 in 3 chance your opponent
was bluffing. Note that I have assumed you are never called with a
worse hand, so you are essentially bluffing here and don't need 77.
You can therefore raise slightly more often with 77, if there is a chance
you will be called by a worse hand.

Cautionary notes: It's not likely your opponent is bluffing, since
he has just bet into several opponents (though it doesn't have to
be likely, just 1 in 3 or so.) Also, your opponent is not
likely to fold better hands, since he is getting great odds to call
your possible bluff. And there's a risk of a 3-bet resteal right back
at you, or of one of the other players calling with a better hand.

Note that other parts of this thread have been in the context of game
theoretic optimal play, where you assume your opponent will play
optimally and you gain when your opponent deviates from optimal play,
though you gain suboptimally. But in this part of the thread we are
talking about exploitive play when you can actually estimate your
opponent's bluff frequency and gain optimally when he deviates from
optimal play.

--
Abdul

robertbbb

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Jun 21, 1999, 3:00:00 AM6/21/99
to
Need to know # outs in trips,straights, flushes,full house, ect. New to
hold'em .

Abdul Jalib <Abd...@PosEV.com> wrote in article
<yerhfo3...@shell9.ba.best.com>...


> Doug <Thats_hersh_foll...@tc.umn.edu> writes:
>
> >
> > In article <19990618122446...@ng-fn1.aol.com>,
> > Foolproof2 <foolp...@aol.com> wrote:
> > >
> > >The problem is how to determine the odds that your opponent is
bluffing,
> > >which Sklansky never explains. For example, in HFAP in several
places
> > >he says things like *...raise if you think you will have the best
hand
> > >55 percent of the time...*
> > >
> > >I have never figured out how you are supposed to determine 55 percent
of the
> > >time.
> >

> > As a beginning poker player, I have to admit that this is one of the
> > things that I am having the hardest time with. I think that in some
part
> > this is at the heart of the math vs personal intuition/experience
> > debate. The problem is that many of the formulas that are supposed to
> > apply to poker require at least one, if not several inputs whose value
> > cannot be accurately measured and therefore can only be arrived at
> > through an educated guess.
>

Abdul Jalib

unread,
Jun 28, 1999, 3:00:00 AM6/28/99
to
"robertbbb" <robert....@foxinternet.net> writes:

> Need to know # outs in trips,straights, flushes,full house, ect. New to
> hold'em .

BASIC CONCEPTS OF SUCKING OUT

If you have the best of it on additional money going into the pot,
you should try to maximize the additional money going into the pot.
If given the money in the pot by the end you have odds to chase, you
should at least call. Keep in mind that betting or raising will
often give you additional ways to win the pot. If you don't have
odds to chase or bluff, you should fold.


BETTING OR RAISING FOR IMMEDIATE PROFIT

Try to get money into the pot if you will win the pot more than your fair
share. You will win more than your fair share of the time when you have
more than the number of outs shown below for the number of opponents:

Betting or Raising
Breakeven Win Chance & Outs for # of Opponents

# Opponents Who Chances to Win You Need Corresponding
Will Be Calling to be Breakeven on Bet Number of Outs
1 1 in 2 23
2 1 in 3 15 1/3
3 1 in 4 11 1/2
4 1 in 5 9 1/5
5 1 in 6 7 2/3
6 1 in 7 6 1/2
7 1 in 8 5 3/4
8 1 in 9 5 1/9


When you have a strong draw, you usually want to keep people in,
so think carefully about how to keep them in while increasing
the pot size. Consider all your options. When no one has bet
you can check-call, bet, or check-raise, and when facing a bet
you can call, raise, or call-reraise.

Your outs are the number of cards that will complete your hand. For
example, if you have JT and the board is KQ23, then any ace or nine
will give you the nut straight, and there are four of each of those,
so you have 8 outs. Here are some of the common draws:


Outs When Drawing One Card (e.g., on the Turn)

Draw on the Turn Outs
4-straight & 4-flush => straight or flush 15
overpair => strong two pair or set 14
set => full house or quads 10
4-flush => flush 9
4-straight => straight 8
4-straight vs 4-flush => straight vs no flush 6
overcards => top pair 6
pair using board card => trips or two pair 5
gutshot straight draw => straight 4
two pair => full house 4
pocket pair => set 2


Your effective outs are your potential outs fudged downwards
to better reflect your actual chances of winning the pot.
For example, if there is a two flush on board and you think the
flush draw is out there, then instead of 8 outs for a straight
draw you effectively have only 6, as the other 2 cards bring in
your opponent's flush draw.

On the flop, if you are planning on taking your draw to the
river, then you effectively have a bit less than double
the number of outs for one card:


Equivalent Outs When Drawing Two Cards (Flop and Turn)

Draws on the Flop and Turn Outs
4-straight & 4-flush => straight or flush 26
overpair => strong two pair, set, house 22
set => full house or quads 15 1/2
4-nut-flush => nut flush 15 1/2
4-baby-flush vs. one => 5-flush 15
4-baby-flush vs. many => only 5-flush 12
nonpair => pair 11 1/2
gut shot straight draw => straight 7 1/2
two pair => full house or quads 7
3-straight & 3-flush => straight or flush 3
3-flush => flush 1 1/2
3-straight => straight 1 1/2


For the runner-runner draws, you need to use both your cards,
except for an ace that makes a 3-flush. You should usually
treat them as just 1 out, not 1 1/2, since a lot of things
can go wrong with them.


CHASING

When you can't make money on additional money going into the pot,
you still have to consider whether the pot will be worth
chasing, considering your chance of winning it. For example,
if the pot were 5 big bets on the turn, and it would cost you
one big bet to call, you are getting 5 to 1 odds to call. Suppose
you had a hand that would win 1 time for every 5 times it lost. In
that case, if you were getting 5 to 1 odds from the pot, then your
call would have an average result of zero, right on the border between
calling and folding. If the pot were any bigger (like if you could
expect a call on the river) then it would be clear case to call, and
if the pot were any smaller, then it would be a clear case to fold.

Before explaining how to figure out precisely whether or not you
should chase, here are some guidelines:

Rules of thumb for calling on the turn: Usually, call one bet with
open-ended straight draws and flush draws, and with a medium pot size
you can call two cold. With a set you should usually be calling all
bets (or raising, of course.) Two overcards are usually no good to
draw with on the turn, except sometimes heads-up. When the pot is
big, you can call with a gutshot straight draw to the nuts.

Rules of thumb for calling on the flop: Call with any draw that you
would call with on the turn, often for two bets cold or more. Call
with gutshots to the nuts if you can be pretty sure you will only have
to pay one bet. Also for one bet, a pair with a backdoor flush draw
is very worthwhile, and so is a backdoor flush draw with a backdoor
straight draw, and similarly for other combinations of weak draws that
together become worthwhile. Be reluctant to call with overcards,
unless heads-up or the board does not have many draws and you are
pretty sure you have the best overcards, like AQ in an unraised pot.

With a little practice, you can be a lot more precise. You can learn
to keep track of the big bets going into the pot almost
subconsciously, and hence you can know the current pot size at all
times. With a little experience, you can estimate the amount of
additional action there will be. And your chance of winning is simply
represented by your number of outs.

Your effective pot size is how much you can expect to win at
the end if you indeed win. It's the current pot size plus
expected action. Generally that will be at least one big
bet bigger than the current pot size, possibly many more big
bets if you expect a lot of action.

You should at least call when your effective outs times
one more than the effective pot size is greater than the number
of unseen cards. The number of unseen cards is usually 46 on the
turn or 47 on the flop. Recast the effective pot size in units of
the number of bets you will need to call.

For example:

Suppose the board is:

Flop Turn
7s 8d 2d 4d

You have: Tc 9c

Preflop an early ultra tight player limps, a middle player calls, you
call late, and both blinds call. On the flop, the small blind bets,
the big blind folds, the early player raises, you call, and the small
blind folds. On the turn, your remaining opponent bets, then turns
over red pocket aces, tells you he knows you are on a straight draw,
explains that you don't have odds to call, and begs you to fold since
he doesn't want to risk losing. What should you do?

There are 6 big bets in the pot now. Unless he is sure about your hand
he will check and call on the river if a nondiamond jack or six comes,
so the effective pot size is more like 7 big bets. Your effective outs
are exactly 6, assuming you cannot bluff him out on the end.

The answer is "it depends." It depends on your opponent. 6*(7+1)=48,
which is greater than 46 (or 44 in this case), so you should call if he
will pay you off, but if he will fold if and only if a nondiamond jack
or six hits, then you get 6*(6+1)=42, and you should fold.


BUYING A FREE CARD

You can raise in late position on the flop with the intention of
checking it through on the turn. Seeing the river card is not
free in this case, but half price. Actually, it's a bit less than
half price, since your flop drawing odds are better than your turn
drawing odds. Savvy opponents are well aware of this play from
flush draws, however, and may thwart it by betting into you on the
turn. Use this play sparingly, mostly when you have big overcards
versus a few weak opponents, and you can always adopt-a-flush-draw
if the flush draw comes in (that is, bet to represent a made flush.)


SEMI-BLUFFING

Your opponents cannot fold if you never bet or raise. Betting or raising
usually is worth at least 4 outs, sometimes 20 or more outs, in terms
of increasing your chance of winning the pot. Sometimes when you
would have to check and fold rather than check and call, you can bet
profitably instead. The combination of a chance of winning with your
draw and a chance of your opponents all folding can make betting (or
raising) more profitable than checking (or folding.) David Sklansky
coined the term "semi-bluffing" to describe this concept.

If you've been betting hard the whole the way, your opponents may not
put you on a draw, and may fold to your bet on the come, or to your
bluff bet on the river, allowing you to steal a large pot. For this
reason, and since you presumably cannot win in a showdown without
making your hand, think twice about taking a free card on the turn,
if you think a bet there or on the river might buy you the pot.


A MADE HAND WITH A DRAW

Sometimes you will have a pair a flush draw, or other combination
of a hand that may be best and a draw. When your made hand is
vulnerable or likely already beaten by fairly weak hands,
you usually should play such a hand very hard, trying to force out
better hands and hands that could draw out on your made hand,
with your draw as a backup in case you get called down by a better
hand. Even with a very strong hand like a made straight with a flush
draw, you might wish to play it hard, hoping to get almost unlimited
reraises from an equivalent straight that you are "freerolling" to
beat with your flush.


CONCLUSION

For novice players, this information can be of great use between
sessions to answer that nagging question, "should I have called?"
This will improve their intuitions in similar situations in the future.
For veteran players, this information can be used in real time
at the tables to make better decisions.


ACKNOWLEDGMENTS

My departed friend Andy Morton came up with the multiplication trick
for determining if you have odds to call. David Sklansky was the
first to publish much of the basic concepts and terminology in
_Theory of Poker_.

--
Abdul

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