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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?

in Holdem, when

you are trying to decide whether to bluff or not. I could not seem to apply

the theory?

Jun 18, 1999, 3:00:00 AM6/18/99

to

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.

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

Jun 18, 1999, 3:00:00 AM6/18/99

to

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).

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>...

Jun 18, 1999, 3:00:00 AM6/18/99

to

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.

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 :-)

Jun 18, 1999, 3:00:00 AM6/18/99

to

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.

>

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.

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.

Jun 19, 1999, 3:00:00 AM6/19/99

to

Foolproof2 wrote:

> The problem is how to determine the odds that your opponent is bluffing, which

> Sklansky never explains.

> 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

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

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 .

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.

>

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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