Theory of Poker Math Question

[Macedon]

To all who have read and digested THEORY OF POKER, I would appreciate some help on some math that was presented on pages 185-186.

In Chapter 19, Sklansky talks about finding the Optimum Bluffing Strategy using Game Theory. Specifically, he states that you should make the odds against you bluffing exactly the same as the pot odds being offered by your bet. That way, your opponent has no idea whether calling or folding is more prudent. The result [from this] is that you produce the optimum strategy against him/her.

Makes sense...right?

But then he introduces the following example:
"Suppose you have a 25% chance of hitting your hand, the pot is $100, and the bet is $100...your opponent is [therefore] getting 2:1 from the pot. Since there is a 25% chance of you making your hand, there should be a 12.5% chance you are bluffing to create the 2-to-1 odds against your bluffing, which is the optimum strategy."

When I do the math, I come up with a different answer. When you add the two percentages together you get 37.5%. That is around 1.7:1 odds, not 2:1. Is Sklansky rounding off?

If you are going by outs, 25% is 3:1, or the equivalent to about 11 outs. 12.5% is 7:1 odds, or a little bit less than 6 outs. Together they add up to 17 outs, or 37%, or 1.7:1. Not 2:1

If you wanted exactly 2:1 odds, shouldn't you add 4 cards (4 outs, or 8.3%) to your 11 out--25% draw, thereby making 2:1 odds (15 outs at 33.3%)?

There is a significant difference between 4 cards and 6 cards, when you are choosing how many cards to bluff on.

Or have I missed something, which I realize is likely?

[Macedon]

Nevermind...I see my error.
Thank you all for your responses....lol

When Sklansky is doing the bluffing by percentages, he compares the 25-to-12.5 to come up with 2:1.
My mistake was adding them together, like he did with the outs and bluffing cards from an earlier example.

Thank you once again for your help....

[MortalWombatDotCom]

This is really more of an aside than anything else, but that advice on optimal bluffing frequency seemd to apply to games like draw or stud, where the last card is hidden from an opponent who has some chance of putting you on a draw, and hardly at all to hold'em, where it is hard to pretend a blank was the card you were drawing to. Can anyone provide a good example situation in hold'em, such that this technique can be used to provide an optimal frequency of bluffing when that situation arises?

[gaming_mouse]

Well, if a total blank comes and you are sure your opponent will call you, then you are right, this doesn't make sense.

But there are plenty of times where you could be drawing to more than one thing. For example, there could be a straight draw and a flush draw on the board. You could be fairly sure your oppo has a made hand, say a high pocket pair.

In this case, to use opitmal bluffing strategy, you would bet whenever you made your staight draw (assuming that's what you were really on), plus x% of the time when the 3rd flush card hits, where x is determined by the method described in the OP's post.

gm

[MortalWombatDotCom]

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Well, if a total blank comes and you are sure your opponent will call you, then you are right, this doesn't make sense.
gm

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total blank or no, any time you are sure your opponent will call you, it doesn't make sense [img]/images/graemlins/smile.gif[/img]

my take is that "optimal bluffing strategy" is really a tool to be used exclusively against players superior to yourself. if you can't figure out how likely they are to call you on a case by case basis, use a theoretical opponent who plays optimally as your model. if you have any decent idea whether a player is more or less likely to fold based on his past behavior in similar situations, or what you showed last time you got called down in a similar situation, or whether your opponent just got AA cracked by 85o, or anything else, "optimal bluffing strategy" is anything but.

and if your opponent usually knows when you are bluffing, "never" is the correct bluffing frequency against that opponent. [img]/images/graemlins/smile.gif[/img]

[gaming_mouse]

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total blank or no, any time you are sure your opponent will call you, it doesn't make sense [img]/images/graemlins/smile.gif[/img]

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true enough.

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my take is that "optimal bluffing strategy" is really a tool to be used exclusively against players superior to yourself.

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Yes, this is pretty much an accepted fact. Sklansky points it out in TOP.

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if you have any decent idea whether a player is more or less likely to fold based on his past behavior in similar situations.... "optimal bluffing strategy" is anything but.

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Well, optimal is always is optimal -- by definition. What you mean is that it is not always maximal: Against a poor opponent, you won't be making as much as you could be. This is a nitpicky point of semantics, but worth pointing out, since optimal is a kind of reserved term from the world of game theory.

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and if your opponent usually knows when you are bluffing, "never" is the correct bluffing frequency against that opponent. [img]/images/graemlins/smile.gif[/img]

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This is not true. The whole point of using a randomized bluffing strategy is that your opponent cannot possibly read you. On the other hand, if you were speaking of playing live money, and if you have some ridiculous tell -- like turning bright red whenever you bluff -- then you are right. But even in that case, you can always make your move without looking at your card (in the case of stud) or without looking at the board (in the case of holdem). All that matters is that the odds against your bluffing match the pot odds. You don't even need to know your own card to accomplish that.

gm

[MortalWombatDotCom]

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and if your opponent usually knows when you are bluffing, "never" is the correct bluffing frequency against that opponent. [img]/images/graemlins/smile.gif[/img]

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This is not true. The whole point of using a randomized bluffing strategy is that your opponent cannot possibly read you. On the other hand, if you were speaking of playing live money, and if you have some ridiculous tell -- like turning bright red whenever you bluff -- then you are right. But even in that case, you can always make your move without looking at your card (in the case of stud) or without looking at the board (in the case of holdem). All that matters is that the odds against your bluffing match the pot odds. You don't even need to know your own card to accomplish that.

gm

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1) randomizing your bluffing eliminates some, but not all, of the information an opponent could profitably use to determine if you are bluffing. some tells are less rediculous than blushing when you bluff and may still be exploited by some non-empty set of opponents. i was in fact talking about real life play and tells of the only-a-few-opponents-will-pick-up-on-them variety and opponents of the one-of-those-few variety. i think that failure to notice that a particular opponent is snapping off more than the "expected" number of your bluffs while paying off fewer than the "expected" number of your value bets is an error on your part. i think that continuing to use randomized bluffs against such an opponent is an error on your part. and i think that such opponents, while thankfully rare, do exist. do you dispute any of these three statements?

2) if you don't look at the last card, then you cannot possibly alter the odds that you are bluffing when you bet by altering the frequency with which you choose to bet. everytime you bet, you are making a "real" bet with probability p, the chance that you made your draw, and you are bluffing with probability 1 - p.

[gaming_mouse]

1) randomizing your bluffing eliminates some, but not all, of the information an opponent could profitably use to determine if you are bluffing.

If you don't look at your the card in question, it eliminates all of it. Aside from that, I agree with everything you say in 1).

2) if you don't look at the last card, then you cannot possibly alter the odds that you are bluffing when you bet by altering the frequency with which you choose to bet

I am pretty sure this is incorrect. Please read over the following and corroborate or refute the reasoning. I think your confusion is coming from looking at it on a case by case basis. Think of how your strategy plays out over many hands. For example:

You are playing holdem and are drawing at a well disguised OESD on the turn (ie, only the river card is yet to come). Your oppo could put you on a whole range of hands, some of which might already have him beat, so bluffing makes sense. Problem is, as in your scenario, this guy can see right through you. No problem. You decide to bet without looking at the river, using randomization.

Say the pot is offering him 10:1 on his final call. Well, you have 8 outs, and there are 45 cards left. We need to determine the percent p such that if you bet out p% of the time on the river -- without looking at the river card -- the odds against your bluffing will be 10:1. Well, given our strategy -- betting p% of the time no matter what -- in the long run we will have:

P(a bet from us is not a bluff) = p*(8/45)

p*(8/45) = (1/11)

p = .511

So if we bet 50% of the time no matter what, we guarantee a wash -- our oppo cannot use our tells against us.

gm

EDIT: Of course, the downside of this is that sometimes you don't bet when you have filled up. But this is the best strategy against a theoretical oppo who can always read us from physical tells.

Also, it would be more accurate to say that we guarantee collecting our equity in the pot, rather than saying that we guarantee "a wash".

[jason1990]

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Say the pot is offering him 10:1 on his final call. Well, you have 8 outs, and there are 45 cards left. We need to determine the percent p such that if you bet out p% of the time on the river -- without looking at the river card -- the odds against your bluffing will be 10:1.

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My understanding of the game theory argument suggests that we want

P(a bet from us is not a bluff) = 10*P(a bet from us is a bluff)

which, if we play as you suggest, yields the equation

p*(8/45) = 10*p*(37/45).

The only solution to this equation is p = 0.

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EDIT: Of course, the downside of this is that sometimes you don't bet when you have filled up.

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Apparently, from the above, the only way to implement the "optimal" strategy without looking at the river card is to *never* bet, which means that this downside doesn't happen sometimes, but in fact it happens always.

[gaming_mouse]

jason,

yes... looking this over, i'm almost sure you're right.

changing the total percentage of times you are bluffing without changing the bluff/non-bluff ration for times when you do bet won't do you any good.

thanks for pointing that out,
gm