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# Poker Math How-To: Optimal Bluffing Frequency

1. ## Poker Math How-To: Optimal Bluffing Frequency

In a no-limit texas hold'em hand, you are first to act with the river card about to come with \$100 behind and the pot after the turn is \$200 (our single opponent has us covered). You hold T 8 on a board of J 9 5 2 and your opponent holds T T. Quite obviously we're value-shoving our \$100 when a Q, 7 or club falls on the river, but how often should we bluff the river for optimal play?

First we need to define what we mean by optimal play. When we play optimally in general, there is no way our opponent can exploit our play. In the described scenario above, our opponent will have two choices at his disposal: to call our bet, or to fold. Our goal is to make both of these options have the same EV, in which case he will have no exploitive option. Our lives are made easier by the simple fact that the EV of folding is always 0. So now we have to figure out how often we should be bluffing so that the EV of him calling is also 0.

From our villain's point of view, he will be needing to call \$100 in a pot of \$300 after we shove the river, getting 3:1 odds. If we are bluffing more often than 25%, he should call since his 3:1 odds will pull a profit on a call. If we are bluffing less often than 25%, he should fold since his 3:1 odds will pull a loss on a call. But what if we bluff exactly 25% of the time? Then a call is breakeven. Zing!

There are 15 cards (nine clubs, 3 more queens and 3 more sevens) that make our hand. If before the river comes we pick out 5 distinct cards (that aren't a club, a queen, or a seven) that we'll bluff on, then we will be bluffing exactly 25% of the time, and our river betting will not be exploitable.
2.  07-19-2008 07:28 PM Hawk Join Date Sep 2007 Posts 1,667 Nice, but I'm not sure how often we should be bluffing rivers at micros (where most people in BC are playing), especially if we know they have a set. I think that bluffing (outside of standard cbets and semibluffs) should be fairly uncommon. I'm only bluffing spots like this against against the better regs I face who I know have some ability to read my hand.
3.  07-19-2008 07:42 PM badgers Join Date Feb 2007 Posts 4,411 Location Spewing your definition of optimal tilts me optimal =/= unexploitable. as the opponent gets better the two will converge. 3k post - Return of the blog!
4. ## Re: Poker Math How-To: Optimal Bluffing Frequency

Originally Posted by spoonitnow
In a no-limit texas hold'em hand, you are first to act with the river card about to come with \$100 behind and the pot after the turn is \$200 (our single opponent has us covered). You hold T 8 on a board of J 9 5 2 and your opponent holds T T. Quite obviously we're value-shoving our \$100 when a Q, 7 or club falls on the river, but how often should we bluff the river for optimal play?

First we need to define what we mean by optimal play. When we play optimally in general, there is no way our opponent can exploit our play. In the described scenario above, our opponent will have two choices at his disposal: to call our bet, or to fold. Our goal is to make both of these options have the same EV, in which case he will have no exploitive option. Our lives are made easier by the simple fact that the EV of folding is always 0. So now we have to figure out how often we should be bluffing so that the EV of him calling is also 0.

From our villain's point of view, he will be needing to call \$100 in a pot of \$300 after we shove the river, getting 3:1 odds. If we are bluffing more often than 25%, he should call since his 3:1 odds will pull a profit on a call. If we are bluffing less often than 25%, he should fold since his 3:1 odds will pull a loss on a call. But what if we bluff exactly 25% of the time? Then a call is breakeven. Zing!

There are 15 cards (nine clubs, 3 more queens and 3 more sevens) that make our hand. If before the river comes we pick out 5 distinct cards (that aren't a club, a queen, or a seven) that we'll bluff on, then we will be bluffing exactly 25% of the time, and our river betting will not be exploitable.
Game theory
Sklansky: Theory of poker
calculate odds you give your opponent on call (example 3-1)
use same ratio in bluffing (example 15outs - 5bluffing cards or 3-1)
adjust on the lower for calling stations, on the higher for nits
choose your bluffing cards before they appear so your bluffing is in fact random
5.  07-19-2008 08:22 PM Guest If we have 15 outs, and a call will break us even, then is it truly a bluff? If.... eh, never mind, I'm just confusing myself.
6.  07-19-2008 08:32 PM asdpikas Join Date Jul 2007 Posts 1,071 Originally Posted by Mezza Morta If we have 15 outs, and a call will break us even, then is it truly a bluff? lets do the math 100 hands if villain folds if we bet, figuring we never bluff 30% of the time we make our hand = 30 wins = +\$6000 10% of the time we bluff = 10 wins = +\$2000 60% of the time we c/f = 0 wins =+\$0 we win \$8000 if villains calls always 30% of the time we make our hand = 30 wins = +\$9000 10% of the time we bluff = 10 losses= -\$1000 60% of the time we c/f = 0 wins =+\$0 we win \$8000 NOTE: This is not break-even, in the sense that we win \$0, we win either way, its breakeven in the sense that whatever villain chooses, he loses the same. Game theory just randomizes our bluffing and ensures our profit. If we were to bluff too much, opponent would be correct to call always. If we were to never bluff opponent would be correct to always fold. If we bluff the "correct" amount of times randomly, opponent has no correct option.
7.  07-19-2008 10:38 PM spoonitnow Resident Asshole Join Date Sep 2005 Posts 14,368 Location North Carolina Originally Posted by badgers your definition of optimal tilts me optimal =/= unexploitable. as the opponent gets better the two will converge. How does "my" definition of optimal tilt you? Optimal strategy cannot be exploited, and your statement here shows your lack of understanding of game theory, no offense. Consider paper-rock-scissors. Optimal strategy is to always randomly pick one of the three choices with equal weight to those three choices, and it cannot be exploited. Any deviation from optimal strategy will create a new strategy that is, by definition, exploitable. It doesn't matter how good or bad our opponent is. Optimal strategy doesn't change.
8.  07-19-2008 10:42 PM Guest Originally Posted by asdpikas Originally Posted by Mezza Morta If we have 15 outs, and a call will break us even, then is it truly a bluff? lets do the math 100 hands if villain folds if we bet, figuring we never bluff 30% of the time we make our hand = 30 wins = +\$6000 10% of the time we bluff = 10 wins = +\$2000 60% of the time we c/f = 0 wins =+\$0 we win \$8000 if villains calls always 30% of the time we make our hand = 30 wins = +\$9000 10% of the time we bluff = 10 losses= -\$1000 60% of the time we c/f = 0 wins =+\$0 we win \$8000 NOTE: This is not break-even, in the sense that we win \$0, we win either way, its breakeven in the sense that whatever villain chooses, he loses the same. Game theory just randomizes our bluffing and ensures our profit. If we were to bluff too much, opponent would be correct to call always. If we were to never bluff opponent would be correct to always fold. If we bluff the "correct" amount of times randomly, opponent has no correct option. Excellent reply sir.
9.  07-19-2008 10:50 PM badgers Join Date Feb 2007 Posts 4,411 Location Spewing Originally Posted by spoonitnow Originally Posted by badgers your definition of optimal tilts me optimal =/= unexploitable. as the opponent gets better the two will converge. How does "my" definition of optimal tilt you? Optimal strategy cannot be exploited, and your statement here shows your lack of understanding of game theory, no offense. Consider paper-rock-scissors. Optimal strategy is to always randomly pick one of the three choices with equal weight to those three choices, and it cannot be exploited. Any deviation from optimal strategy will create a new strategy that is, by definition, exploitable. It doesn't matter how good or bad our opponent is. Optimal strategy doesn't change. No. I understand game theory well enough to understand what you're saying in your post. That does not make this strategy optimal. It makes it unexploitable, the two are not the same. Against a player more inclined to call, an optimal strategy would include fewer bluffs than an unexploitable strategy as we are trying to capitalise on villains mistakes. Just because our strategy is now exploitable does not mean that it is sub-optimal since an optimal strategy will constantly adjust to our opponents tendancies. 3k post - Return of the blog!
11.  07-20-2008 12:36 AM bjsaust Join Date May 2007 Posts 6,352 Location Ballarat, Australia I really like this series of articles that addresses what you two are debating: http://cardsharp.org/game-theory-im-not-a-fan/ Links at the top to 3 different articles where he uses different strategies to analyse situations, then that article I linked goes on to discuss the pluses and minuses of each strategy. Just dipping my toes back in.
12.  07-20-2008 12:57 AM spoonitnow Resident Asshole Join Date Sep 2005 Posts 14,368 Location North Carolina Originally Posted by bjsaust I really like this series of articles that addresses what you two are debating: http://cardsharp.org/game-theory-im-not-a-fan/ Links at the top to 3 different articles where he uses different strategies to analyse situations, then that article I linked goes on to discuss the pluses and minuses of each strategy. Well there's nothing really to debate, just a slight misunderstanding on badger's part about difference phrases that happen to contain the word optimal, but that could happen to anyone.
13. ## Re: Poker Math How-To: Optimal Bluffing Frequency

Originally Posted by asdpikas
Game theory
Sklansky: Theory of poker
calculate odds you give your opponent on call (example 3-1)
use same ratio in bluffing (example 15outs - 5bluffing cards or 3-1)
adjust on the lower for calling stations, on the higher for nits
choose your bluffing cards before they appear so your bluffing is in fact random
14.  07-20-2008 08:07 AM ponyboy Join Date Jan 2005 Posts 390 Originally Posted by Hawkfan79 Nice, but I'm not sure how often we should be bluffing rivers at micros (where most people in BC are playing), especially if we know they have a set. I think that bluffing (outside of standard cbets and semibluffs) should be fairly uncommon. I'm only bluffing spots like this against against the better regs I face who I know have some ability to read my hand. In my experience thus far this seems to be the case. A set is enough for a donk to call you down. However, with bet sizing you can induce them to at least think about it hard, especially with shorter stacks. A lot of people have trouble shoving themselves all in with a set when the board has a lot of scare cards on it.
15.  07-20-2008 02:43 PM badgers Join Date Feb 2007 Posts 4,411 Location Spewing IDK I would still prefer to call them optimal and unexploitable because the two are clearly not the same in all but a theoretical context (ie. none of our opponents are perfect). I should probably have phrased all my responses to this thread better after all this is a very good thread and very relevant to stuff I've been working on lately so good stuff spoon. 3k post - Return of the blog!
16.  07-20-2008 04:05 PM asdpikas Join Date Jul 2007 Posts 1,071 Originally Posted by ponyboy Originally Posted by Hawkfan79 Nice, but I'm not sure how often we should be bluffing rivers at micros (where most people in BC are playing), especially if we know they have a set. I think that bluffing (outside of standard cbets and semibluffs) should be fairly uncommon. I'm only bluffing spots like this against against the better regs I face who I know have some ability to read my hand. In my experience thus far this seems to be the case. A set is enough for a donk to call you down. However, with bet sizing you can induce them to at least think about it hard, especially with shorter stacks. A lot of people have trouble shoving themselves all in with a set when the board has a lot of scare cards on it. This concept of game theory based bluffing is aplicable to many situations, so the fact that most donks will always call with a set is not important, you could say villain has AA and its the same thing.
17.  07-20-2008 06:02 PM spoonitnow Resident Asshole Join Date Sep 2005 Posts 14,368 Location North Carolina Originally Posted by badgers IDK I would still prefer to call them optimal and unexploitable because the two are clearly not the same in all but a theoretical context (ie. none of our opponents are perfect). I should probably have phrased all my responses to this thread better after all this is a very good thread and very relevant to stuff I've been working on lately so good stuff spoon. I know man, trust me. The way the terminology comes off in most branches of mathematics is aggravating as fuck at times.

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