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.

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.

your definition of optimal tilts me

optimal =/= unexploitable.

as the opponent gets better the two will converge.

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

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.

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.

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.

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.

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.

Originally Posted by badgers

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.

You're arguing the wrong optimal here, almost literal apples and oranges.

The one you're talking about ----> Optimal Exploitive Strategy: A strategy which yields the highest possible EV against your opponent’s strategy.

The one the thread is about ----> Game Theory Optimal: A strategy that yields the highest possible EV (or: “is optimal”) if your opponent always chooses the best possible counter-strategy.

Optimal play is decided without knowing how your opponent plays. All exploitive strategies are deviations from optimal play that give us a higher EV than optimal play against a specific opponent's strategy. What's referred to as the "optimal exploitive strategy" just means the best-scoring exploitive strategy, or the exploitive strategy with the highest EV.

And that's the source of what's bothering you.

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.

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.

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

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.

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.

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.

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.