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This article will 
discuss how we can use game theory concepts to defend unexploitably vs our opponent's aggression. It's important to understand the examples we use will be extremely simplified models of NLHE.
discuss how we can use game theory concepts to defend unexploitably vs our opponent's aggression. It's important to understand the examples we use will be extremely simplified models of NLHE. Also while we will focus primarily on our defending frequency, this can only be used as a rough guide. The frequencies we defend are generally a by-product of looking at the equities of our range and the expected value of certain lines.
In other words we will take a shortcut here by estimating our defending frequencies independent of our equity. Finding a correct range construction is immensely more complex than this. As such the strategies outlined in this article can be used as an introduction to game theory concepts – but more knowledge is required to begin solving the game of poker.
In other words we will take a shortcut here by estimating our defending frequencies independent of our equity. Finding a correct range construction is immensely more complex than this. As such the strategies outlined in this article can be used as an introduction to game theory concepts – but more knowledge is required to begin solving the game of poker.
GTO: Break Even Point of a Bluff
The first required skill for unexploitable defense is being able to calculate the break-even point of a bluff.
There is a simple formula for this.
% of total pot invested = % of time bluff needs to work to break even
Example 1 – Villain cbets 7bb into a 10bb pot. How often does his bluff need to work to break even?
The total pot-size here is 17bb. Villain is investing 7bb out of those 17bb. So which percentage of the pot is he investing? We simply divide villain's investment by the total pot-size.
7/17 = 0.412
We can now move the decimal point 2 places to the right to convert to a percentage. Villain's bluff needs to work 41.2% of the time to break-even. In other words if villain's bluff succeeds more than 41.2% of the time he is generating automatic profit even if he gives up on every turn. He would be exploiting his opponent in this instance– and the goal of this article is to prevent him from doing so.
Example 2 – Villain cbets 7bb into a 10bb pot. How can we prevent him from generating automatic profit?
How can we prevent villains flop bet from succeeding more than 41% of the time? We need to be defending at least 59% of the time on average. The exact defending frequency would depend on:
- how we defend (ie raising or just calling)
- the board texture (how harmful is giving free cards)
- any inherent range advantage (the relative equities described in the introduction)
Generally speaking the more we defend by just calling, and the wetter the board texture, the higher above that 59% threshold we need to defend. Calculating this is very complex, so at this stage we should be content to add between 0-5% to the defending frequency based on an estimate regarding the above variables. We'd pick 0% addition to the threshold if we defended by raising 100% of the time and never smooth-calling.
In a situation where we have an inherent disadvantage, we may choose to defend less than the 59% threshold and allow our opponent to generate automatic profit
However, assuming a situation where we have an inherent disadvantage (lack of position, equity disadvantage) we may choose to defend less than the 59% threshold and allow our opponent to generate automatic profit. More on this later.GTO: Defending on Dry Textures
Example 3 - We 
face a cbet OOP on an A72r board. We decide to check-raise bluff with a decent suited connector and a backdoor draw. What is your initial reaction to this line?
face a cbet OOP on an A72r board. We decide to check-raise bluff with a decent suited connector and a backdoor draw. What is your initial reaction to this line?I'm expecting a range of reactions to this one. Some of you probably feel it is OK. Some of you had negative reaction to this hand because we are raising on a dry texture. You possibly realise that this is incorrect from a GTO point of view. But why?
The most common answer is often not the correct one. People generally answer that it's incorrect because we don't represent a legit value hand. Or they say that it's impossible to balance our raising range on this texture.
Neither of these is the actual reason. It is completely possible to balance a raising range on a bone-dry texture and become completely unexploitable. The problem lies not with our raising range but with our calling range.
If we start raising all sets and decent top-pairs for value, our calling range quickly becomes very weak and capped. This is because we had way less decent hands to begin with on a dry texture, and we need to protect all of our weak hands by slowplaying the monsters. On a drawy texture we can still raise a balanced range and have enough decent made hands to protect our calling range.
If we start raising all sets and decent top-pairs for value, our calling range quickly becomes very weak and capped. This is because we had way less decent hands to begin with on a dry texture, and we need to protect all of our weak hands by slowplaying the monsters. On a drawy texture we can still raise a balanced range and have enough decent made hands to protect our calling range.
So generally, when facing a bet on a dry texture we should use our entire defending range as calls rather than raises.
Steps:
- work out villain's break even point on a bluff, and hence how often we need to defend so that we don't give our opponent automatic-profit
- calculate how many combos of hands we get to the flop with
- work out how many total combos we need to defend
- select our defending range from the hands which have the best backdoor potential.
Note that raw equity is far less important than playability. For example if the board texture is K72r then a hand like A8o has a decent amount of raw equity vs an average range. More equity in fact than a hand like TJs with a backdoor flush draw.
However. seeing as TJs improves on over half the turn cards it's a much better hand to defend. Our turn playability will be significantly higher.
On a drawy texture we can still raise a balanced range and have enough decent made hands to protect our calling range.
However. seeing as TJs improves on over half the turn cards it's a much better hand to defend. Our turn playability will be significantly higher.
GTO: Defending on Drawy Textures
The main difference on 
drawy textures is that we now have a raising range alongside our calling range which makes range construction a little more complex.
drawy textures is that we now have a raising range alongside our calling range which makes range construction a little more complex.The initial process is similar however. We calculate how frequently we need to defend and decide which hands are in our defending range.
Next we should select the hands we feel are strong enough to raise for value. From here we can calculate how many bluff combos we need, but this will depend on which bluff:value ratio is correct here. See the article “GTO Aggression in Poker” for more info on calculating this.
As a rough guide, for most standard situations with 100bb stacks we will be using a 2:1 bluff:value ratio on the flop, 1:1 on the turn, and 2:1 on the river. Assuming we are on the flop and use a 2:1 bluff:value ratio we will have twice as many bluff combos as value hands.
From here we can use a process of deduction to calculate how many flatting combos that would leave us. The final step is to decide which hands make the most sense as bluffs and which make the most sense as calls.
From here we can use a process of deduction to calculate how many flatting combos that would leave us. The final step is to decide which hands make the most sense as bluffs and which make the most sense as calls.
Steps:
- work out villain's break even point on a bluff, and hence how often we need to defend so that we don't give our opponent automatic-profit
- calculate how many combos of hands we get to the flop with
- work out how many total combos we need to defend
- decide which combos are strong enough to raise for value
- decide on a bluff:value ratio and calculate how many bluff combos we will use
- calculate number of flatting combos by process of deduction
- select flatting range, followed by bluffing range
GTO: Inherent Disadvantage
Keep in mind that these are only guidelines and don't necessarily give the GTO optimal solution. Poker is far more complex than this.
There are certain situations where we have inherent disadvantage and therefore it is correct for us to allow our opponent to generate automatic profit. This is not necessarily a leak, since when the situation is reversed we should be able to generate automatic profit from our opponent.
In other words, the idea behind GTO play is not to create an equilibrium in isolated situations but an equilibrium across the entire spectrum of possible scenarios.
In other words, the idea behind GTO play is not to create an equilibrium in isolated situations but an equilibrium across the entire spectrum of possible scenarios.
To give one example, we should expect to give our opponents a measure of automatic profit when we are facing a cbet OOP. It's too difficult to overcome our positional disadvantage. However we also expect that when we are the one IP that we should generate automatic profit from our opponent. If he tries to prevent us from generating automatic profit he will end up defending a range which is too wide to be supported OOP.
These are only guidelines and don't necessarily give the GTO optimal solution. Poker is far more complex than this.
To give another example – it's common for player's river fold-to-cbet to be around 60%. At first glance this may seem extremely high. We know that if opponent bets 2/3rds pot on the river that he is actually generating automatic profit in this example. However this is acceptable from a GTO point of view because one player has a clear range advantage.
The aggressor is barreling 3 streets and hence has an uncapped range. The defender is calling down passively rather than raising. He is very often likely capped. As a result of his inherent range disadvantage it is acceptable for him to overfold in this situation, expecting that when the roles are reversed his opponent will also overfold, keeping the equilibrium in place.
