## Understanding the Balance of Folding and Calling

In the game of poker, finding the right balance between folding and calling is crucial. Folding too much puts us at risk of being exploited by opponents who bluff relentlessly. On the other hand, calling too frequently often leads to becoming a losing player in the long run.

For instance, imagine folding every hand except for the absolute strongest ones. While this approach may seem safe, it allows opponents to take advantage of us by bluffing with a wide range of hands, knowing that we will only continue with the very best.

Conversely, if we call every bet thrown our way without careful consideration, we fall into the trap of becoming a calling station. This means that our opponents can profitably value bet us with their strong hands, knowing that we will pay off their bets even when we have weaker holdings.

### Striking the Perfect Balance

By employing Game Theory Optimal (GTO) analysis, we can determine the ideal defense frequency that prevents our opponents from bluffing profitably while also avoiding the pitfalls of over-calling. This optimal frequency creates a balanced strategy that maximizes our chances of success.

The perfect middle ground is a delicate balance that ensures we defend with a frequency that neither allows opponents to exploit us nor leaves us vulnerable to excessive value betting. It is a strategic approach that keeps our opponents guessing and maximizes our long-term profitability.

Achieving optimal balance in poker strategy is crucial for success.

### The Break-Even Point of Bluffs

In poker, understanding the break-even point of bluffs is crucial to prevent opponents from exploiting us and to make informed decisions. The break-even point refers to the percentage of the total pot that an opponent must invest in their bluff for it to be profitable.

To calculate the break-even point, let’s consider an example: an aggressor makes a bluff of \$100 into a \$100 pot on the river. How often does their bluff need to work in order to be profitable?

We can determine this by looking at the percentage of the total final pot that the aggressor is investing. In this case, the total final pot is \$200 (including the aggressor’s bet), and the aggressor is putting in \$100. Therefore, their bluff requires a success rate of \$100/\$200, or 50%, to break even.

Understanding the break-even point allows us to gauge the profitability of an opponent’s bluff and make better decisions in response. It provides a reference point for evaluating the risk-reward balance of our actions and helps us find the optimal middle ground between folding too much and calling recklessly.

By being aware of the break-even point, we can avoid falling into the traps of opponents who bluff relentlessly or exploit us by extracting value with strong hands. Striking the right balance between folding and calling is crucial for successful poker play.

Remember, the break-even point depends on the specific situation and bet sizing. Analyzing these factors will help us make more informed decisions at the poker table and enhance our overall game.

Understanding the break-even point of bluffs in poker strategy.

### Applying the Perfect Polarization Model

In a previous post, we introduced the perfect polarization model, and now it’s time to apply it in practice. This model helps us understand the dynamics of folding frequency and its impact on the profitability of bluffs. Let’s dive deeper into this concept and explore its implications.

When we fold more frequently than our opponent’s break-even threshold, we become vulnerable to their relentless bluffs. On the other hand, if we fold less frequently than the break-even threshold, our opponent can exploit us by removing all bluffs from their strategy.  This later situation is often summed up with “Never bluff a a calling station.” Striking the right balance is crucial.

The break-even threshold represents the point at which a bluff becomes profitable. We noted above that, when the aggressor makes a \$100 bluff into a \$100 pot on the river, the break-even threshold is calculated by looking at the percentage of the total pot being invested. In this case, the aggressor is investing \$100 out of a total pot of \$200, which translates to 50%.

Now, let’s explore the implications of folding frequency. If we fold exactly 50% of the time, we find ourselves in the perfect middle ground of the perfect polarization model. At this frequency, there is nothing the aggressor can do to exploit us, as their bluffs no longer hold any advantage.

However, folding more frequently than 50% allows our opponent to profitably bluff with all their air hands, taking advantage of our excessive folding tendencies. Conversely, if we fold less frequently than 50%, the aggressor can eliminate all bluffs from their strategy, making it extremely challenging for us to extract value with strong hands.

Therefore, finding the right folding frequency is crucial to preventing our opponents from exploiting us and maintaining a balanced strategy. Striking the perfect middle ground ensures that our opponent’s bluffs are no longer profitable, safeguarding our overall profitability in the game.

### Attempting to Counter and Exploring EV

In the world of poker, there is a common belief that the aggressor’s expected value (EV) is fixed and cannot be changed. However, let’s take a closer look at this notion and explore the EV of different strategies.

### The EV of Bluffing and Value Betting

To understand the concept of EV, let’s examine the EV of bluffing and value betting.

When the aggressor makes a bluff, their EV depends on how often their bluff needs to work in order to be profitable. On the other hand, when value betting, the aggressor aims to extract value from their strong hands.

It’s worth noting that value betting can be profitable even in the context of game theory optimal (GTO) poker. The goal of the defender is not to prevent the aggressor from profiting entirely. That would be an impossible task. Instead, the defender’s objective is to limit the aggressor’s maximum profit potential.

This makes sense, there are just certain situations and board run outs that are profitable for one seat vs the other.  However, a skilled player in the disadvantageous seat can still minimize their losses. Sometimes, I will tell myself that I “won” a hand because the opponent should have felted me, but did not recognize their advantage and did not maximize.

### The Defender’s Role

The defender’s role is not solely focused on preventing the aggressor from profiting. Rather, it’s about minimizing the aggressor’s profit beyond what they are entitled to given the situation.

By understanding this perspective, it becomes clear that the defender’s goal is to not fold excessively while not calling too frequently. Finding the right balance is crucial to prevent exploitation and maximize their own EV.

While the aggressor’s EV can be modified, the defender’s primary objective is to restrict the aggressor’s maximum profit, rather than completely negating their profitability.

### Calculating the EV of the Aggressor’s Range

When analyzing the EV (expected value) of a betting range in poker, it’s important to consider the entire range of hands the aggressor could have, rather than focusing on individual holdings. This provides a more comprehensive understanding of the overall strategy.

Let’s consider a scenario where the aggressor decides to remove all bluffs from their betting range thus only including value hands. This means that every bet made by the aggressor is meant to extract value rather than bluffing.

What happens to the EV of the betting range in this case? The overall strategy’s EV may change due to the altered composition of the range and actually can lower the EV.

Exploring the perfect polarization model in poker strategy.

### Implications for the EV of the Range and Overall Strategy

At first glance, it may seem like removing all bluffs from the range would increase the EV. However, this is not necessarily the case. While the EV of the range vs an unobservant opponent who calls too much may increase, the overall strategy’s EV could suffer due to the reduced frequency of bets and increased predictability.

Consider the aggressor’s EV when using a balanced strategy with a range consisting of both value hands and bluffs. Let’s assume that the EV of this balanced range is \$100. Now, the aggressor moves to only bet the value hands in the range. Let’s decide the defender correctly adjusts their strategy to fold 100% of the time against the aggressor’s pure value bet strategy.

The overall EV of the strategy decreases since the aggressor is winning only what is already in  the pot with his value hands (since they do not get called) and they are winning nothing with their potential bluffs.  Essentially, they are “only winning the hands they should” based on the cards.

Going the other direction, if the aggressor decides to bet with 100% of their range: that is with the value and the bluff hands, the observant defender will start calling 100% of the time.  The possible bluffs will be larger than the value hands so the aggressor will be burning money.

Both of these “pure” strategies have a very simple exploit.  However, the “mixed” strategy does better.  Imagine the “value only” strategy mixes in one bluff a year.  No one is going to notice that and that bluff will get paid making it a little bit better than the no bluff strategy.

Looking at the other side of this situation, if the ultra-aggressive player neglects to make one bluff from his bet-100% strategy, that will lose a little bit less than normal.  If you sneak towards the middle, there is a perfect point where you make “more than your cards are entitled to” and that is what GTO is all about.

Mastering folding, calling, and defense frequency in poker.

## Maximizing EV Across the Entire Strategy

Maximizing EV across an entire strategy is only possible by learning to maximize each sub-game. The river bluff-to-value for the attacker for a purely polarized range is the cleanest to model mathematically.

Step back a street, the betting ranges on the turn can be chosen such that no matter which river card comes, the turn betting range is chosen to maximize across all possible rivers.  Push that logic back one street at a time and you have a crazy big optimization problem.  That is what GTO is all about.

GTO starts on the flop betting a subset of the pre-flop range to make sure that among the 47 possible turns and then the 46 possible rivers, when considered together, the overall potential of the range is the best.

## Conclusions for the Defender

In the perfect polarization model, the defender’s optimal strategy is crucial to prevent the aggressor from profiting excessively. After careful analysis, it is clear that folding a percentage of your potential range based on the bet to pot ratio is optimal.

Folding at this frequency makes the aggressor’s bluffs break even, preventing them from profiting consistently. It ensures that the defender doesn’t fold too often, allowing the aggressor to exploit them relentlessly. Likewise, folding less frequently opens the door for the aggressor to remove bluffs and exploit the defender.

While it may be tempting for the defender to adjust their strategy and fold more often than optimal, it’s important to consider the potential for counter-exploitation. Aggressively folding may reduce the EV of the aggressor’s value bets, but it leaves the defender vulnerable to aggressive bluffing.

Conversely, calling more often to make the aggressor’s bluffs unprofitable also carries risks. The defender becomes susceptible to the aggressor’s exploitation by removing all -EV bluffs from their strategy.

Therefore, it becomes evident that folding at the ratio dictated by risk-to-reward provides the defender with the optimal approach in our perfect polarization model. This frequency ensures a balanced strategy, mitigates the risk of being exploited, and maintains a level playing field.

Maximizing expected value (EV) across various poker strategies.

## The Gems

In this article, we explored important concepts that can significantly impact your poker game. Let’s recap the key takeaways:

1. Finding the Perfect Middle Ground: As players, we need to strike a balance between folding too much and calling too often. By utilizing GTO analysis, we can find the optimal frequency to prevent opponents from bluffing profitably while avoiding being exploited ourselves.
2. Understanding Opponent Bet Sizing: The size of our opponent’s bet provides valuable information. Larger bet sizes generally indicate stronger hands, while smaller bets might be indicative of bluffs or weaker holdings. It’s crucial to pay attention to these bet sizes and adjust our strategies accordingly.
3. Calculating the Break-Even Point: To determine the profitability of a bluff, we can calculate the break-even point. This is the threshold at which the bluff needs to work in terms of percentage of the total pot invested. Understanding the break-even point helps us make informed decisions on whether to call or fold against bluffs.
4. Defending to Make Opponents’ Bluffs 0EV: The defender’s goal is to prevent opponents from making excessive profits. By defending at a frequency that makes their bluffs break even, we minimize their ability to exploit us. Striking this balance is crucial to maintaining a strong defensive strategy.

Remember, these concepts are just the tip of the iceberg when it comes to mastering poker strategy. By implementing these principles into your game and continually honing your skills, you’ll be on your way to becoming a formidable player at the tables.