The 2008 Bitcoin white paper solves the classic computer science consensus problem.

Issue Statement

The Byzantine Generals Problem (BGP) is called after an allegory in which several generals must collaborate and attack a city at the same time to win (figure 1-left). Any general who retreats at the last minute loses the fight (figure 1-right). Thus, precise messengers and no rogue generals are essential. This is difficult without a trusted central authority.

In their 1982 publication, Leslie Lamport, Robert Shostak, and Marshall Please termed this topic the Byzantine Generals Problem to simplify distributed computer systems.

Consensus in a distributed computer network is the issue. Reaching a consensus on which systems work (and stay in the network) and which don't makes maintaining a network tough (i.e., needs to be removed from network). Challenges include unreliable communication routes between systems and mis-reporting systems.

Solving BGP can let us construct machine learning solutions without single points of failure or trusted central entities. One server hosts model parameters while numerous workers train the model. This study describes fault-tolerant Distributed Byzantine Machine Learning.

Bitcoin invented a mechanism for a distributed network of nodes to agree on which transactions should go into the distributed ledger (blockchain) without a trusted central body. It solved BGP implementation. Satoshi Nakamoto, the pseudonymous bitcoin creator, solved the challenge by cleverly combining cryptography and consensus mechanisms.

Disclaimer

This is not financial advice. It discusses a unique computer science solution.

Bitcoin

Bitcoin's white paper begins:

“A purely peer-to-peer version of electronic cash would allow online payments to be sent directly from one party to another without going through a financial institution.” Source: https://www.ussc.gov/sites/default/files/pdf/training/annual-national-training-seminar/2018/Emerging_Tech_Bitcoin_Crypto.pdf

Bitcoin's main parts:

1. The open-source and versioned bitcoin software that governs how nodes, miners, and the bitcoin token operate.

2. The native kind of token, known as a bitcoin token, may be created by mining (up to 21 million can be created), and it can be transferred between wallet addresses in the bitcoin network.

3. Distributed Ledger, which contains exact copies of the database (or "blockchain") containing each transaction since the first one in January 2009.

4. distributed network of nodes (computers) running the distributed ledger replica together with the bitcoin software. They broadcast the transactions to other peer nodes after validating and accepting them.

5. Proof of work (PoW) is a cryptographic requirement that must be met in order for a miner to be granted permission to add a new block of transactions to the blockchain of the cryptocurrency bitcoin. It takes the form of a valid hash digest. In order to produce new blocks on average every 10 minutes, Bitcoin features a built-in difficulty adjustment function that modifies the valid hash requirement (length of nonce). PoW requires a lot of energy since it must continually generate new hashes at random until it satisfies the criteria.

6. The competing parties known as miners carry out continuous computing processing to address recurrent cryptography issues. Transaction fees and some freshly minted (mined) bitcoin are the rewards they receive. The amount of hashes produced each second—or hash rate—is a measure of mining capacity.

Cryptography, decentralization, and the proof-of-work consensus method are Bitcoin's most unique features.

Bitcoin uses encryption

Bitcoin employs this established cryptography.

1. Hashing

2. digital signatures based on asymmetric encryption

Hashing (SHA-256) (SHA-256)

Hashing converts unique plaintext data into a digest. Creating the plaintext from the digest is impossible. Bitcoin miners generate new hashes using SHA-256 to win block rewards.

A new hash is created from the current block header and a variable value called nonce. To achieve the required hash, mining involves altering the nonce and re-hashing.

The block header contains the previous block hash and a Merkle root, which contains hashes of all transactions in the block. Thus, a chain of blocks with increasing hashes links back to the first block. Hashing protects new transactions and makes the bitcoin blockchain immutable. After a transaction block is mined, it becomes hard to fabricate even a little entry.

Asymmetric Cryptography Digital Signatures

Asymmetric cryptography (public-key encryption) requires each side to have a secret and public key. Public keys (wallet addresses) can be shared with the transaction party, but private keys should not. A message (e.g., bitcoin payment record) can only be signed by the owner (sender) with the private key, but any node or anybody with access to the public key (visible in the blockchain) can verify it. Alex will submit a digitally signed transaction with a desired amount of bitcoin addressed to Bob's wallet to a node to send bitcoin to Bob. Alex alone has the secret keys to authorize that amount. Alex's blockchain public key allows anyone to verify the transaction.

Solution

Now, apply bitcoin to BGP. BGP generals resemble bitcoin nodes. The generals' consensus is like bitcoin nodes' blockchain block selection. Bitcoin software on all nodes can:

Check transactions (i.e., validate digital signatures)

2. Accept and propagate just the first miner to receive the valid hash and verify it accomplished the task. The only way to guess the proper hash is to brute force it by repeatedly producing one with the fixed/current block header and a fresh nonce value.

Thus, PoW and a dispersed network of nodes that accept blocks from miners that solve the unfalsifiable cryptographic challenge solve consensus.

Suppose:

1. Unreliable nodes

2. Unreliable miners

Bitcoin accepts the longest chain if rogue nodes cause divergence in accepted blocks. Thus, rogue nodes must outnumber honest nodes in accepting/forming the longer chain for invalid transactions to reach the blockchain. As of November 2022, 7000 coordinated rogue nodes are needed to takeover the bitcoin network.

Dishonest miners could also try to insert blocks with falsified transactions (double spend, reverse, censor, etc.) into the chain. This requires over 50% (51% attack) of miners (total computational power) to outguess the hash and attack the network. Mining hash rate exceeds 200 million (source). Rewards and transaction fees encourage miners to cooperate rather than attack. Quantum computers may become a threat.

Visit my Quantum Computing post.

Quantum computers—what are they? Quantum computers will have a big influence. towardsdatascience.com

Nodes have more power than miners since they can validate transactions and reject fake blocks. Thus, the network is secure if honest nodes are the majority.

Summary

Table 1 compares three Byzantine Generals Problem implementations.

Bitcoin white paper and implementation solved the consensus challenge of distributed systems without central governance. It solved the illusive Byzantine Generals Problem.

Resources

Resources

1. https://en.wikipedia.org/wiki/Byzantine_fault

2. Source-code for Bitcoin Core Software — https://github.com/bitcoin/bitcoin

3. Bitcoin white paper — https://bitcoin.org/bitcoin.pdf

4. https://en.wikipedia.org/wiki/Bitcoin

5. https://www.microsoft.com/en-us/research/publication/byzantine-generals-problem/

6. https://www.microsoft.com/en-us/research/uploads/prod/2016/12/The-Byzantine-Generals-Problem.pdf

7. https://en.wikipedia.org/wiki/Hash_function

8. https://en.wikipedia.org/wiki/Merkle_tree

9. https://en.wikipedia.org/wiki/SHA-2

10. https://en.wikipedia.org/wiki/Public-key_cryptography

11. https://en.wikipedia.org/wiki/Digital_signature

12. https://en.wikipedia.org/wiki/Proof_of_work

13. https://en.wikipedia.org/wiki/Quantum_cryptography

14. https://dci.mit.edu/bitcoin-security-initiative

15. https://dci.mit.edu/51-attacks

16. Genuinely Distributed Byzantine Machine Learning, El-Mahdi El-Mhamdi et al., 2020. ACM, New York, NY, https://doi.org/10.1145/3382734.3405695

Python Heatmap Technical Indicator

Heatmaps provide an instant overview. They can be used with correlations or to predict reactions or confirm the trend in trading. This article covers RSI heatmap creation.

The Market System

Market regime:

• Bullish trend: The market tends to make higher highs, which indicates that the overall trend is upward.

• Sideways: The market tends to fluctuate while staying within predetermined zones.

• Bearish trend: The market has the propensity to make lower lows, indicating that the overall trend is downward.

Most tools detect the trend, but we cannot predict the next state. The best way to solve this problem is to assume the current state will continue and trade any reactions, preferably in the trend.

If the EURUSD is above its moving average and making higher highs, a trend-following strategy would be to wait for dips before buying and assuming the bullish trend will continue.

Indicator of Relative Strength

J. Welles Wilder Jr. introduced the RSI, a popular and versatile technical indicator. Used as a contrarian indicator to exploit extreme reactions. Calculating the default RSI usually involves these steps:

• Determine the difference between the closing prices from the prior ones.

• Distinguish between the positive and negative net changes.

• Create a smoothed moving average for both the absolute values of the positive net changes and the negative net changes.

• Take the difference between the smoothed positive and negative changes. The Relative Strength RS will be the name we use to describe this calculation.

• To obtain the RSI, use the normalization formula shown below for each time step.

The 13-period RSI and black GBPUSD hourly values are shown above. RSI bounces near 25 and pauses around 75. Python requires a four-column OHLC array for RSI coding.

import numpy as np
def add_column(data, times):
    
    for i in range(1, times + 1):
    
        new = np.zeros((len(data), 1), dtype = float)
        
        data = np.append(data, new, axis = 1)
    return data
def delete_column(data, index, times):
    
    for i in range(1, times + 1):
    
        data = np.delete(data, index, axis = 1)
    return data
def delete_row(data, number):
    
    data = data[number:, ]
    
    return data
def ma(data, lookback, close, position): 
    
    data = add_column(data, 1)
    
    for i in range(len(data)):
           
            try:
                
                data[i, position] = (data[i - lookback + 1:i + 1, close].mean())
            
            except IndexError:
                
                pass
            
    data = delete_row(data, lookback)
    
    return data
def smoothed_ma(data, alpha, lookback, close, position):
    
    lookback = (2 * lookback) - 1
    
    alpha = alpha / (lookback + 1.0)
    
    beta  = 1 - alpha
    
    data = ma(data, lookback, close, position)
    data[lookback + 1, position] = (data[lookback + 1, close] * alpha) + (data[lookback, position] * beta)
    for i in range(lookback + 2, len(data)):
        
            try:
                
                data[i, position] = (data[i, close] * alpha) + (data[i - 1, position] * beta)
        
            except IndexError:
                
                pass
            
    return data
def rsi(data, lookback, close, position):
    
    data = add_column(data, 5)
    
    for i in range(len(data)):
        
        data[i, position] = data[i, close] - data[i - 1, close]
     
    for i in range(len(data)):
        
        if data[i, position] > 0:
            
            data[i, position + 1] = data[i, position]
            
        elif data[i, position] < 0:
            
            data[i, position + 2] = abs(data[i, position])
            
    data = smoothed_ma(data, 2, lookback, position + 1, position + 3)
    data = smoothed_ma(data, 2, lookback, position + 2, position + 4)
    data[:, position + 5] = data[:, position + 3] / data[:, position + 4]
    
    data[:, position + 6] = (100 - (100 / (1 + data[:, position + 5])))
    data = delete_column(data, position, 6)
    data = delete_row(data, lookback)
    return data

Make sure to focus on the concepts and not the code. You can find the codes of most of my strategies in my books. The most important thing is to comprehend the techniques and strategies.

My weekly market sentiment report uses complex and simple models to understand the current positioning and predict the future direction of several major markets. Check out the report here:

Using the Heatmap to Find the Trend

RSI trend detection is easy but useless. Bullish and bearish regimes are in effect when the RSI is above or below 50, respectively. Tracing a vertical colored line creates the conditions below. How:

• When the RSI is higher than 50, a green vertical line is drawn.

• When the RSI is lower than 50, a red vertical line is drawn.

Zooming out yields a basic heatmap, as shown below.

Plot code:

def indicator_plot(data, second_panel, window = 250):
    fig, ax = plt.subplots(2, figsize = (10, 5))
    sample = data[-window:, ]
    for i in range(len(sample)):
        ax[0].vlines(x = i, ymin = sample[i, 2], ymax = sample[i, 1], color = 'black', linewidth = 1)  
        if sample[i, 3] > sample[i, 0]:
            ax[0].vlines(x = i, ymin = sample[i, 0], ymax = sample[i, 3], color = 'black', linewidth = 1.5)  
        if sample[i, 3] < sample[i, 0]:
            ax[0].vlines(x = i, ymin = sample[i, 3], ymax = sample[i, 0], color = 'black', linewidth = 1.5)  
        if sample[i, 3] == sample[i, 0]:
            ax[0].vlines(x = i, ymin = sample[i, 3], ymax = sample[i, 0], color = 'black', linewidth = 1.5)  
    ax[0].grid() 
    for i in range(len(sample)):
        if sample[i, second_panel] > 50:
            ax[1].vlines(x = i, ymin = 0, ymax = 100, color = 'green', linewidth = 1.5)  
        if sample[i, second_panel] < 50:
            ax[1].vlines(x = i, ymin = 0, ymax = 100, color = 'red', linewidth = 1.5)  
    ax[1].grid()
indicator_plot(my_data, 4, window = 500)

Call RSI on your OHLC array's fifth column. 4. Adjusting lookback parameters reduces lag and false signals. Other indicators and conditions are possible.

Another suggestion is to develop an RSI Heatmap for Extreme Conditions.

Contrarian indicator RSI. The following rules apply:

• Whenever the RSI is approaching the upper values, the color approaches red.

• The color tends toward green whenever the RSI is getting close to the lower values.

Zooming out yields a basic heatmap, as shown below.

Plot code:

import matplotlib.pyplot as plt
def indicator_plot(data, second_panel, window = 250):
    fig, ax = plt.subplots(2, figsize = (10, 5))
    sample = data[-window:, ]
    for i in range(len(sample)):
        ax[0].vlines(x = i, ymin = sample[i, 2], ymax = sample[i, 1], color = 'black', linewidth = 1)  
        if sample[i, 3] > sample[i, 0]:
            ax[0].vlines(x = i, ymin = sample[i, 0], ymax = sample[i, 3], color = 'black', linewidth = 1.5)  
        if sample[i, 3] < sample[i, 0]:
            ax[0].vlines(x = i, ymin = sample[i, 3], ymax = sample[i, 0], color = 'black', linewidth = 1.5)  
        if sample[i, 3] == sample[i, 0]:
            ax[0].vlines(x = i, ymin = sample[i, 3], ymax = sample[i, 0], color = 'black', linewidth = 1.5)  
    ax[0].grid() 
    for i in range(len(sample)):
        if sample[i, second_panel] > 90:
            ax[1].vlines(x = i, ymin = 0, ymax = 100, color = 'red', linewidth = 1.5)  
        if sample[i, second_panel] > 80 and sample[i, second_panel] < 90:
            ax[1].vlines(x = i, ymin = 0, ymax = 100, color = 'darkred', linewidth = 1.5)  
        if sample[i, second_panel] > 70 and sample[i, second_panel] < 80:
            ax[1].vlines(x = i, ymin = 0, ymax = 100, color = 'maroon', linewidth = 1.5)  
        if sample[i, second_panel] > 60 and sample[i, second_panel] < 70:
            ax[1].vlines(x = i, ymin = 0, ymax = 100, color = 'firebrick', linewidth = 1.5) 
        if sample[i, second_panel] > 50 and sample[i, second_panel] < 60:
            ax[1].vlines(x = i, ymin = 0, ymax = 100, color = 'grey', linewidth = 1.5) 
        if sample[i, second_panel] > 40 and sample[i, second_panel] < 50:
            ax[1].vlines(x = i, ymin = 0, ymax = 100, color = 'grey', linewidth = 1.5) 
        if sample[i, second_panel] > 30 and sample[i, second_panel] < 40:
            ax[1].vlines(x = i, ymin = 0, ymax = 100, color = 'lightgreen', linewidth = 1.5)
        if sample[i, second_panel] > 20 and sample[i, second_panel] < 30:
            ax[1].vlines(x = i, ymin = 0, ymax = 100, color = 'limegreen', linewidth = 1.5) 
        if sample[i, second_panel] > 10 and sample[i, second_panel] < 20:
            ax[1].vlines(x = i, ymin = 0, ymax = 100, color = 'seagreen', linewidth = 1.5)  
        if sample[i, second_panel] > 0 and sample[i, second_panel] < 10:
            ax[1].vlines(x = i, ymin = 0, ymax = 100, color = 'green', linewidth = 1.5)
    ax[1].grid()
indicator_plot(my_data, 4, window = 500)

Dark green and red areas indicate imminent bullish and bearish reactions, respectively. RSI around 50 is grey.

Summary

To conclude, my goal is to contribute to objective technical analysis, which promotes more transparent methods and strategies that must be back-tested before implementation.

Technical analysis will lose its reputation as subjective and unscientific.

When you find a trading strategy or technique, follow these steps:

• Put emotions aside and adopt a critical mindset.

• Test it in the past under conditions and simulations taken from real life.

• Try optimizing it and performing a forward test if you find any potential.

• Transaction costs and any slippage simulation should always be included in your tests.

• Risk management and position sizing should always be considered in your tests.

After checking the above, monitor the strategy because market dynamics may change and make it unprofitable.

The same way Amazon killed bookstores. You just can’t see it yet.

Shultz globalized coffee. Before Starbucks, coffee sucked.

All accounts say 1970s coffee was awful.

Starbucks had three stores selling ground Indonesian coffee in the 1980s.

What a show!

A year after joining the company at 29, Shultz traveled to Italy for R&D.

He noticed the coffee shops' sense of theater and community and realized Starbucks was in the wrong business.

Integrating coffee and destination created a sense of community in the store.

Brilliant!

He told Starbucks' founders about his experience.

They disapproved.

For two years.

Shultz left and opened an Italian coffee shop chain like any good entrepreneur.

Starbucks ran into financial trouble, so the founders offered to sell to Shultz.

Shultz bought Starbucks in 1987 for $3.8 million, including six stores and a payment plan.

Starbucks is worth $100.79Billion, per Google Finance.

26,500 times Shultz's initial investment

Starbucks is releasing its own NFT Platform under Shultz and his early Vision.

This year, Starbucks Odyssey launches. The new digital experience combines a Loyalty Rewards program with NFT.

The side chain Polygon-based platform doesn't require a Crypto Wallet. Customers can earn and buy digital assets to unlock incentives and experiences.

They've removed all friction, making it more immersive and convenient than a coffee shop.

Brilliant!

NFTs are the access coupon to their digital community, but they don't highlight the technology.

They prioritize consumer experience by adding non-technical users to Web3. Their collectables are called journey stamps, not NFTs.

No mention of bundled gas fees.

Brady Brewer, Starbucks' CMO, said;

“It happens to be built on blockchain and web3 technologies, but the customer — to be honest — may very well not even know that what they’re doing is interacting with blockchain technology. It’s just the enabler,”

Rewards members will log into a web app using their loyalty program credentials to access Starbucks Odyssey. They won't know about blockchain transactions.

Join the waitlist here

Starbucks has just dealt its rivals a devastating blow.

It generates more than ten times the revenue of its closest competitor Costa Coffee.

The coffee giant is booming.

Starbucks is ahead of its competitors. No wonder.

They have an innovative, adaptable leadership team.

Starbucks' DNA challenges the narrative, especially when others reject their ideas.

I’m off for a cappuccino.

"GTO" (Game Theory Optimal) has been used a lot in poker recently. To clarify its meaning and application, the aim of this article is to define what it is, when to use it when playing, what strategies to apply for how to play GTO poker, for beginner and more advanced players!

Poker GTO

In poker, you can choose between two main winning strategies:

Exploitative play maximizes expected value (EV) by countering opponents' sub-optimal plays and weaker tendencies. Yes, playing this way opens you up to being exploited, but the weaker opponents you're targeting won't change their game to counteract this, allowing you to reap maximum profits over the long run.

GTO (Game-Theory Optimal): You try to play perfect poker, which forces your opponents to make mistakes (which is where almost all of your profit will be derived from). It mixes bluffs or semi-bluffs with value bets, clarifies bet sizes, and more.

GTO vs. Exploitative: Which is Better in Poker?

Before diving into GTO poker strategy, it's important to know which of these two play styles is more profitable for beginners and advanced players. The simple answer is probably both, but usually more exploitable.

Most players don't play GTO poker and can be exploited in their gameplay and strategy, allowing for more profits to be made using an exploitative approach. In fact, it’s only in some of the largest games at the highest stakes that GTO concepts are fully utilized and seen in practice, and even then, exploitative plays are still sometimes used.

Knowing, understanding, and applying GTO poker basics will create a solid foundation for your poker game. It's also important to understand GTO so you can deviate from it to maximize profits.

GTO Poker Strategy

According to Ed Miller's book "Poker's 1%," the most fundamental concept that only elite poker players understand is frequency, which could be in relation to cbets, bluffs, folds, calls, raises, etc.

GTO poker solvers (downloadable online software) give solutions for how to play optimally in any given spot and often recommend using mixed strategies based on select frequencies.

In a river situation, a solver may tell you to call 70% of the time and fold 30%. It may also suggest calling 50% of the time, folding 35% of the time, and raising 15% of the time (with a certain range of hands).

Frequencies are a fundamental and often unrecognized part of poker, but they run through these 5 GTO concepts.

1. Preflop ranges

To compensate for positional disadvantage, out-of-position players must open tighter hand ranges.

Premium starting hands aren't enough, though. Considering GTO poker ranges and principles, you want a good, balanced starting hand range from each position with at least some hands that can make a strong poker hand regardless of the flop texture (low, mid, high, disconnected, etc).

Below is a GTO preflop beginner poker chart for online 6-max play, showing which hand ranges one should open-raise with. Table positions are color-coded (see key below).

NOTE: For GTO play, it's advisable to use a mixed strategy for opening in the small blind, combining open-limps and open-raises for various hands. This cannot be illustrated with the color system used for the chart.

Choosing which hands to play is often a math problem, as discussed below.

Other preflop GTO poker charts include which hands to play after a raise, which to 3bet, etc. Solvers can help you decide which preflop hands to play (call, raise, re-raise, etc.).

2. Pot Odds

Always make +EV decisions that profit you as a poker player. Understanding pot odds (and equity) can help.

Postflop Pot Odds

Let’s say that we have JhTh on a board of 9h8h2s4c (open-ended straight-flush draw).  We have $40 left and $50 in the pot. He has you covered and goes all-in. As calling or folding are our only options, playing GTO involves calculating whether a call is +EV or –EV. (The hand was empty.)

Any remaining heart, Queen, or 7 wins the hand. This means we can improve 15 of 46 unknown cards, or 32.6% of the time.

What if our opponent has a set? The 4h or 2h could give us a flush, but it could also give the villain a boat. If we reduce outs from 15 to 14.5, our equity would be 31.5%.

We must now calculate pot odds.

(bet/(our bet+pot)) = pot odds

= $50 / ($40 + $90)

= $40 / $130

= 30.7%

To make a profitable call, we need at least 30.7% equity. This is a profitable call as we have 31.5% equity (even if villain has a set). Yes, we will lose most of the time, but we will make a small profit in the long run, making a call correct.

Pot odds aren't just for draws, either. If an opponent bets 50% pot, you get 3 to 1 odds on a call, so you must win 25% of the time to be profitable. If your current hand has more than 25% equity against your opponent's perceived range, call.

Preflop Pot Odds

Preflop, you raise to 3bb and the button 3bets to 9bb. You must decide how to act. In situations like these, we can actually use pot odds to assist our decision-making.

This pot is:

(our open+3bet size+small blind+big blind)

(3bb+9bb+0.5bb+1bb)

= 13.5

This means we must call 6bb to win a pot of 13.5bb, which requires 30.7% equity against the 3bettor's range.

Three additional factors must be considered:

Being out of position on our opponent makes it harder to realize our hand's equity, as he can use his position to put us in tough spots. To profitably continue against villain's hand range, we should add 7% to our equity.

Implied Odds / Reverse Implied Odds: The ability to win or lose significantly more post-flop (than pre-flop) based on our remaining stack.

While statistics on 3bet stats can be gained with a large enough sample size (i.e. 8% 3bet stat from button), the numbers don't tell us which 8% of hands villain could be 3betting with. Both polarized and depolarized charts below show 8% of possible hands.

7.4% of hands are depolarized.

Polarized Hand range (7.54%):

Each hand range has different contents. We don't know if he 3bets some hands and calls or folds others.

Using an exploitable strategy can help you play a hand range correctly. The next GTO concept will make things easier.

3. Minimum Defense Frequency:

This concept refers to the % of our range we must continue with (by calling or raising) to avoid being exploited by our opponents. This concept is most often used off-table and is difficult to apply in-game.

These beginner GTO concepts will help your decision-making during a hand, especially against aggressive opponents.

MDF formula:

MDF = POT SIZE/(POT SIZE+BET SIZE)

Here's a poker GTO chart of common bet sizes and minimum defense frequency.

Take the number of hand combos in your starting hand range and use the MDF to determine which hands to continue with. Choose hands with the most playability and equity against your opponent's betting range.

Say you open-raise HJ and BB calls. Qh9h6c flop. Your opponent leads you for a half-pot bet. MDF suggests keeping 67% of our range.

Using the above starting hand chart, we can determine that the HJ opens 254 combos:

We must defend 67% of these hands, or 170 combos, according to MDF. Hands we should keep include:

Flush draws

Open-Ended Straight Draws

Gut-Shot Straight Draws

Overcards

Any Pair or better

So, our flop continuing range could be:

Some highlights:

Fours and fives have little chance of improving on the turn or river.

We only continue with AX hearts (with a flush draw) without a pair or better.

We'll also include 4 AJo combos, all of which have the Ace of hearts, and AcJh, which can block a backdoor nut flush combo.

Let's assume all these hands are called and the turn is blank (2 of spades). Opponent bets full-pot. MDF says we must defend 50% of our flop continuing range, or 85 of 170 combos, to be unexploitable. This strategy includes our best flush draws, straight draws, and made hands.

Here, we keep combining:

Nut flush draws

Pair + flush draws

GS + flush draws

Second Pair, Top Kicker+

One combo of JJ that doesn’t block the flush draw or backdoor flush draw.

On the river, we can fold our missed draws and keep our best made hands. When calling with weaker hands, consider blocker effects and card removal to avoid overcalling and decide which combos to continue.

4. Poker GTO Bet Sizing

To avoid being exploited, balance your bluffs and value bets. Your betting range depends on how much you bet (in relation to the pot). This concept only applies on the river, as draws (bluffs) on the flop and turn still have equity (and are therefore total bluffs).

On the flop, you want a 2:1 bluff-to-value-bet ratio. On the flop, there won't be as many made hands as on the river, and your bluffs will usually contain equity. The turn should have a "bluffing" ratio of 1:1. Use the chart below to determine GTO river bluff frequencies (relative to your bet size):

This chart relates to your opponent's pot odds. If you bet 50% pot, your opponent gets 3:1 odds and must win 25% of the time to call. Poker GTO theory suggests including 25% bluff combinations in your betting range so you're indifferent to your opponent calling or folding.

Best river bluffs don't block hands you want your opponent to have (or not have). For example, betting with missed Ace-high flush draws is often a mistake because you block a missed flush draw you want your opponent to have when bluffing on the river (meaning that it would subsequently be less likely he would have it, if you held two of the flush draw cards). Ace-high usually has some river showdown value.

If you had a 3-flush on the river and wanted to raise, you could bluff raise with AX combos holding the bluff suit Ace. Blocking the nut flush prevents your opponent from using that combo.

5. Bet Sizes and Frequency

GTO beginner strategies aren't just bluffs and value bets. They show how often and how much to bet in certain spots. Top players have benefited greatly from poker solvers, which we'll discuss next.

GTO Poker Software

In recent years, various poker GTO solvers have been released to help beginner, intermediate, and advanced players play balanced/GTO poker in various situations.

PokerSnowie and PioSolver are popular GTO and poker study programs.

While you can't compute players' hand ranges and what hands to bet or check with in real time, studying GTO play strategies with these programs will pay off. It will improve your poker thinking and understanding.

Solvers can help you balance ranges, choose optimal bet sizes, and master cbet frequencies.

GTO Poker Tournament

Late-stage tournaments have shorter stacks than cash games. In order to follow GTO poker guidelines, Nash charts have been created, tweaked, and used for many years (and also when to call, depending on what number of big blinds you have when you find yourself shortstacked).

The charts are for heads-up push/fold. In a multi-player game, the "pusher" chart can only be used if play is folded to you in the small blind. The "caller" chart can only be used if you're in the big blind and assumes a small blind "pusher" (with a much wider range than if a player in another position was open-shoving).

Divide the pusher chart's numbers by 2 to see which hand to use from the Button. Divide the original chart numbers by 4 to find the CO's pushing range. Some of the figures will be impossible to calculate accurately for the CO or positions to the right of the blinds because the chart's highest figure is "20+" big blinds, which is also used for a wide range of hands in the push chart.

Both of the GTO charts below are ideal for heads-up play, but exploitable HU shortstack strategies can lead to more +EV decisions against certain opponents. Following the charts will make your play GTO and unexploitable.

Poker pro Max Silver created the GTO push/fold software SnapShove. (It's accessible online at www.snapshove.com or as iOS or Android apps.)

Players can access GTO shove range examples in the full version. (You can customize the number of big blinds you have, your position, the size of the ante, and many other options.)

In Conclusion

Due to the constantly changing poker landscape, players are always improving their skills. Exploitable strategies often yield higher profit margins than GTO-based approaches, but knowing GTO beginner and advanced concepts can give you an edge for a few reasons.

It creates a solid gameplay base.

Having a baseline makes it easier to exploit certain villains.

You can avoid leveling wars with your opponents by making sound poker decisions based on GTO strategy.

It doesn't require assuming opponents' play styles.

Not results-oriented.

This is just the beginning of GTO and poker theory. Consider investing in the GTO poker solver software listed above to improve your game.