An approximate introduction to how zk-SNARKs are possible (part 1)
You can make a proof for the statement "I know a secret number such that if you take the word ‘cow', add the number to the end, and SHA256 hash it 100 million times, the output starts with 0x57d00485aa". The verifier can verify the proof far more quickly than it would take for them to run 100 million hashes themselves, and the proof would also not reveal what the secret number is.
In the context of blockchains, this has 2 very powerful applications: Perhaps the most powerful cryptographic technology to come out of the last decade is general-purpose succinct zero knowledge proofs, usually called zk-SNARKs ("zero knowledge succinct arguments of knowledge"). A zk-SNARK allows you to generate a proof that some computation has some particular output, in such a way that the proof can be verified extremely quickly even if the underlying computation takes a very long time to run. The "ZK" part adds an additional feature: the proof can keep some of the inputs to the computation hidden.
You can make a proof for the statement "I know a secret number such that if you take the word ‘cow', add the number to the end, and SHA256 hash it 100 million times, the output starts with 0x57d00485aa". The verifier can verify the proof far more quickly than it would take for them to run 100 million hashes themselves, and the proof would also not reveal what the secret number is.
In the context of blockchains, this has two very powerful applications:
- Scalability: if a block takes a long time to verify, one person can verify it and generate a proof, and everyone else can just quickly verify the proof instead
- Privacy: you can prove that you have the right to transfer some asset (you received it, and you didn't already transfer it) without revealing the link to which asset you received. This ensures security without unduly leaking information about who is transacting with whom to the public.
But zk-SNARKs are quite complex; indeed, as recently as in 2014-17 they were still frequently called "moon math". The good news is that since then, the protocols have become simpler and our understanding of them has become much better. This post will try to explain how ZK-SNARKs work, in a way that should be understandable to someone with a medium level of understanding of mathematics.
Why ZK-SNARKs "should" be hard
Let us take the example that we started with: we have a number (we can encode "cow" followed by the secret input as an integer), we take the SHA256 hash of that number, then we do that again another 99,999,999 times, we get the output, and we check what its starting digits are. This is a huge computation.
A "succinct" proof is one where both the size of the proof and the time required to verify it grow much more slowly than the computation to be verified. If we want a "succinct" proof, we cannot require the verifier to do some work per round of hashing (because then the verification time would be proportional to the computation). Instead, the verifier must somehow check the whole computation without peeking into each individual piece of the computation.
One natural technique is random sampling: how about we just have the verifier peek into the computation in 500 different places, check that those parts are correct, and if all 500 checks pass then assume that the rest of the computation must with high probability be fine, too?
Such a procedure could even be turned into a non-interactive proof using the Fiat-Shamir heuristic: the prover computes a Merkle root of the computation, uses the Merkle root to pseudorandomly choose 500 indices, and provides the 500 corresponding Merkle branches of the data. The key idea is that the prover does not know which branches they will need to reveal until they have already "committed to" the data. If a malicious prover tries to fudge the data after learning which indices are going to be checked, that would change the Merkle root, which would result in a new set of random indices, which would require fudging the data again... trapping the malicious prover in an endless cycle.
But unfortunately there is a fatal flaw in naively applying random sampling to spot-check a computation in this way: computation is inherently fragile. If a malicious prover flips one bit somewhere in the middle of a computation, they can make it give a completely different result, and a random sampling verifier would almost never find out.
It only takes one deliberately inserted error, that a random check would almost never catch, to make a computation give a completely incorrect result.
If tasked with the problem of coming up with a zk-SNARK protocol, many people would make their way to this point and then get stuck and give up. How can a verifier possibly check every single piece of the computation, without looking at each piece of the computation individually? There is a clever solution.
see part 2
(Edited)
More on Web3 & Crypto

Amelie Carver
3 years ago
Web3 Needs More Writers to Educate Us About It
WRITE FOR THE WEB3
Why web3’s messaging is lost and how crypto winter is growing growth seeds
People interested in crypto, blockchain, and web3 typically read Bitcoin and Ethereum's white papers. It's a good idea. Documents produced for developers and academia aren't always the ideal resource for beginners.
Given the surge of extremely technical material and the number of fly-by-nights, rug pulls, and other scams, it's little wonder mainstream audiences regard the blockchain sector as an expensive sideshow act.
What's the solution?
Web3 needs more than just builders.
After joining TikTok, I followed Amy Suto of SutoScience. Amy switched from TV scriptwriting to IT copywriting years ago. She concentrates on web3 now. Decentralized autonomous organizations (DAOs) are seeking skilled copywriters for web3.
Amy has found that web3's basics are easy to grasp; you don't need technical knowledge. There's a paradigm shift in knowing the basics; be persistent and patient.
Apple is positioning itself as a data privacy advocate, leveraging web3's zero-trust ethos on data ownership.
Finn Lobsien, who writes about web3 copywriting for the Mirror and Twitter, agrees: acronyms and abstractions won't do.
Web3 preached to the choir. Curious newcomers have only found whitepapers and scams when trying to learn why the community loves it. No wonder people resist education and buy-in.
Due to the gender gap in crypto (Crypto Bro is not just a stereotype), it attracts people singing to the choir or trying to cash in on the next big thing.
Last year, the industry was booming, so writing wasn't necessary. Now that the bear market has returned (for everyone, but especially web3), holding readers' attention is a valuable skill.
White papers and the Web3
Why does web3 rely so much on non-growth content?
Businesses must polish and improve their messaging moving into the 2022 recession. The 2021 tech boom provided such a sense of affluence and (unsustainable) growth that no one needed great marketing material. The market found them.
This was especially true for web3 and the first-time crypto believers. Obviously. If they knew which was good.
White papers help. White papers are highly technical texts that walk a reader through a product's details. How Does a White Paper Help Your Business and That White Paper Guy discuss them.
They're meant for knowledgeable readers. Investors and the technical (academic/developer) community read web3 white papers. White papers are used when a product is extremely technical or difficult to assist an informed reader to a conclusion. Web3 uses them most often for ICOs (initial coin offerings).
White papers for web3 education help newcomers learn about the web3 industry's components. It's like sending a first-grader to the Annotated Oxford English Dictionary to learn to read. It's a reference, not a learning tool, for words.
Newcomers can use platforms that teach the basics. These included Coinbase's Crypto Basics tutorials or Cryptochicks Academy, founded by the mother of Ethereum's inventor to get more women utilizing and working in crypto.
Discord and Web3 communities
Discord communities are web3's opposite. Discord communities involve personal communications and group involvement.
Online audience growth begins with community building. User personas prefer 1000 dedicated admirers over 1 million lukewarm followers, and the language is much more easygoing. Discord groups are renowned for phishing scams, compromised wallets, and incorrect information, especially since the crypto crisis.
White papers and Discord increase industry insularity. White papers are complicated, and Discord has a high risk threshold.
Web3 and writing ads
Copywriting is emotional, but white papers are logical. It uses the brain's quick-decision centers. It's meant to make the reader invest immediately.
Not bad. People think sales are sleazy, but they can spot the poor things.
Ethical copywriting helps you reach the correct audience. People who gain a following on Medium are likely to have copywriting training and a readership (or three) in mind when they publish. Tim Denning and Sinem Günel know how to identify a target audience and make them want to learn more.
In a fast-moving market, copywriting is less about long-form content like sales pages or blogs, but many organizations do. Instead, the copy is concise, individualized, and high-value. Tweets, email marketing, and IM apps (Discord, Telegram, Slack to a lesser extent) keep engagement high.
What does web3's messaging lack? As DAOs add stricter copyrighting, narrative and connecting tales seem to be missing.
Web3 is passionate about constructing the next internet. Now, they can connect their passion to a specific audience so newcomers understand why.

Nabil Alouani
3 years ago
Why Cryptocurrency Is Not Dead Despite the FTX Scam
A fraud, free-market, antifragility tale
Crypto's only rival is public opinion.
In less than a week, mainstream media, bloggers, and TikTokers turned on FTX's founder.
While some were surprised, almost everyone with a keyboard and a Twitter account predicted the FTX collapse. These financial oracles should have warned the 1.2 million people Sam Bankman-Fried duped.
After happening, unexpected events seem obvious to our brains. It's a bug and a feature because it helps us cope with disasters and makes our reasoning suck.
Nobody predicted the FTX debacle. Bloomberg? Politicians. Non-famous. No cryptologists. Who?
When FTX imploded, taking billions of dollars with it, an outrage bomb went off, and the resulting shockwave threatens the crypto market's existence.
As someone who lost more than $78,000 in a crypto scam in 2020, I can only understand people’s reactions. When the dust settles and rationality returns, we'll realize this is a natural occurrence in every free market.
What specifically occurred with FTX? (Skip if you are aware.)
FTX is a cryptocurrency exchange where customers can trade with cash. It reached #3 in less than two years as the fastest-growing platform of its kind.
FTX's performance helped make SBF the crypto poster boy. Other reasons include his altruistic public image, his support for the Democrats, and his company Alameda Research.
Alameda Research made a fortune arbitraging Bitcoin.
Arbitrage trading uses small price differences between two markets to make money. Bitcoin costs $20k in Japan and $21k in the US. Alameda Research did that for months, making $1 million per day.
Later, as its capital grew, Alameda expanded its trading activities and began investing in other companies.
Let's now discuss FTX.
SBF's diabolic master plan began when he used FTX-created FTT coins to inflate his trading company's balance sheets. He used inflated Alameda numbers to secure bank loans.
SBF used money he printed himself as collateral to borrow billions for capital. Coindesk exposed him in a report.
One of FTX's early investors tweeted that he planned to sell his FTT coins over the next few months. This would be a minor event if the investor wasn't Binance CEO Changpeng Zhao (CZ).
The crypto space saw a red WARNING sign when CZ cut ties with FTX. Everyone with an FTX account and a brain withdrew money. Two events followed. FTT fell from $20 to $4 in less than 72 hours, and FTX couldn't meet withdrawal requests, spreading panic.
SBF reassured FTX users on Twitter. Good assets.
He lied.
SBF falsely claimed FTX had a liquidity crunch. At the time of his initial claims, FTX owed about $8 billion to its customers. Liquidity shortages are usually minor. To get cash, sell assets. In the case of FTX, the main asset was printed FTT coins.
Sam wouldn't get out of trouble even if he slashed the discount (from $20 to $4) and sold every FTT. He'd flood the crypto market with his homemade coins, causing the price to crash.
SBF was trapped. He approached Binance about a buyout, which seemed good until Binance looked at FTX's books.
Binance's tweet ended SBF, and he had to apologize, resign as CEO, and file for bankruptcy.
Bloomberg estimated Sam's net worth to be zero by the end of that week. 0!
But that's not all. Twitter investigations exposed fraud at FTX and Alameda Research. SBF used customer funds to trade and invest in other companies.
Thanks to the Twitter indie reporters who made the mainstream press look amateurish. Some Twitter detectives didn't sleep for 30 hours to find answers. Others added to existing threads. Memes were hilarious.
One question kept repeating in my bald head as I watched the Blue Bird. Sam, WTF?
Then I understood.
SBF wanted that FTX becomes a bank.
Think about this. FTX seems healthy a few weeks ago. You buy 2 bitcoins using FTX. You'd expect the platform to take your dollars and debit your wallet, right?
No. They give I-Owe-Yous.
FTX records owing you 2 bitcoins in its internal ledger but doesn't credit your account. Given SBF's tricks, I'd bet on nothing.
What happens if they don't credit my account with 2 bitcoins? Your money goes into FTX's capital, where SBF and his friends invest in marketing, political endorsements, and buying other companies.
Over its two-year existence, FTX invested in 130 companies. Once they make a profit on their purchases, they'll pay you and keep the rest.
One detail makes their strategy dumb. If all FTX customers withdraw at once, everything collapses.
Financially savvy people think FTX's collapse resembles a bank run, and they're right. SBF designed FTX to operate like a bank.
You expect your bank to open a drawer with your name and put $1,000 in it when you deposit $1,000. They deposit $100 in your drawer and create an I-Owe-You for $900. What happens to $900?
Let's sum it up: It's boring and headache-inducing.
When you deposit money in a bank, they can keep 10% and lend the rest. Fractional Reserve Banking is a popular method. Fractional reserves operate within and across banks.
Fractional reserve banking generates $10,000 for every $1,000 deposited. People will pay off their debt plus interest.
As long as banks work together and the economy grows, their model works well.
SBF tried to replicate the system but forgot two details. First, traditional banks need verifiable collateral like real estate, jewelry, art, stocks, and bonds, not digital coupons. Traditional banks developed a liquidity buffer. The Federal Reserve (or Central Bank) injects massive cash into troubled banks.
Massive cash injections come from taxpayers. You and I pay for bankers' mistakes and annual bonuses. Yes, you may think banking is rigged. It's rigged, but it's the best financial game in 150 years. We accept its flaws, including bailouts for too-big-to-fail companies.
Anyway.
SBF wanted Binance's bailout. Binance said no, which was good for the crypto market.
Free markets are resilient.
Nassim Nicholas Taleb coined the term antifragility.
“Some things benefit from shocks; they thrive and grow when exposed to volatility, randomness, disorder, and stressors and love adventure, risk, and uncertainty. Yet, in spite of the ubiquity of the phenomenon, there is no word for the exact opposite of fragile. Let us call it antifragile. Antifragility is beyond resilience or robustness. The resilient resists shocks and stays the same; the antifragile gets better.”
The easiest way to understand how antifragile systems behave is to compare them with other types of systems.
Glass is like a fragile system. It snaps when shocked.
Similar to rubber, a resilient system. After a stressful episode, it bounces back.
A system that is antifragile is similar to a muscle. As it is torn in the gym, it gets stronger.
Time-changed things are antifragile. Culture, tech innovation, restaurants, revolutions, book sales, cuisine, economic success, and even muscle shape. These systems benefit from shocks and randomness in different ways, but they all pay a price for antifragility.
Same goes for the free market and financial institutions. Taleb's book uses restaurants as an example and ends with a reference to the 2008 crash.
“Restaurants are fragile. They compete with each other. But the collective of local restaurants is antifragile for that very reason. Had restaurants been individually robust, hence immortal, the overall business would be either stagnant or weak and would deliver nothing better than cafeteria food — and I mean Soviet-style cafeteria food. Further, it [the overall business] would be marred with systemic shortages, with once in a while a complete crisis and government bailout.”
Imagine the same thing with banks.
Independent banks would compete to offer the best services. If one of these banks fails, it will disappear. Customers and investors will suffer, but the market will recover from the dead banks' mistakes.
This idea underpins a free market. Bitcoin and other cryptocurrencies say this when criticizing traditional banking.
The traditional banking system's components never die. When a bank fails, the Federal Reserve steps in with a big taxpayer-funded check. This hinders bank evolution. If you don't let banking cells die and be replaced, your financial system won't be antifragile.
The interdependence of banks (centralization) means that one bank's mistake can sink the entire fleet, which brings us to SBF's ultimate travesty with FTX.
FTX has left the cryptocurrency gene pool.
FTX should be decentralized and independent. The super-star scammer invested in more than 130 crypto companies and linked them, creating a fragile banking-like structure. FTX seemed to say, "We exist because centralized banks are bad." But we'll be good, unlike the centralized banking system.
FTX saved several companies, including BlockFi and Voyager Digital.
FTX wanted to be a crypto bank conglomerate and Federal Reserve. SBF wanted to monopolize crypto markets. FTX wanted to be in bed with as many powerful people as possible, so SBF seduced politicians and celebrities.
Worst? People who saw SBF's plan flaws praised him. Experts, newspapers, and crypto fans praised FTX. When billions pour in, it's hard to realize FTX was acting against its nature.
Then, they act shocked when they realize FTX's fall triggered a domino effect. Some say the damage could wipe out the crypto market, but that's wrong.
Cell death is different from body death.
FTX is out of the game despite its size. Unfit, it fell victim to market natural selection.
Next?
The challengers keep coming. The crypto economy will improve with each failure.
Free markets are antifragile because their fragile parts compete, fostering evolution. With constructive feedback, evolution benefits customers and investors.
FTX shows that customers don't like being scammed, so the crypto market's health depends on them. Charlatans and con artists are eliminated quickly or slowly.
Crypto isn't immune to collapse. Cryptocurrencies can go extinct like biological species. Antifragility isn't immortality. A few more decades of evolution may be enough for humans to figure out how to best handle money, whether it's bitcoin, traditional banking, gold, or something else.
Keep your BS detector on. Start by being skeptical of this article's finance-related claims. Even if you think you understand finance, join the conversation.
We build a better future through dialogue. So listen, ask, and share. When you think you can't find common ground with the opposing view, remember:
Sam Bankman-Fried lied.

Shan Vernekar
3 years ago
How the Ethereum blockchain's transactions are carried out
Overview
Ethereum blockchain is a network of nodes that validate transactions. Any network node can be queried for blockchain data for free. To write data as a transition requires processing and writing to each network node's storage. Fee is paid in ether and is also called as gas.
We'll examine how user-initiated transactions flow across the network and into the blockchain.
Flow of transactions
A user wishes to move some ether from one external account to another. He utilizes a cryptocurrency wallet for this (like Metamask), which is a browser extension.
The user enters the desired transfer amount and the external account's address. He has the option to choose the transaction cost he is ready to pay.
Wallet makes use of this data, signs it with the user's private key, and writes it to an Ethereum node. Services such as Infura offer APIs that enable writing data to nodes. One of these services is used by Metamask. An example transaction is shown below. Notice the “to” address and value fields.
var rawTxn = {
nonce: web3.toHex(txnCount),
gasPrice: web3.toHex(100000000000),
gasLimit: web3.toHex(140000),
to: '0x633296baebc20f33ac2e1c1b105d7cd1f6a0718b',
value: web3.toHex(0),
data: '0xcc9ab24952616d6100000000000000000000000000000000000000000000000000000000'
};The transaction is written to the target Ethereum node's local TRANSACTION POOL. It informed surrounding nodes of the new transaction, and those nodes reciprocated. Eventually, this transaction is received by and written to each node's local TRANSACTION pool.
The miner who finds the following block first adds pending transactions (with a higher gas cost) from the nearby TRANSACTION POOL to the block.
The transactions written to the new block are verified by other network nodes.
A block is added to the main blockchain after there is consensus and it is determined to be genuine. The local blockchain is updated with the new node by additional nodes as well.
Block mining begins again next.
The image above shows how transactions go via the network and what's needed to submit them to the main block chain.
References
ethereum.org/transactions How Ethereum transactions function, their data structure, and how to send them via app. ethereum.org
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Liz Martin
3 years ago
A Search Engine From Apple?
Apple's search engine has long been rumored. Recent Google developments may confirm the rumor. Is Apple about to become Google's biggest rival?
Here's a video:
People noted Apple's changes in 2020. AppleBot, a web crawler that downloads and caches Internet content, was more active than in the last five years.
Apple hired search engine developers, including ex-Googlers, such as John Giannandrea, Google's former search chief.
Apple also changed the way iPhones search. With iOS 14, Apple's search results arrived before Google's.
These facts fueled rumors that Apple was developing a search engine.
Apple and Google Have a Contract
Many skeptics said Apple couldn't compete with Google. This didn't affect the company's competitiveness.
Apple is the only business with the resources and scale to be a Google rival, with 1.8 billion active devices and a $2 trillion market cap.
Still, people doubted that due to a license deal. Google pays Apple $8 to $12 billion annually to be the default iPhone and iPad search engine.
Apple can't build an independent search product under this arrangement.
Why would Apple enter search if it's being paid to stay out?
Ironically, this partnership has many people believing Apple is getting into search.
A New Default Search Engine May Be Needed
Google was sued for antitrust in 2020. It is accused of anticompetitive and exclusionary behavior. Justice wants to end Google's monopoly.
Authorities could restrict Apple and Google's licensing deal due to its likely effect on market competitiveness. Hence Apple needs a new default search engine.
Apple Already Has a Search Engine
The company already has a search engine, Spotlight.
Since 2004, Spotlight has aired. It was developed to help users find photos, documents, apps, music, and system preferences.
Apple's search engine could do more than organize files, texts, and apps.
Spotlight Search was updated in 2014 with iOS 8. Web, App Store, and iTunes searches became available. You could find nearby places, movie showtimes, and news.
This search engine has subsequently been updated and improved. Spotlight added rich search results last year.
If you search for a TV show, movie, or song, photos and carousels will appear at the top of the page.
This resembles Google's rich search results.
When Will the Apple Search Engine Be Available?
When will Apple's search launch? Robert Scoble says it's near.
Scoble tweeted a number of hints before this year's Worldwide Developer Conference.
Scoble bases his prediction on insider information and deductive reasoning. January 2023 is expected.
Will you use Apple's search engine?
Vanessa Karel
3 years ago
10 hard lessons from founding a startup.
Here is the ugly stuff, read this if you have a founder in your life or are trying to become one. Your call.
#1 You'll try to talk yourself to sleep, but it won't always work.
As founders, we're all driven. Good and bad, you're restless. Success requires resistance and discipline. Your startup will be on your mind 24/7, and not everyone will have the patience to listen to your worries, ideas, and coffee runs. You become more self-sufficient than ever before.
#2 No one will understand what you're going through unless they've been a founder.
Some of my closest friends don't understand the work that goes into starting a business, and we can't blame them.
#3 You'll feel alienated.
Your problems aren't common; calling your bestie won't help. You must search hard for the right resources. It alienates you from conversations you no longer relate to. (No 4th of July, no long weekends!)
#4 Since you're your "own boss," people assume you have lots of free time.
Do you agree? I was on a webinar with lots of new entrepreneurs, and one woman said, "I started my own business so I could have more time for myself." This may be true for some lucky people, and you can be flexible with your schedule. If you want your business to succeed, you'll probably be its slave for a while.
#5 No time for illness or family emergencies.
Both last month. Oh, no! Physically and emotionally withdrawing at the worst times will give you perspective. I learned this the hard way because I was too stubborn to postpone an important interview. I thought if I rested all day and only took one call, I'd be fine. Nope. I had a fever and my mind wasn't as sharp, so my performance and audience interaction suffered. Nope. Better to delay than miss out.
Oh, and setting a "OoO" makes you cringe.
#6 Good luck with your mental health, perfectionists.
When building a startup, it's difficult to accept that there won't be enough time to do everything. You can't make them all, not perfectly. You must learn to accept things that are done but not perfect.
#7 As a founder, you'll make mistakes, but you'll want to make them quickly so you can learn.
Hard lessons are learned quicker. You'll need to pivot and try new things often; some won't work, and it's best to discover them sooner rather than later.
#8 Pyramid schemes abound.
I didn't realize how bad it was until I started a company. You must spy and constantly research. As a founder, you'll receive many emails from people claiming to "support" you. Be wary and keep your eyes open. When it's too good to be true. Some "companies" will try to get you to pay for "competitions" to "pitch at events." Don't do it.
#9 Keep your competitor research to a minimum.
Actually, competition is good. It means there's a market for those solutions. However, this can be mentally exhausting too. Learn about their geography and updates, but that's it.
#10 You'll feel guilty taking vacation.
I don't know what to say, but I no longer enjoy watching TV, and that's okay. Pay attention to things that enrich you, bring you joy, and have fun. It boosts creativity.
Being a startup founder may be one of the hardest professional challenges you face, but it's also a great learning experience. Your passion will take you places you never imagined and open doors to opportunities you wouldn't have otherwise. You'll meet amazing people. No regrets, no complaints. It's a roller coaster, but the good days are great.
Miss anything? Comment below

Matthew Cluff
3 years ago
GTO Poker 101
"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.
