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Vitalik

Vitalik

3 years ago

An approximate introduction to how zk-SNARKs are possible (part 2)

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? But it turns out that there is a clever solution.

Polynomials

Polynomials are a special class of algebraic expressions of the form:

  • x+5
  • x^4
  • x^3+3x^2+3x+1
  • 628x^{271}+318x^{270}+530x^{269}+…+69x+381

i.e. they are a sum of any (finite!) number of terms of the form cx^k

There are many things that are fascinating about polynomials. But here we are going to zoom in on a particular one: polynomials are a single mathematical object that can contain an unbounded amount of information (think of them as a list of integers and this is obvious). The fourth example above contained 816 digits of tau, and one can easily imagine a polynomial that contains far more.

Furthermore, a single equation between polynomials can represent an unbounded number of equations between numbers. For example, consider the equation A(x)+ B(x) = C(x). If this equation is true, then it's also true that:

  • A(0)+B(0)=C(0)
  • A(1)+B(1)=C(1)
  • A(2)+B(2)=C(2)
  • A(3)+B(3)=C(3)

And so on for every possible coordinate. You can even construct polynomials to deliberately represent sets of numbers so you can check many equations all at once. For example, suppose that you wanted to check:

  • 12+1=13
  • 10+8=18
  • 15+8=23
  • 15+13=28

You can use a procedure called Lagrange interpolation to construct polynomials A(x) that give (12,10,15,15) as outputs at some specific set of coordinates (eg. (0,1,2,3)), B(x) the outputs (1,8,8,13) on thos same coordinates, and so forth. In fact, here are the polynomials:

  • A(x)=-2x^3+\frac{19}{2}x^2-\frac{19}{2}x+12
  • B(x)=2x^3-\frac{19}{2}x^2+\frac{29}{2}x+1
  • C(x)=5x+13

Checking the equation A(x)+B(x)=C(x) with these polynomials checks all four above equations at the same time.

Comparing a polynomial to itself

You can even check relationships between a large number of adjacent evaluations of the same polynomial using a simple polynomial equation. This is slightly more advanced. Suppose that you want to check that, for a given polynomial F, F(x+2)=F(x)+F(x+1) with the integer range {0,1…89} (so if you also check F(0)=F(1)=1, then F(100) would be the 100th Fibonacci number)

As polynomials, F(x+2)-F(x+1)-F(x) would not be exactly zero, as it could give arbitrary answers outside the range x={0,1…98}. But we can do something clever. In general, there is a rule that if a polynomial P is zero across some set S=\{x_1,x_2…x_n\} then it can be expressed as P(x)=Z(x)*H(x), where Z(x)=(x-x_1)*(x-x_2)*…*(x-x_n) and H(x) is also a polynomial. In other words, any polynomial that equals zero across some set is a (polynomial) multiple of the simplest (lowest-degree) polynomial that equals zero across that same set.

Why is this the case? It is a nice corollary of polynomial long division: the factor theorem. We know that, when dividing P(x) by Z(x), we will get a quotient Q(x) and a remainder R(x) is strictly less than that of Z(x). Since we know that P is zero on all of S, it means that R has to be zero on all of S as well. So we can simply compute R(x) via polynomial interpolation, since it's a polynomial of degree at most n-1 and we know n values (the zeros at S). Interpolating a polynomial with all zeroes gives the zero polynomial, thus R(x)=0 and H(x)=Q(x).

Going back to our example, if we have a polynomial F that encodes Fibonacci numbers (so F(x+2)=F(x)+F(x+1) across x=\{0,1…98\}), then I can convince you that F actually satisfies this condition by proving that the polynomial P(x)=F(x+2)-F(x+1)-F(x) is zero over that range, by giving you the quotient:
H(x)=\frac{F(x+2)-F(x+1)-F(x)}{Z(x)}
Where Z(x) = (x-0)*(x-1)*…*(x-98).
You can calculate Z(x) yourself (ideally you would have it precomputed), check the equation, and if the check passes then F(x) satisfies the condition!

Now, step back and notice what we did here. We converted a 100-step-long computation into a single equation with polynomials. Of course, proving the N'th Fibonacci number is not an especially useful task, especially since Fibonacci numbers have a closed form. But you can use exactly the same basic technique, just with some extra polynomials and some more complicated equations, to encode arbitrary computations with an arbitrarily large number of steps.

see part 3

(Edited)

Hackernoon

Hackernoon

3 years ago


👏 Awesome post! When is part 3 coming?

Trent Lapinski

Trent Lapinski

3 years ago

Very complex topic, great explanation

More on Web3 & Crypto

Isobel Asher Hamilton

Isobel Asher Hamilton

3 years ago

$181 million in bitcoin buried in a dump. $11 million to get them back

$181 million in bitcoin buried in a dump

James Howells lost 8,000 bitcoins. He has $11 million to get them back.

His life altered when he threw out an iPhone-sized hard drive.

Howells, from the city of Newport in southern Wales, had two identical laptop hard drives squirreled away in a drawer in 2013. One was blank; the other had 8,000 bitcoins, currently worth around $181 million.

He wanted to toss out the blank one, but the drive containing the Bitcoin went to the dump.

He's determined to reclaim his 2009 stash.

Howells, 36, wants to arrange a high-tech treasure hunt for bitcoins. He can't enter the landfill.

James Howells lost 8,000 bitcoins

Newport's city council has rebuffed Howells' requests to dig for his hard drive for almost a decade, stating it would be expensive and environmentally destructive.

I got an early look at his $11 million idea to search 110,000 tons of trash. He expects submitting it to the council would convince it to let him recover the hard disk.

110,000 tons of trash, 1 hard drive

Finding a hard disk among heaps of trash may seem Herculean.

Former IT worker Howells claims it's possible with human sorters, robot dogs, and an AI-powered computer taught to find hard drives on a conveyor belt.

His idea has two versions, depending on how much of the landfill he can search.

His most elaborate solution would take three years and cost $11 million to sort 100,000 metric tons of waste. Scaled-down version costs $6 million and takes 18 months.

He's created a team of eight professionals in AI-powered sorting, landfill excavation, garbage management, and data extraction, including one who recovered Columbia's black box data.

The specialists and their companies would be paid a bonus if they successfully recovered the bitcoin stash.

Howells: "We're trying to commercialize this project."

Howells claimed rubbish would be dug up by machines and sorted near the landfill.

Human pickers and a Max-AI machine would sort it. The machine resembles a scanner on a conveyor belt.

Remi Le Grand of Max-AI told us it will train AI to recognize Howells-like hard drives. A robot arm would select candidates.

Howells has added security charges to his scheme because he fears people would steal the hard drive.

He's budgeted for 24-hour CCTV cameras and two robotic "Spot" canines from Boston Dynamics that would patrol at night and look for his hard drive by day.

Howells said his crew met in May at the Celtic Manor Resort outside Newport for a pitch rehearsal.

Richard Hammond's narrative swings from banal to epic.

Richard Hammond filmed the meeting and created a YouTube documentary on Howells.

Hammond said of Howells' squad, "They're committed and believe in him and the idea."

Hammond: "It goes from banal to gigantic." "If I were in his position, I wouldn't have the strength to answer the door."

Howells said trash would be cleaned and repurposed after excavation. Reburying the rest.

"We won't pollute," he declared. "We aim to make everything better."

The Newport, Wales, landfill from the air. Darren Britton / Wales News

After the project is finished, he hopes to develop a solar or wind farm on the dump site. The council is unlikely to accept his vision soon.

A council representative told us, "Mr. Howells can't convince us of anything." "His suggestions constitute a significant ecological danger, which we can't tolerate and are forbidden by our permit."

Will the recovered hard drive work?

The "platter" is a glass or metal disc that holds the hard drive's data. Howells estimates 80% to 90% of the data will be recoverable if the platter isn't damaged.

Phil Bridge, a data-recovery expert who consulted Howells, confirmed these numbers.

If the platter is broken, Bridge adds, data recovery is unlikely.

Bridge says he was intrigued by the proposal. "It's an intriguing case," he added. Helping him get it back and proving everyone incorrect would be a great success story.

Who'd pay?

Swiss and German venture investors Hanspeter Jaberg and Karl Wendeborn told us they would fund the project if Howells received council permission.

Jaberg: "It's a needle in a haystack and a high-risk investment."

Howells said he had no contract with potential backers but had discussed the proposal in Zoom meetings. "Until Newport City Council gives me something in writing, I can't commit," he added.

Suppose he finds the bitcoins.

Howells said he would keep 30% of the data, worth $54 million, if he could retrieve it.

A third would go to the recovery team, 30% to investors, and the remainder to local purposes, including gifting £50 ($61) in bitcoin to each of Newport's 150,000 citizens.

Howells said he opted to spend extra money on "professional firms" to help convince the council.

What if the council doesn't approve?

If Howells can't win the council's support, he'll sue, claiming its actions constitute a "illegal embargo" on the hard drive. "I've avoided that path because I didn't want to cause complications," he stated. I wanted to cooperate with Newport's council.

Howells never met with the council face-to-face. He mentioned he had a 20-minute Zoom meeting in May 2021 but thought his new business strategy would help.

He met with Jessica Morden on June 24. Morden's office confirmed meeting.

After telling the council about his proposal, he can only wait. "I've never been happier," he said. This is our most professional operation, with the best employees.

The "crypto proponent" buys bitcoin every month and sells it for cash.

Howells tries not to think about what he'd do with his part of the money if the hard disk is found functional. "Otherwise, you'll go mad," he added.


This post is a summary. Read the full article here.

ANDREW SINGER

ANDREW SINGER

3 years ago

Crypto seen as the ‘future of money’ in inflation-mired countries

Crypto as the ‘future of money' in inflation-stricken nations

Citizens of devalued currencies “need” crypto. “Nice to have” in the developed world.

According to Gemini's 2022 Global State of Crypto report, cryptocurrencies “evolved from what many considered a niche investment into an established asset class” last year.

More than half of crypto owners in Brazil (51%), Hong Kong (51%), and India (54%), according to the report, bought cryptocurrency for the first time in 2021.

The study found that inflation and currency devaluation are powerful drivers of crypto adoption, especially in emerging market (EM) countries:

“Respondents in countries that have seen a 50% or greater devaluation of their currency against the USD over the last decade were more than 5 times as likely to plan to purchase crypto in the coming year.”

Between 2011 and 2021, the real lost 218 percent of its value against the dollar, and 45 percent of Brazilians surveyed by Gemini said they planned to buy crypto in 2019.

The rand (South Africa's currency) has fallen 103 percent in value over the last decade, second only to the Brazilian real, and 32 percent of South Africans expect to own crypto in the coming year. Mexico and India, the third and fourth highest devaluation countries, followed suit.

Compared to the US dollar, Hong Kong and the UK currencies have not devalued in the last decade. Meanwhile, only 5% and 8% of those surveyed in those countries expressed interest in buying crypto.

What can be concluded? Noah Perlman, COO of Gemini, sees various crypto use cases depending on one's location. 

‘Need to have' investment in countries where the local currency has devalued against the dollar, whereas in the developed world it is still seen as a ‘nice to have'.

Crypto as money substitute

As an adjunct professor at New York University School of Law, Winston Ma distinguishes between an asset used as an inflation hedge and one used as a currency replacement.

Unlike gold, he believes Bitcoin (BTC) is not a “inflation hedge”. They acted more like growth stocks in 2022. “Bitcoin correlated more closely with the S&P 500 index — and Ether with the NASDAQ — than gold,” he told Cointelegraph. But in the developing world, things are different:

“Inflation may be a primary driver of cryptocurrency adoption in emerging markets like Brazil, India, and Mexico.”

According to Justin d'Anethan, institutional sales director at the Amber Group, a Singapore-based digital asset firm, early adoption was driven by countries where currency stability and/or access to proper banking services were issues. Simply put, he said, developing countries want alternatives to easily debased fiat currencies.

“The larger flows may still come from institutions and developed countries, but the actual users may come from places like Lebanon, Turkey, Venezuela, and Indonesia.”

“Inflation is one of the factors that has and continues to drive adoption of Bitcoin and other crypto assets globally,” said Sean Stein Smith, assistant professor of economics and business at Lehman College.

But it's only one factor, and different regions have different factors, says Stein Smith. As a “instantaneously accessible, traceable, and cost-effective transaction option,” investors and entrepreneurs increasingly recognize the benefits of crypto assets. Other places promote crypto adoption due to “potential capital gains and returns”.

According to the report, “legal uncertainty around cryptocurrency,” tax questions, and a general education deficit could hinder adoption in Asia Pacific and Latin America. In Africa, 56% of respondents said more educational resources were needed to explain cryptocurrencies.

Not only inflation, but empowering our youth to live better than their parents without fear of failure or allegiance to legacy financial markets or products, said Monica Singer, ConsenSys South Africa lead. Also, “the issue of cash and remittances is huge in Africa, as is the issue of social grants.”

Money's future?

The survey found that Brazil and Indonesia had the most cryptocurrency ownership. In each country, 41% of those polled said they owned crypto. Only 20% of Americans surveyed said they owned cryptocurrency.

These markets are more likely to see cryptocurrencies as the future of money. The survey found:

“The majority of respondents in Latin America (59%) and Africa (58%) say crypto is the future of money.”
Brazil (66%), Nigeria (63%), Indonesia (61%), and South Africa (57%). Europe and Australia had the fewest believers, with Denmark at 12%, Norway at 15%, and Australia at 17%.

Will the Ukraine conflict impact adoption?

The poll was taken before the war. Will the devastating conflict slow global crypto adoption growth?

With over $100 million in crypto donations directly requested by the Ukrainian government since the war began, Stein Smith says the war has certainly brought crypto into the mainstream conversation.

“This real-world demonstration of decentralized money's power could spur wider adoption, policy debate, and increased use of crypto as a medium of exchange.”
But the war may not affect all developing nations. “The Ukraine war has no impact on African demand for crypto,” Others loom larger. “Yes, inflation, but also a lack of trust in government in many African countries, and a young demographic very familiar with mobile phones and the internet.”

A major success story like Mpesa in Kenya has influenced the continent and may help accelerate crypto adoption. Creating a plan when everyone you trust fails you is directly related to the African spirit, she said.

On the other hand, Ma views the Ukraine conflict as a sort of crisis check for cryptocurrencies. For those in emerging markets, the Ukraine-Russia war has served as a “stress test” for the cryptocurrency payment rail, he told Cointelegraph.

“These emerging markets may see the greatest future gains in crypto adoption.”
Inflation and currency devaluation are persistent global concerns. In such places, Bitcoin and other cryptocurrencies are now seen as the “future of money.” Not in the developed world, but that could change with better regulation and education. Inflation and its impact on cash holdings are waking up even Western nations.

Read original post here.

CyberPunkMetalHead

CyberPunkMetalHead

3 years ago

Developed an automated cryptocurrency trading tool for nearly a year before unveiling it this month.

Overview

I'm happy to provide this important update. We've worked on this for a year and a half, so I'm glad to finally write it. We named the application AESIR because we’ve love Norse Mythology. AESIR automates and runs trading strategies.

  • Volatility, technical analysis, oscillators, and other signals are currently supported by AESIR.

  • Additionally, we enhanced AESIR's ability to create distinctive bespoke signals by allowing it to analyze many indicators and produce a single signal.

  • AESIR has a significant social component that allows you to copy the best-performing public setups and use them right away.

Enter your email here to be notified when AEISR launches.

Views on algorithmic trading

First, let me clarify. Anyone who claims algorithmic trading platforms are money-printing plug-and-play devices is a liar. Algorithmic trading platforms are a collection of tools.

A trading algorithm won't make you a competent trader if you lack a trading strategy and yolo your funds without testing. It may hurt your trade. Test and alter your plans to account for market swings, but comprehend market signals and trends.

Status Report

Throughout closed beta testing, we've communicated closely with users to design a platform they want to use.

To celebrate, we're giving you free Aesir Viking NFTs and we cover gas fees.

Why use a trading Algorithm?

  • Automating a successful manual approach

  • experimenting with and developing solutions that are impossible to execute manually

One AESIR strategy lets you buy any cryptocurrency that rose by more than x% in y seconds.

AESIR can scan an exchange for coins that have gained more than 3% in 5 minutes. It's impossible to manually analyze over 1000 trading pairings every 5 minutes. Auto buy dips or DCA around a Dip

Sneak Preview

Here's the Leaderboard, where you can clone the best public settings.

As a tiny, self-funded team, we're excited to unveil our product. It's a beta release, so there's still more to accomplish, but we know where we stand.

If this sounds like a project that you might want to learn more about, you can sign up to our newsletter and be notified when AESIR launches.

Useful Links:

Join the Discord | Join our subreddit | Newsletter | Mint Free NFT

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Matthew Cluff

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.

Hudson Rennie

Hudson Rennie

2 years ago

My Work at a $1.2 Billion Startup That Failed

Sometimes doing everything correctly isn't enough.

Image via: glassdoor.com licensed under CC BY 2.0

In 2020, I could fix my life.

After failing to start a business, I owed $40,000 and had no work.

A $1.2 billion startup on the cusp of going public pulled me up.

Ironically, it was getting ready for an epic fall — with the world watching.

Life sometimes helps. Without a base, even the strongest fall. A corporation that did everything right failed 3 months after going public.

First-row view.

Apple is the creator of Adore.

Out of respect, I've altered the company and employees' names in this account, despite their failure.

Although being a publicly traded company, it may become obvious.

We’ll call it “Adore” — a revolutionary concept in retail shopping.

Two Apple execs established Adore in 2014 with a focus on people-first purchasing.

Jon and Tim:

  • The concept for the stylish Apple retail locations you see today was developed by retail expert Jon Swanson, who collaborated closely with Steve Jobs.

  • Tim Cruiter is a graphic designer who produced the recognizable bouncing lamp video that appears at the start of every Pixar film.

The dynamic duo realized their vision.

“What if you could combine the convenience of online shopping with the confidence of the conventional brick-and-mortar store experience.”

Adore's mobile store concept combined traditional retail with online shopping.

Adore brought joy to 70+ cities and 4 countries over 7 years, including the US, Canada, and the UK.

Being employed on the ground floor, with world dominance and IPO on the horizon, was exciting.

I started as an Adore Expert.

I delivered cell phones, helped consumers set them up, and sold add-ons.

As the company grew, I became a Virtual Learning Facilitator and trained new employees across North America using Zoom.

In this capacity, I gained corporate insider knowledge. I worked with the creative team and Jon and Tim.

Image via Instagram: @goenjoy

It's where I saw company foundation fissures. Despite appearances, investors were concerned.

The business strategy was ground-breaking.

Even after seeing my employee stocks fall from a home down payment to $0 (when Adore filed for bankruptcy), it's hard to pinpoint what went wrong.

Solid business model, well-executed.

Jon and Tim's chase for public funding ended in glory.

Here’s the business model in a nutshell:

Buying cell phones is cumbersome. You have two choices:

  1. Online purchase: not knowing what plan you require or how to operate your device.

  2. Enter a store, which can be troublesome and stressful.

Apple, AT&T, and Rogers offered Adore as a free delivery add-on. Customers could:

  • Have their phone delivered by UPS or Canada Post in 1-2 weeks.

  • Alternately, arrange for a person to visit them the same day (or sometimes even the same hour) to assist them set up their phone and demonstrate how to use it (transferring contacts, switching the SIM card, etc.).

Each Adore Expert brought a van with extra devices and accessories to customers.

Happy customers.

Here’s how Adore and its partners made money:

Adores partners appreciated sending Experts to consumers' homes since they improved customer satisfaction, average sale, and gadget returns.

**Telecom enterprises have low customer satisfaction. The average NPS is 30/100. Adore's global NPS was 80.

Adore made money by:

  • a set cost for each delivery

  • commission on sold warranties and extras

Consumer product applications seemed infinite.

A proprietary scheduling system (“The Adore App”), allowed for same-day, even same-hour deliveries.

It differentiates Adore.

They treated staff generously by:

  • Options on stock

  • health advantages

  • sales enticements

  • high rates per hour

Four-day workweeks were set by experts.

Being hired early felt like joining Uber, Netflix, or Tesla. We hoped the company's stocks would rise.

Exciting times.

I smiled as I greeted more than 1,000 new staff.

I spent a decade in retail before joining Adore. I needed a change.

After a leap of faith, I needed a lifeline. So, I applied for retail sales jobs in the spring of 2019.

The universe typically offers you what you want after you accept what you need. I needed a job to settle my debt and reach $0 again.

And the universe listened.

After being hired as an Adore Expert, I became a Virtual Learning Facilitator. Enough said.

After weeks of economic damage from the pandemic.

This employment let me work from home during the pandemic. It taught me excellent business skills.

I was active in brainstorming, onboarding new personnel, and expanding communication as we grew.

This job gave me vital skills and a regular paycheck during the pandemic.

It wasn’t until January of 2022 that I left on my own accord to try to work for myself again — this time, it’s going much better.

Adore was perfect. We valued:

  • Connection

  • Discovery

  • Empathy

Everything we did centered on compassion, and we held frequent Justice Calls to discuss diversity and work culture.

The last day of onboarding typically ended in tears as employees felt like they'd found a home, as I had.

Like all nice things, the wonderful vibes ended.

First indication of distress

My first day at the workplace was great.

Fun, intuitive, and they wanted creative individuals, not salesman.

While sales were important, the company's vision was more important.

“To deliver joy through life-changing mobile retail experiences.”

Thorough, forward-thinking training. We had a module on intuition. It gave us role ownership.

We were flown cross-country for training, gave feedback, and felt like we made a difference. Multiple contacts responded immediately and enthusiastically.

The atmosphere was genuine.

Making money was secondary, though. Incredible service was a priority.

Jon and Tim answered new hires' questions during Zoom calls during onboarding. CEOs seldom meet new hires this way, but they seemed to enjoy it.

All appeared well.

But in late 2021, things started changing.

Adore's leadership changed after its IPO. From basic values to sales maximization. We lost communication and were forced to fend for ourselves.

Removed the training wheels.

It got tougher to gain instructions from those above me, and new employees told me their roles weren't as advertised.

External money-focused managers were hired.

Instead of creative types, we hired salespeople.

With a new focus on numbers, Adore's uniqueness began to crumble.

Via Zoom, hundreds of workers were let go.

So.

Early in 2022, mass Zoom firings were trending. A CEO firing 900 workers over Zoom went viral.

Adore was special to me, but it became a headline.

30 June 2022, Vice Motherboard published Watch as Adore's CEO Fires Hundreds.

It described a leaked video of Jon Swanson laying off all staff in Canada and the UK.

They called it a “notice of redundancy”.

The corporation couldn't pay its employees.

I loved Adore's underlying ideals, among other things. We called clients Adorers and sold solutions, not add-ons.

But, like anything, a company is only as strong as its weakest link. And obviously, the people-first focus wasn’t making enough money.

There were signs. The expansion was presumably a race against time and money.

Adore finally declared bankruptcy.

Adore declared bankruptcy 3 months after going public. It happened in waves, like any large-scale fall.

  • Initial key players to leave were

  • Then, communication deteriorated.

  • Lastly, the corporate culture disintegrated.

6 months after leaving Adore, I received a letter in the mail from a Law firm — it was about my stocks.

Adore filed Chapter 11. I had to sue to collect my worthless investments.

I hoped those stocks will be valuable someday. Nope. Nope.

Sad, I sighed.

$1.2 billion firm gone.

I left the workplace 3 months before starting a writing business. Despite being mediocre, I'm doing fine.

I got up as Adore fell.

Finally, can we scale kindness?

I trust my gut. Changes at Adore made me leave before it sank.

Adores' unceremonious slide from a top startup to bankruptcy is astonishing to me.

The company did everything perfectly, in my opinion.

  • first to market,

  • provided excellent service

  • paid their staff handsomely.

  • was responsible and attentive to criticism

The company wasn't led by an egotistical eccentric. The crew had centuries of cumulative space experience.

I'm optimistic about the future of work culture, but is compassion scalable?

Jared Heyman

Jared Heyman

2 years ago

The survival and demise of Y Combinator startups

I've written a lot about Y Combinator's success, but as any startup founder or investor knows, many startups fail.

Rebel Fund invests in the top 5-10% of new Y Combinator startups each year, so we focus on identifying and supporting the most promising technology startups in our ecosystem. Given the power law dynamic and asymmetric risk/return profile of venture capital, we worry more about our successes than our failures. Since the latter still counts, this essay will focus on the proportion of YC startups that fail.

Since YC's launch in 2005, the figure below shows the percentage of active, inactive, and public/acquired YC startups by batch.

As more startups finish, the blue bars (active) decrease significantly. By 12 years, 88% of startups have closed or exited. Only 7% of startups reach resolution each year.

YC startups by status after 12 years:

Half the startups have failed, over one-third have exited, and the rest are still operating.

In venture investing, it's said that failed investments show up before successful ones. This is true for YC startups, but only in their early years.

Below, we only present resolved companies from the first chart. Some companies fail soon after establishment, but after a few years, the inactive vs. public/acquired ratio stabilizes around 55:45. After a few years, a YC firm is roughly as likely to quit as fail, which is better than I imagined.

I prepared this post because Rebel investors regularly question me about YC startup failure rates and how long it takes for them to exit or shut down.

Early-stage venture investors can overlook it because 100x investments matter more than 0x investments.

YC founders can ignore it because it shouldn't matter if many of their peers succeed or fail ;)