Integrity
Write
Loading...
Liam Vaughan

Liam Vaughan

3 years ago

Investors can bet big on almost anything on a new prediction market.

Kalshi allows five-figure bets on the Grammys, the next Covid wave, and future SEC commissioners. Worst-case scenario

On Election Day 2020, two young entrepreneurs received a call from the CFTC chairman. Luana Lopes Lara and Tarek Mansour spent 18 months trying to start a new type of financial exchange. Instead of betting on stock prices or commodity futures, people could trade instruments tied to real-world events, such as legislation, the weather, or the Oscar winner.

Heath Tarbert, a Trump appointee, shouted "Congratulations." "You're competing with 1840s-era markets. I'm sure you'll become a powerhouse too."

Companies had tried to introduce similar event markets in the US for years, but Tarbert's agency, the CFTC, said no, arguing they were gambling and prone to cheating. Now the agency has reversed course, approving two 24-year-olds who will have first-mover advantage in what could become a huge new asset class. Kalshi Inc. raised $30 million from venture capitalists within weeks of Tarbert's call, his representative says. Mansour, 26, believes this will be bigger than crypto.

Anyone who's read The Wisdom of Crowds knows prediction markets' potential. Well-designed markets can help draw out knowledge from disparate groups, and research shows that when money is at stake, people make better predictions. Lopes Lara calls it a "bullshit tax." That's why Google, Microsoft, and even the US Department of Defense use prediction markets internally to guide decisions, and why university-linked political betting sites like PredictIt sometimes outperform polls.

Regulators feared Wall Street-scale trading would encourage investors to manipulate reality. If the stakes are high enough, traders could pressure congressional staffers to stall a bill or bet on whether Kanye West's new album will drop this week. When Lopes Lara and Mansour pitched the CFTC, senior regulators raised these issues. Politically appointed commissioners overruled their concerns, and one later joined Kalshi's board.

Will Kanye’s new album come out next week? Yes or no?

Kalshi's victory was due more to lobbying and legal wrangling than to Silicon Valley-style innovation. Lopes Lara and Mansour didn't invent anything; they changed a well-established concept's governance. The result could usher in a new era of market-based enlightenment or push Wall Street's destructive tendencies into the real world.

If Kalshi's founders lacked experience to bolster their CFTC application, they had comical youth success. Lopes Lara studied ballet at the Brazilian Bolshoi before coming to the US. Mansour won France's math Olympiad. They bonded over their work ethic in an MIT computer science class.

Lopes Lara had the idea for Kalshi while interning at a New York hedge fund. When the traders around her weren't working, she noticed they were betting on the news: Would Apple hit a trillion dollars? Kylie Jenner? "It was anything," she says.

Are mortgage rates going up? Yes or no?

Mansour saw the business potential when Lopes Lara suggested it. He interned at Goldman Sachs Group Inc., helping investors prepare for the UK leaving the EU. Goldman sold clients complex stock-and-derivative combinations. As he discussed it with Lopes Lara, they agreed that investors should hedge their risk by betting on Brexit itself rather than an imperfect proxy.

Lopes Lara and Mansour hypothesized how a marketplace might work. They settled on a "event contract," a binary-outcome instrument like "Will inflation hit 5% by the end of the month?" The contract would settle at $1 (if the event happened) or zero (if it didn't), but its price would fluctuate based on market sentiment. After a good debate, a politician's election odds may rise from 50 to 55. Kalshi would charge a commission on every trade and sell data to traders, political campaigns, businesses, and others.

In October 2018, five months after graduation, the pair flew to California to compete in a hackathon for wannabe tech founders organized by the Silicon Valley incubator Y Combinator. They built a website in a day and a night and presented it to entrepreneurs the next day. Their prototype barely worked, but they won a three-month mentorship program and $150,000. Michael Seibel, managing director of Y Combinator, said of their idea, "I had to take a chance!"

Will there be another moon landing by 2025?

Seibel's skepticism was rooted in America's historical wariness of gambling. Roulette, poker, and other online casino games are largely illegal, and sports betting was only legal in a few states until May 2018. Kalshi as a risk-hedging platform rather than a bookmaker seemed like a good idea, but convincing the CFTC wouldn't be easy. In 2012, the CFTC said trading on politics had no "economic purpose" and was "contrary to the public interest."

Lopes Lara and Mansour cold-called 60 Googled lawyers during their time at Y Combinator. Everyone advised quitting. Mansour recalls the pain. Jeff Bandman, a former CFTC official, helped them navigate the agency and its characters.

When they weren’t busy trying to recruit lawyers, Lopes Lara and Mansour were meeting early-stage investors. Alfred Lin of Sequoia Capital Operations LLC backed Airbnb, DoorDash, and Uber Technologies. Lin told the founders their idea could capitalize on retail trading and challenge how the financial world manages risk. "Come back with regulatory approval," he said.

In the US, even small bets on most events were once illegal. Under the Commodity Exchange Act, the CFTC can stop exchanges from listing contracts relating to "terrorism, assassination, war" and "gaming" if they are "contrary to the public interest," which was often the case.

Will subway ridership return to normal? Yes or no?

In 1988, as academic interest in the field grew, the agency allowed the University of Iowa to set up a prediction market for research purposes, as long as it didn't make a profit or advertise and limited bets to $500. PredictIt, the biggest and best-known political betting platform in the US, also got an exemption thanks to an association with Victoria University of Wellington in New Zealand. Today, it's a sprawling marketplace with its own subculture and lingo. PredictIt users call it "Rules Cuck Panther" when they lose on a technicality. Major news outlets cite PredictIt's odds on Discord and the Star Spangled Gamblers podcast.

CFTC limits PredictIt bets to $850. To keep traders happy, PredictIt will often run multiple variations of the same question, listing separate contracts for two dozen Democratic primary candidates, for example. A trader could have more than $10,000 riding on a single outcome. Some of the site's traders are current or former campaign staffers who can answer questions like "How many tweets will Donald Trump post from Nov. 20 to 27?" and "When will Anthony Scaramucci's role as White House communications director end?"

According to PredictIt co-founder John Phillips, politicians help explain the site's accuracy. "Prediction markets work well and are accurate because they attract people with superior information," he said in a 2016 podcast. “In the financial stock market, it’s called inside information.”

Will Build Back Better pass? Yes or no?

Trading on nonpublic information is illegal outside of academia, which presented a dilemma for Lopes Lara and Mansour. Kalshi's forecasts needed to be accurate. Kalshi must eliminate insider trading as a regulated entity. Lopes Lara and Mansour wanted to build a high-stakes PredictIt without the anarchy or blurred legal lines—a "New York Stock Exchange for Events." First, they had to convince regulators event trading was safe.

When Lopes Lara and Mansour approached the CFTC in the spring of 2019, some officials in the Division of Market Oversight were skeptical, according to interviews with people involved in the process. For all Kalshi's talk of revolutionizing finance, this was just a turbocharged version of something that had been rejected before.

The DMO couldn't see the big picture. The staff review was supposed to ensure Kalshi could complete a checklist, "23 Core Principles of a Designated Contract Market," which included keeping good records and having enough money. The five commissioners decide. With Trump as president, three of them were ideologically pro-market.

Lopes Lara, Mansour, and their lawyer Bandman, an ex-CFTC official, answered the DMO's questions while lobbying the commissioners on Zoom about the potential of event markets to mitigate risks and make better decisions. Before each meeting, they would write a script and memorize it word for word.

Will student debt be forgiven? Yes or no?

Several prediction markets that hadn't sought regulatory approval bolstered Kalshi's case. Polymarket let customers bet hundreds of thousands of dollars anonymously using cryptocurrencies, making it hard to track. Augur, which facilitates private wagers between parties using blockchain, couldn't regulate bets and hadn't stopped users from betting on assassinations. Kalshi, by comparison, argued it was doing everything right. (The CFTC fined Polymarket $1.4 million for operating an unlicensed exchange in January 2022. Polymarket says it's now compliant and excited to pioneer smart contract-based financial solutions with regulators.

Kalshi was approved unanimously despite some DMO members' concerns about event contracts' riskiness. "Once they check all the boxes, they're in," says a CFTC insider.

Three months after CFTC approval, Kalshi announced funding from Sequoia, Charles Schwab, and Henry Kravis. Sequoia's Lin, who joined the board, said Tarek, Luana, and team created a new way to invest and engage with the world.

The CFTC hadn't asked what markets the exchange planned to run since. After approval, Lopes Lara and Mansour had the momentum. Kalshi's March list of 30 proposed contracts caused chaos at the DMO. The division handles exchanges that create two or three new markets a year. Kalshi’s business model called for new ones practically every day.

Uncontroversial proposals included weather and GDP questions. Others, on the initial list and later, were concerning. DMO officials feared Covid-19 contracts amounted to gambling on human suffering, which is why war and terrorism markets are banned. (Similar logic doomed ex-admiral John Poindexter's Policy Analysis Market, a Bush-era plan to uncover intelligence by having security analysts bet on Middle East events.) Regulators didn't see how predicting the Grammy winners was different from betting on the Patriots to win the Super Bowl. Who, other than John Legend, would need to hedge the best R&B album winner?

Event contracts raised new questions for the DMO's product review team. Regulators could block gaming contracts that weren't in the public interest under the Commodity Exchange Act, but no one had defined gaming. It was unclear whether the CFTC had a right or an obligation to consider whether a contract was in the public interest. How was it to determine public interest? Another person familiar with the CFTC review says, "It was a mess." The agency didn't comment.

CFTC staff feared some event contracts could be cheated. Kalshi wanted to run a bee-endangerment market. The DMO pushed back, saying it saw two problems symptomatic of the asset class: traders could press government officials for information, and officials could delay adding the insects to the list to cash in.

The idea that traders might manipulate prediction markets wasn't paranoid. In 2013, academics David Rothschild and Rajiv Sethi found that an unidentified party lost $7 million buying Mitt Romney contracts on Intrade, a now-defunct, unlicensed Irish platform, in the runup to the 2012 election. The authors speculated that the trader, whom they dubbed the “Romney Whale,” may have been looking to boost morale and keep donations coming in.

Kalshi said manipulation and insider trading are risks for any market. It built a surveillance system and said it would hire a team to monitor it. "People trade on events all the time—they just use options and other instruments. This brings everything into the open, Mansour says. Kalshi didn't include election contracts, a red line for CFTC Democrats.

Lopes Lara and Mansour were ready to launch kalshi.com that summer, but the DMO blocked them. Product reviewers were frustrated by spending half their time on an exchange that represented a tiny portion of the derivatives market. Lopes Lara and Mansour pressed politically appointed commissioners during the impasse.

Tarbert, the chairman, had moved on, but Kalshi found a new supporter in Republican Brian Quintenz, a crypto-loving former hedge fund manager. He was unmoved by the DMO's concerns, arguing that speculation on Kalshi's proposed events was desirable and the agency had no legal standing to prevent it. He supported a failed bid to allow NFL futures earlier this year. Others on the commission were cautious but supportive. Given the law's ambiguity, they worried they'd be on shaky ground if Kalshi sued if they blocked a contract. Without a permanent chairman, the agency lacked leadership.

To block a contract, DMO staff needed a majority of commissioners' support, which they didn't have in all but a few cases. "We didn't have the votes," a reviewer says, paraphrasing Hamilton. By the second half of 2021, new contract requests were arriving almost daily at the DMO, and the demoralized and overrun division eventually accepted defeat and stopped fighting back. By the end of the year, three senior DMO officials had left the agency, making it easier for Kalshi to list its contracts unimpeded.

Today, Kalshi is growing. 32 employees work in a SoHo office with big windows and exposed brick. Quintenz, who left the CFTC 10 months after Kalshi was approved, is on its board. He joined because he was interested in the market's hedging and risk management opportunities.

Mid-May, the company's website had 75 markets, such as "Will Q4 GDP be negative?" Will NASA land on the moon by 2025? The exchange recently reached 2 million weekly contracts, a jump from where it started but still a small number compared to other futures exchanges. Early adopters are PredictIt and Polymarket fans. Bets on the site are currently capped at $25,000, but Kalshi hopes to increase that to $100,000 and beyond.

With the regulatory drawbridge down, Lopes Lara and Mansour must move quickly. Chicago's CME Group Inc. plans to offer index-linked event contracts. Kalshi will release a smartphone app to attract customers. After that, it hopes to partner with a big brokerage. Sequoia is a major investor in Robinhood Markets Inc. Robinhood users could have access to Kalshi so that after buying GameStop Corp. shares, they'd be prompted to bet on the Oscars or the next Fed commissioner.

Some, like Illinois Democrat Sean Casten, accuse Robinhood and its competitors of gamifying trading to encourage addiction, but Kalshi doesn't seem worried. Mansour says Kalshi's customers can't bet more than they've deposited, making debt difficult. Eventually, he may introduce leveraged bets.

Tension over event contracts recalls another CFTC episode. Brooksley Born proposed regulating the financial derivatives market in 1994. Alan Greenspan and others in the government opposed her, saying it would stifle innovation and push capital overseas. Unrestrained, derivatives grew into a trillion-dollar industry until 2008, when they sparked the financial crisis.

Today, with a midterm election looming, it seems reasonable to ask whether Kalshi plans to get involved. Elections have historically been the biggest draw in prediction markets, with 125 million shares traded on PredictIt for 2020. “We can’t discuss specifics,” Mansour says. “All I can say is, you know, we’re always working on expanding the universe of things that people can trade on.”

Any election contracts would need CFTC approval, which may be difficult with three Democratic commissioners. A Republican president would change the equation.

More on Economics & Investing

Sam Hickmann

Sam Hickmann

3 years ago

What is this Fed interest rate everybody is talking about that makes or breaks the stock market?

The Federal Funds Rate (FFR) is the target interest rate set by the Federal Reserve System (Fed)'s policy-making body (FOMC). This target is the rate at which the Fed suggests commercial banks borrow and lend their excess reserves overnight to each other.

The FOMC meets 8 times a year to set the target FFR. This is supposed to promote economic growth. The overnight lending market sets the actual rate based on commercial banks' short-term reserves. If the market strays too far, the Fed intervenes.

Banks must keep a certain percentage of their deposits in a Federal Reserve account. A bank's reserve requirement is a percentage of its total deposits. End-of-day bank account balances averaged over two-week reserve maintenance periods are used to determine reserve requirements.

If a bank expects to have end-of-day balances above what's needed, it can lend the excess to another institution.

The FOMC adjusts interest rates based on economic indicators that show inflation, recession, or other issues that affect economic growth. Core inflation and durable goods orders are indicators.

In response to economic conditions, the FFR target has changed over time. In the early 1980s, inflation pushed it to 20%. During the Great Recession of 2007-2009, the rate was slashed to 0.15 percent to encourage growth.

Inflation picked up in May 2022 despite earlier rate hikes, prompting today's 0.75 percent point increase. The largest increase since 1994. It might rise to around 3.375% this year and 3.1% by the end of 2024.

Desiree Peralta

Desiree Peralta

2 years ago

How to Use the 2023 Recession to Grow Your Wealth Exponentially

This season's three best money moves.

Photo by Tima Miroshnichenko

“Millionaires are made in recessions.” — Time Capital

We're in a serious downturn, whether or not we're in a recession.

97% of business owners are decreasing costs by more than 10%, and all markets are down 30%.

If you know what you're doing and analyze the markets correctly, this is your chance to become a millionaire.

In any recession, there are always excellent possibilities to seize. Real estate, crypto, stocks, enterprises, etc.

What you do with your money could influence your future riches.

This article analyzes the three key markets, their circumstances for 2023, and how to profit from them.

Ways to make money on the stock market.

If you're conservative like me, you should invest in an index fund. Most of these funds are down 10-30% of ATH:

Prices comparitions between funds, — By Google finance

In earlier recessions, most money index funds lost 20%. After this downturn, they grew and passed the ATH in subsequent months.

Now is the greatest moment to invest in index funds to grow your money in a low-risk approach and make 20%.

If you want to be risky but wise, pick companies that will get better next year but are struggling now.

Even while we can't be 100% confident of a company's future performance, we know some are strong and will have a fantastic year.

Microsoft (down 22%), JPMorgan Chase (15.6%), Amazon (45%), and Disney (33.8%).

These firms give dividends, so you can earn passively while you wait.

So I consider that a good strategy to make wealth in the current stock market is to create two portfolios: one based on index funds to earn 10% to 20% profit when the corrections end, and the other based on individual stocks of popular and strong companies to earn 20%-30% return and dividends while you wait.

How to profit from the downturn in the real estate industry.

With rising mortgage rates, it's the worst moment to buy a home if you don't want to be eaten by banks. In the U.S., interest rates are double what they were three years ago, so buying now looks foolish.

Interest rates chart — by Bankrate

Due to these rates, property prices are falling, but that won't last long since individuals will take advantage.

According to historical data, now is the ideal moment to buy a house for the next five years and perhaps forever.

House prices since 1970 — By Trading Economics

If you can buy a house, do it. You can refinance the interest at a lower rate with acceptable credit, but not the house price.

Take advantage of the housing market prices now because you won't find a decent deal when rates normalize.

How to profit from the cryptocurrency market.

This is the riskiest market to tackle right now, but it could offer the most opportunities if done appropriately.

The most powerful cryptocurrencies are down more than 60% from last year: $68,990 for BTC and $4,865 for ETH.

If you focus on those two coins, you can make 30%-60% without waiting for them to return to their ATH, and they're low enough to be a solid investment.

I don't encourage trying other altcoins because the crypto market is in crisis and you can lose everything if you're greedy.

Still, the main Cryptos are a good investment provided you store them in an external wallet and follow financial gurus' security advice.

Last thoughts

We can't anticipate a recession until it ends. We can't forecast a market or asset's lowest point, therefore waiting makes little sense.

If you want to develop your wealth, assess the money prospects on all the marketplaces and initiate long-term trades.

Many millionaires are made during recessions because they don't fear negative figures and use them to scale their money.

Jan-Patrick Barnert

Jan-Patrick Barnert

3 years ago

Wall Street's Bear Market May Stick Around

If history is any guide, this bear market might be long and severe.

This is the S&P 500 Index's fourth such incident in 20 years. The last bear market of 2020 was a "shock trade" caused by the Covid-19 pandemic, although earlier ones in 2000 and 2008 took longer to bottom out and recover.

Peter Garnry, head of equities strategy at Saxo Bank A/S, compares the current selloff to the dotcom bust of 2000 and the 1973-1974 bear market marked by soaring oil prices connected to an OPEC oil embargo. He blamed high tech valuations and the commodity crises.

"This drop might stretch over a year and reach 35%," Garnry wrote.

Here are six bear market charts.

Time/depth

The S&P 500 Index plummeted 51% between 2000 and 2002 and 58% during the global financial crisis; it took more than 1,000 trading days to recover. The former took 638 days to reach a bottom, while the latter took 352 days, suggesting the present selloff is young.

Valuations

Before the tech bubble burst in 2000, valuations were high. The S&P 500's forward P/E was 25 times then. Before the market fell this year, ahead values were near 24. Before the global financial crisis, stocks were relatively inexpensive, but valuations dropped more than 40%, compared to less than 30% now.

Earnings

Every stock crash, especially earlier bear markets, returned stocks to fundamentals. The S&P 500 decouples from earnings trends but eventually recouples.

Support

Central banks won't support equity investors just now. The end of massive monetary easing will terminate a two-year bull run that was among the strongest ever, and equities may struggle without cheap money. After years of "don't fight the Fed," investors must embrace a new strategy.

Bear Haunting Bear

If the past is any indication, rising government bond yields are bad news. After the financial crisis, skyrocketing rates and a falling euro pushed European stock markets back into bear territory in 2011.

Inflation/rates

The current monetary policy climate differs from past bear markets. This is the first time in a while that markets face significant inflation and rising rates.


This post is a summary. Read full article here

You might also like

The woman

The woman

2 years ago

The best lesson from Sundar Pichai is that success and stress don't mix.

His regular regimen teaches stress management.

Made by the author with AI

In 1995, an Indian graduate visited the US. He obtained a scholarship to Stanford after graduating from IIT with a silver medal. First flight. His ticket cost a year's income. His head was full.

Pichai Sundararajan is his full name. He became Google's CEO and a world leader. Mr. Pichai transformed technology and inspired millions to dream big.

This article reveals his daily schedule.

Mornings

While many of us dread Mondays, Mr. Pichai uses the day to contemplate.

A typical Indian morning. He awakens between 6:30 and 7 a.m. He avoids working out in the mornings.

Mr. Pichai oversees the internet, but he reads a real newspaper every morning.

Pichai mentioned that he usually enjoys a quiet breakfast during which he reads the news to get a good sense of what’s happening in the world. Pichai often has an omelet for breakfast and reads while doing so. The native of Chennai, India, continues to enjoy his daily cup of tea, which he describes as being “very English.”

Pichai starts his day. BuzzFeed's Mat Honan called the CEO Banana Republic dad.

Overthinking in the morning is a bad idea. It's crucial to clear our brains and give ourselves time in the morning before we hit traffic.

Mr. Pichai's morning ritual shows how to stay calm. Wharton Business School found that those who start the day calmly tend to stay that way. It's worth doing regularly.

And he didn't forget his roots.

Afternoons

He has a busy work schedule, as you can imagine. Running one of the world's largest firm takes time, energy, and effort. He prioritizes his work. Monitoring corporate performance and guaranteeing worker efficiency.

Sundar Pichai spends 7-8 hours a day to improve Google. He's noted for changing the company's culture. He wants to boost employee job satisfaction and performance.

His work won him recognition within the company.

Pichai received a 96% approval rating from Glassdoor users in 2017.

Mr. Pichai stresses work satisfaction. Each day is a new canvas for him to find ways to enrich people's job and personal lives.

His work offers countless lessons. According to several profiles and press sources, the Google CEO is a savvy negotiator. Mr. Pichai's success came from his strong personality, work ethic, discipline, simplicity, and hard labor.

Evenings

His evenings are spent with family after a busy day. Sundar Pichai's professional and personal lives are balanced. Sundar Pichai is a night owl who re-energizes about 9 p.m.

However, he claims to be most productive after 10 p.m., and he thinks doing a lot of work at that time is really useful. But he ensures he sleeps for around 7–8 hours every day. He enjoys long walks with his dog and enjoys watching NSDR on YouTube. It helps him in relaxing and sleep better.

His regular routine teaches us what? Work wisely, not hard, discipline, vision, etc. His stress management is key. Leading one of the world's largest firm with 85,000 employees is scary.

The pressure to achieve may ruin a day. Overworked employees are more likely to make mistakes or be angry with coworkers, according to the Family Work Institute. They can't handle daily problems, making the house more stressful than the office.

Walking your dog, having fun with friends, and having hobbies are as vital as your office.

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

Julie Zhuo

Julie Zhuo

2 years ago

Comparing poor and excellent managers

10-sketch explanation

Choosing Tasks

Bringing News

carrying out 1:1s

providing critique

Managing Turbulence