Strong Theorems on Coin Tossing

“The generation of random numbers is too important to be left to chance.”
– Robert R. Coveyou

In the paper “Strong theorems on coin tossing,” mathematician Pál Révész tells a story about a high-school class. The teacher leaves the room, and the children are told to divide into two groups.

In the first group, students have to flip a coin two hundred times and record whether a flip lands heads or tails. In the other group, students have to come up with random heads-or-tails sequences, but without using an actual coin; instead, they have to “generate” random numbers in their head.

Once all the students each have 200 lines recorded, the teacher returns and tries to guess which student belongs to which group. Most of the time, the teacher guesses quite well.

Their secret is this – for the average person, it doesn’t feel right to put down five, or even four, consecutive heads or tails. That many runs just doesn’t sound plausible. However, a statistician knows that when a uniformly random coin generates a sequence of 200, it’s very likely to have runs of six or more. Its probability is close to 97 percent.

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The Daisy’s Trick

“She loves me,” says a man in love, picking off the first petal of a daisy. “She loves me not,” he says a moment later, picking off the next petal. And so, he continues removing the petals, until the last of the flower is gone and the results are in. The daisy helped the player determine whether or not he’s loved by the girl he desires.

But how did the daisy know that?

The daisy’s trick is that it gets a little help from the man himself. People play the “she-loves-me-she-loves-me-not” game with all sorts of questions, and the answer is not always 50-50. And sure enough, if we observe someone de-petaling a daisy in this way, we will usually see them adjust the rules midway through.

“I’m not even sure this is really a daisy,” a player might say as they get halfway through a flower and start to suspect that the result might be unfavorable in the end. They then go on to find another daisy to restart the game and discard the results of the first one. This improves their chances – now it’s only one in two flowers that has to have an even number of petals.

Our internal compass knows exactly how many times we are allowed to start again to balance out the game’s probabilities. If we give ourselves a 9-in-10 chance to be loved back, we might end up “cheating” three to four times, but not more. After that, if the last daisy has bad news, we’ll probably accept the verdict.

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Equal Jobs, Slightly Less Pay

After the hack of Sony Picture’s internal emails, the world learned that American Hustle actress Jennifer Lawrence had gotten fewer back-end points than her male counterparts in the movie.

Lawrence largely blamed herself when she wrote about this experience on her website: “I failed as a negotiator because I gave up early. I didn’t want to keep fighting over millions of dollars that, frankly, due to two franchises, I don’t need,” she wrote. “But if I’m honest with myself, I would be lying if I didn’t say there was an element of wanting to be liked that influenced my decision to close the deal without a real fight.”

The actress’s experience is not unfamiliar to researchers. Studies tell us that, in many situations, women face a higher social risk than men if they initiate negotiations. This can help explain why women are more likely to avoid transactional behavior.

Women report greater anxiety about negotiation and are less likely to perceive situations as negotiable. It’s easy to see how this is bad news for women; proverbs such as “You’ll never get what you don’t ask for” exist for a reason. If even a small percentage of uninvited negotiations are successful, according to the rule of compounding interests, those differences in starting salaries can lead to substantial compensation gaps over time.

Gender inequality lowers the quality of life for both men and women. Women suffer the largest and most direct cost of these inequalities, but the costs span across society and families. It’s not a women’s issue nor is it only a men’s; it’s a problem that needs everyone’s attention and participation.

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Riches in the Niches

There’s a scene in the movie A Beautiful Mind, where the mathematician John Nash and his colleagues are about to party with a group of girls. Nash, the protagonist, goes into a lengthy mathematical explanation when informing his nerdy friends, in essence, why he should be the only one hitting on the hot girl. 

In the movie, the “pigeon scene” is a good excuse for introducing “the bargaining problem,” one of John Nash’s key achievements in game theory.

But Andy Warhol helps us understand the concept in a much simpler way. In The Philosophy of Andy Warhol (From A to B and Back Again), the artist explains how living in New York City changes people. On beautiful, sunny days, Central Park gets so crowded that you can’t even see through the throng. On a rainy Sunday morning, however, no one wants to get out and walk the empty streets, so you can have them all to yourself.

As Warhol puts it, “Living in New York City gives people real incentives to want things that nobody else wants — to want all the leftover things. There are so many people here to compete with that changing your tastes to what other people don’t want is your only hope of getting anything.”

Opportunity changes people’s preferences and tastes. The world is a dynamic marketplace, where supply-and-demand changes constantly.

Finding a romantic partner is not just about your own checklist either. It’s also about everyone else’s. Everything is changing all the time, which, of course, is just another kind of disaster. Culture matters, the zeitgeist matters, and our friends’ opinions matter — sometimes more than our own.

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No Such Thing As a Stupid Question

In a conversation, people balance competing concerns. They want to gather information, and at the same time, they want to create a favorable impression. Impression management is an important social skill; you don’t want to look like someone who is always nosing around deeply personal secrets. And sometimes, you don’t want to look like you don’t know the answer.

“There are naive questions, tedious questions, ill-phrased questions, questions put after inadequate self-criticism. But every question is a cry to understand the world. There is no such thing as a dumb question.” —Carl Sagan, The Demon-Haunted World

The idea that people should feel comfortable asking dumb questions has strong support. Economically speaking, a community wins if its members have all the information they need to make good decisions. By allowing people to ask stupid questions, a community strengthens its ability to share knowledge.

And our questions convey information. Listeners will learn about us from our questions. If nothing else, our questions will tell a listener what we’re interested in.

Questions are not just useful for gathering information; they are a brilliant conversation tool.

  1. Questions can be used to express interest in others. People like to talk about themselves, and they like when someone shows interest in them. Simply being curious about someone will build rapport and help you earn social credit.
  2. Questions create opportunities to share information. You can direct the conversation by asking someone about a particular topic, and they might respond by asking a complementary question.
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Enemy of my Enemy of my Enemy

When Apple launched its “Think Different” campaign in the ’90s, it was to drive a wedge between computer enthusiasts. “I’m a Mac, you’re a PC” was designed to make people pick a side. Apple’s enemy is the boring computer that only geeks use. 

Fighting against something is emotionally more powerful than fighting for a cause. Supporting a cause is a wish without a deadline. But an unpredictable enemy? That’s an imminent threat that someone needs to deal with, right now.

Harley Davidson said that their bikes were about freedom and adventure. This message was a declaration of war on Japanese bike makers and their customers, who saw motorcycles as being about speed and power.

The search engine DuckDuckGo has multiple layers of enemies, starting with every other search provider that’s tracking people. Google is their enemy and so is Microsoft’s Bing, and so is every advertiser and ad network preying on user data.

Do some people still prefer Google’s customized search results? Of course, they do. A good antagonist needs to polarize and cause debate. When you’re building a brand, it needs to evoke strong opinions; otherwise, it won’t have customers who care.

The enemy technique works exactly because the world is complex. People from all corners of the globe can form groups, based on only a few of their specific preferences.

Picking sides can often feel like choosing between right and wrong, but in reality, most of the divisions are arbitrary. To break the spell (to keep calm and carry on being responsible) it’s good to remind ourselves of what Niels Bohr wrote: “The opposite of a correct statement is a false statement. But the opposite of a profound truth may well be another profound truth.”

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Buying Product-Market Fit

Like many of you reading this, I run a bunch of pet projects on the side. Pretty much by definition, but it’s worth pointing out that all of these side projects are before reaching product-market fit.

A founder’s only job before reaching product-market fit is, to find product-market fit.

We don’t “focus on” building a great product, we don’t “focus on” getting customers, we don’t “focus on” changing the world.

We focus on building a business. A defensible business is the only way to survive in the long term.

All the other things are a side effect. Startups build great products, because that helps building a great business.

As someone who really enjoys looking for shortcuts, I often look for buying other’s side projects. This can help me getting to product-market fit cheaper. So before I build anything, I look at the free and open source projects out there. There’s always a chance to get lucky looking on Github first.

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Don’t Require Users to Sign Up

Isn’t the ultimate best signup experience, when you don’t need to sign up at all?

I use Doodle to schedule activities with my friends, because Doodle is super simple to use. You assemble a list of options and then share a secret link with your friends. That’s all you need to find out which time slots work for everyone.

Compare that to almost any other calendar scheduling service, where you’re greeted with a login wall. Some of the services don’t even tell us what they do before asking us to sign up.

These two options are the digital world equivalent of approaching a castle with either,

  • A massive moat and the drawbridge all the way up, or a
  • Red carpet nicely rolled out for the guests, and maybe even someone nice guiding them in.

This year we’ve seen people starting to use the verb “zooming” instead of “skypeing” when referring to video calling someone. “Let’s have a Skype call” has been replaced by “I’ll send you a Zoom link” and it’s for a good reason.

It’s still quite fiddly to send someone a working Skype link: you can only do it from within the Skype app, and the recipient too needs a Skype account with the application downloaded. But it’s two clicks to send someone a Zoom link, which allows anyone to join in and chat away in seconds — even in a browser tab if they must.

This is how Zoom went from an unknown company to the signature product of 2020, and did so on a market ruled by giants like Microsoft or Google. Zoom is greeting its users with a red carpet rather than a drawbridge.

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Queue as Rorschach Test

The longest line I ever stood in took six hours, and it was to watch the tennis tournament at Wimbledon. It’s a long queue, but not an uncomfortable one. There’s plenty of space and people are nice – the crowd’s energy only grew as we got closer to the gates.

Tickets for the three other Grand Slam tennis tournaments can get pretty expensive, but Wimbledon organizers decided that everyone should pay the same fair price. The downside: if you want a ticket, you need to wake up early. The upside: you don’t need money to watch the games the games in London; you just need to be a committed enough tennis fan to brave the queue.

The weirdest queue I know is at Berlin’s infamous nightclub, the Berghain. Potential guests can’t be sure whether their hour-long wait will pay off until the very last minute. That’s because the club tries to establish a “good mix of people” each night to provide the best possible experience. So the bouncers are hyper-selective about who they let in, and it will be a function of the other people in the queue. Sometimes you’ll get in, and other times you’ll be turned down.

Whether a queue is the best system to organize resources, and whether it’s fair or not – most of that is our own perception. But what we think of the system says as much about us as it does about the queue itself.

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The Third Wave

My issue with lockdowns is that they are solving the wrong problem. Yes, I understand that we need to keep the R number low to help hospitals fight the virus.

But we just wasted half a year doing nothing. Maybe nothing is an overstatement and we did a lot of hoping that there won’t be a second wave, I don’t know. From where I’m sitting it’s nothing.

From where I’m sitting, this is an optimization problem.

  1. We have X number of people who will catch this thing.
  2. Those people need the best chance to fight it off, for which they need hospitals, nurses, drugs and what not.
  3. Let’s get all of those things.

Closing shops, restaurants, schools, offices and what not, and then sending people on furlough is all fine — if we do it for a reason. If we do this to win time, and then waste all of that time? I’d really love to opt out please. Put your lockdown up yours.

Yes, close down an airport — but then build a makeshift hospital there. Yes, send hospitality workers home for half a year — and hire them to support nurses, hospitals or the elderly.

Governments, your job is not only to optimize.

Your job is to optimize for the right thing.

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