Topic

Math

The art of quantifying our world.

Positive vs. Negative Sum Games

If I have an extra cup of coffee, and you have an extra two dollars, we both come out ahead when I agree to sell it to you.

This might sound painfully obvious, but it's an example of the idea I most wish I had been taught in school: the concept of positive and zero-sum games.

A zero-sum game is one in which the total gained by the winner is equal to the loss endured by the loser. Sports are the most frequently cited example of zero-sum games, since only one team can win. Poker is also zero-sum because it's winner-takes-all. You either lose all your money or win everyone else's.

But this concept isn't limited to games in the strict sense of the word. Politics is also typically a zero-sum game, since only one person can win the race.

Zero-sum games are ferocious, because in the long run, if you don't come out ahead, you lose everything.

Positive-sum games, on the other hand, are the opposite. Two parties playing a positive-sum game both stand to gain something from the interaction, regardless of who "wins."

International trade, for example, is positive sum. If one state has surplus oil and another has surplus corn, both states win by trading if they each want what the other has to offer.

And a lyricist and composer agreeing to collaborate on a new musical are also engaging in a positive-sum game. Perhaps neither can write the next Tony-award-winning musical on their own, but by teaming up, they dramatically increase the potential upside of their efforts.

All else being equal, if you have the choice between playing a negative-sum game and a positive-sum one, you should probably choose the latter.

In a capitalist society, it's easy to fall into the trap of viewing the world as a zero-sum competition: Either you have this dollar, or I do.

But this mindset is a trap. So many areas of our society that seem zero-sum on the surface are really positive-sum.

Have you ever noticed the way in which businesses in competition with one another tend to cluster together? That's because they're often more financially successful in proximity to each other. You might see a Starbucks around the corner from an indie cafe (or even across the street from another Starbucks), or multiple clothing retailers in the same mall.

There are multiple reasons for this, but by aggregating demand for a particular good or service in the optimal part of town, businesses can ultimately attract more customers.

When I worked at a circus school, it was tempting to see ourselves as playing a zero-sum game with the other school in town. But in reality, relatively few people actively seek out circus classes, and most of our new students came from word-of-mouth. This means that the more circus schools there are, the more people will hear that it's a fun way to stay active and learn unusual skills.

Did we lose out on some students to our "competitor?" Absolutely. But ultimately, we were working together to grow the pie – to get more people to realize that they didn't have to grow up in a circus to learn trapeze.

Games don't always fall neatly into one or the other category, and you can even play them in ways that make them fall closer to the positive-sum side.

For example, a couple negotiating a divorce can decide to make some compromises early in the process to minimize how much they lose to lawyer fees. In doing so, they avoid a purely zero-sum fight and each walk away with more money on average.

When you start to recognize how much of life can be played in positive-sum way, you may discover that the range of possibilities to you opens up.

Chickens and Goats

In 1937, Claude Shannon changed the world forever when he published his master's thesis, A Symbolic Analysis of Relay and Switching Circuits.

It's widely considered among the most important masters theses of the century, and for good reason: By combining Boolean algebra (which breaks down problems into true/false variables and operators like and/or) and circuit design, Shannon demonstrated that a series of circuits could solve math problems.

If that sounds like a primitive computer to you, you're correct. This insight forms part of the bedrock of modern computers and helped usher in the entire field of information theory.

Jimmy Soni & Rob Goodman write in their extraordinary biography of Shannon, A Mind at Play:

[B]ecause Boole had shown how to resolve logic into a series of binary, true-false decisions, any system capable of representing binaries has access to the entire logical universe he described. “The laws of thought” had been extended to the inanimate world. [Emphasis added]

Today, we take for granted the idea that information can be conveyed regardless of its medium through a series of ones and zeroes (i.e. true and false statements).

You could even, given enough time, transmit the entirety of Beethoven's Symphony No. 5 to someone half a mile away by tossing chickens and goats off a bridge. If the person on shore understood that chickens = true and goats = false, they could eventually reconstruct an entire recording of the piece.

Claude Shannon was unquestionably a genius. And his discovery follows a familiar pattern. Here's Andy Benoit writing about Peyton Manning for Vault:

[M]ost geniuses—especially those who lead others—prosper not by deconstructing intricate complexities but by exploiting unrecognized simplicities.

Like so many insights, the idea that circuits could be designed to represent Boolean algebra seems simple, even obvious, now.

But no one had fully articulated it before Shannon took the work of an obscure logician and paired it with electrical currents.

In doing so, he paved the way for the digital revolution and the Information Age.

See also: Latticework of Ideas

Pizza Party

You're having dinner with a friend and need to decide how much pizza to get.

The local pizza shop has two specials, and you can choose either two 12" pizzas or one 18" pizza for the same price.

Which is the better deal?

Even though the 18" pie is only 50% wider, it actually has more pizza than the other two combined.

Just because something seems like more doesn't mean it is.

Queuing Theory

John Cook shares a profound reminder on the importance of having slack in a system:

Suppose a small bank has only one teller. Customers take an average of 10 minutes to serve and they arrive at the rate of 5.8 per hour. What will the expected waiting time be?

Assuming customers arrive at random times and take a random amount of time to serve (averaging 10 minutes), the mean wait time will be 5 hours.

But what will the wait time be when you add a second bank teller?

You might be tempted to assume that the wait time simply gets cut in half. But in fact, the second teller drops the wait time to 3 minutes. That's an astounding 99% reduction. 🤯

Why is this true? It's because this example is modeled off of the real world, where arrival and service times are truly random. If everyone arrived at regular intervals and took exactly 10 minutes to serve, there would be no waiting.

When we're designing a system, it's easy to fall into the trap of assuming optimal conditions.

But the world is messy, and by building in slack, we acknowledge that best laid plans often go awry.

Hat tip to Kottke, who provides additional examples and anecdotes of this.

Simpson's Paradox

Simpson's paradox is a statistical phenomenon that occurs when a pattern across groups of data disappears when those groups are combined.

A famous example of this is UC Berkeley's admission rates for its 1973 class. The school discovered that it had admitted 44% of male applicants and 35% of female applicants.

On the surface, this looked like a considerable gender bias, but when they examined the data more closely, they discovered that women tended to apply to departments with more competitive rates of admission, while men tended to apply to less competitive departments.

Not only that, but because 6 departments were biased towards women, while only 4 departments were biased towards men, that year's class skewed in favor of accepting women, even though they had a lower overall acceptance rate.

This idea can be tricky to grok at first, so here's another example:

In baseball, player A can have a lower batting average than player B two years in a row. But if there is a discrepancy between the number of at bats they have, player A can have a higher batting average over the course of both years.

Here's how this played out for Derek Jeter and David Justice in the 1995-96 seasons:

Source: Wikipedia

Justice had a higher batting average both years, but Jeter had more at bats, so when the data are combined, Justice's lead disappears.

The common thread across both these examples is that there are hidden variables at play. The competitiveness of each of UC Berkeley's departments and the number of at bats are both concealed by the summarized data.

Simpson's paradox is an important reminder that our intuition is important when analyzing something.

Our intuition helps us figure out which questions to ask, and knowing what to ask is half the battle.

Compounding

As humans, we are terrible at understanding compounding effects. We tend to think in linear terms, but this blinds us to just how powerful compounding is.

A classic way to illustrate this is the following scenario: A genie appears and offers you either a million dollars now, or a sum of money every day for a month, starting with one penny today and doubling the amount you receive every day.

Which offer should you accept?

That million dollar offer is tempting, but the second option is significantly more lucrative: By day 30, you would net a whopping $10,737,418.23.

The compounding plan would initially be tough from a budgeting perspective, though. You'd have to wait a whole week to make your first dollar, and halfway through the month you would still only have $327.67.

It's hard to internalize just how dramatically something that grows like this can change over time, in part because it requires so much patience and delayed gratification.

But the truth is, many of the most important things in life compound. From relationships to wisdom, your investments in many areas grow exponentially over time. If you consistently put in the work, you'll reap the rewards.

For example, going to the gym once isn't going to have much effect. But if you go five times a week for six months, you'll see substantial changes in your fitness. The impact of that first workout actually increases over time if you maintain the habit.

If you want your efforts to compound, just keep going, and don't give up too early.

The Monty Hall Problem

Imagine you're on a game show facing three closed doors. The host tells you that one door has a car behind it, but the other two have goats.

You're asked to pick a door, in hopes of winning the new car. After you do so, the host opens one of the other doors that has a goat behind it. She then gives you the option to switch your choice to the other remaining closed door.

Here's the brain teaser: Are your odds of winning the car better if you switch your choice to the other door?

Two closed doors with question marks on them, next to an open door with a goat behind it.
Cepheus, Public domain, via Wikimedia Commons

This is known as the Monty Hall problem. It was popularized in Marilyn vos Savant's column in Parade magazine in 1990, and the solution is so unintuitive, thousands of people wrote letters of disagreement to her after she published it.

If you're like me, your instinct is that switching shouldn't matter. After the host opens a door, you have a 50/50 chance either way of picking the car, right?

Wrong.

In fact, your odds of winning the car are overwhelmingly better if you switch doors: A 2/3 chance if you switch, and a 1/3 chance if you don't. Though it might not seem like it at first, you have a lot more information than you did previously.

Understanding why this is true is easier if you consider a version of the problem with 100 doors: If you pick one door, and the host opens 98 of the remaining doors, should you switch to the other remaining closed door?

Which door seems more likely to have a car behind it? Your random pick or the door the host intentionally left closed?

Your chance of picking the correct door the first time was 1%. The host isn't opening doors at random – she knows which door the car is behind and is only opening doors with goats behind them.

So if you switch your choice to the other door, you have a 99% chance of winning the car, because so many of the wrong doors have already been opened.

No matter how many doors you imagine the problem with, your chance of picking correctly is always inverted if you switch once the host reveals every door except one.

Jim Frost calls this a statistical illusion. Just like an optical illusion can trick your brain into seeing something impossible, this problem can deceive you into thinking that the original solution is 50/50.

Here's another scenario that illustrates why this illusion is so compelling: Imagine you walk in the room after the host opens the door to reveal a goat. Since you don't know which door the contestant initially picked, your odds of picking the correct door at this point are 50/50.

It's a coin toss for you, because you have less information. But the contestant, who knows which door they initially picked, still has the better odds if they switch, because they know which door the host chose not to reveal.

So why do we care about this? On the surface, it's an inherently interesting problem, because it's a bit of an illusion. But it's also representative of something we experience regularly: When you're presented with multiple options and make a decision, be prepared to change your mind if you receive more information – even if it goes against your intuition.

It might make all the difference.

Context Matters

I'll never forget the season I scored half of my soccer team's goals.

To be honest, I was not a particularly good soccer player. The actual number of goals I scored that season?

One.

Context matters everywhere, but especially with statistics. It's easy to be misled by stats because they seem so objective, but they're easy to use in a dishonest way, even when they’re technically true.

A 100% year-over-year increase in people getting eaten by mountain lions sounds scary. But if last year that number was three, we probably don't have much to fear.

The average human has half a uterus, but that's not a useful representation of our anatomy if there aren't many people close to the average.

And just because we've observed the sun rising 100% of the time so far, that doesn't mean we can extrapolate that out forever.

Quantifying the world is just as much art as it is science.

Strathern’s Insight

Economist Charles Goodhart proposed the following rule in a 1984 article: "Any observed statistical regularity will tend to collapse once pressure is placed upon it for control purposes."'

The statement is applicable in many places beyond statistics, of course, which the anthropologist Marilyn Strathern pointed out in her 1997 paper: "When a measure becomes a target, it ceases to be a good measure."

This observation is worth keeping in mind every time we try to measure something with a goal in mind.

The number on the scale does not necessarily reflect your overall health.

The number of likes your brand has on social media isn't necessarily a measure of its success.

The number of books you read in a year doesn't necessarily represent how much you've learned.

Another example: When you put pressure on the police to reduce crime, they may discover that it's easier to simply downgrade the severity of reported crimes than to address the systemic problems that lead to them in the first place.

It's often hard to know if we're making progress towards something without measuring it. But as Strathern so insightfully noted, once our goal is a number, the strategies we have for reaching that number may distract us from our actual goal.

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