Category: Numeracy

Alex Bellos: Every Number Tells a Story

From The Grapes of Math By Alex Borros
From The Grapes of Math By Alex Bellos

“We depend on numbers to make sense of the world,
and have done so ever since we started to count.”
— Alex Bellos

***

The earliest symbols used for numbers go back to about 5000 years ago Sumer (modern day Iraq). They didn't really look far for names. Ges, the word for one, also meant man. Min, the word for two, also meant women. At first, numbers served a practical purpose, mostly things like counting sheep and determining taxes.

“Yet numbers also revealed abstract patterns,” writes Alex Bellos in his fascinating book The Grapes of Math: How Life Reflects Numbers and Numbers Reflect Life, which, he continues, “made them objects of deep contemplation. Perhaps the earliest mathematical discovery was that numbers come in two types, even and odd: those that can be halved cleanly, such as 2, 4 and 6, and those that cannot, such as 1, 3 and 5.

Pythagoras and Number Gender

Pythagoras, the Greek teacher, who lived in the sixth century BC and is most famous for his theorem about triangles, agreed with the Sumerians on number gender. He believed that odd numbers were masculine and even ones were feminine. This is where it gets interesting … why did he think that? It was, he believed, a resistance to splitting in two that embodied strength. The ability to be divisible by two, in his eyes was a weakness. He believed odd numbers were master over even. Christianity agrees with the gender theory, God created Adam first and Eve second. The latter being the sin.

Large Numbers

Numbers originally accounted for practical and countable things, such as sheep and teeth. Things get interesting as quantities increase because we don't use numbers in the same way.

We approximate using a “round number” as a place mark. It is easier and more convenient. When I say, for example, that there were a hundred people at the market, I don’t mean that there were exactly one hundred people there. … Big numbers are understood approximately, small ones precisely, and these two systems interact uneasily. It is clearly nonsensical to say that next year the universe will be “13.7 billion and one” years old. It will remain 13.7 billion years old for the rest of our lives.

Round numbers usually end in zero.

The word round is used because a round number represents the completion of a full counting cycle, not because zero is a circle. There are ten digits in our number system, so any combination of cycles will always be divisible by ten. Because we are so used to using round numbers for big numbers, when we encounter a big number that is nonround— say, 754,156,293— it feels discrepant.

Manoj Thomas, a psychologist at Cornell University, argues that we are uneasy with large, non-round numbers, which causes us to see them as smaller than they are and carries with it practical implications when, say, selling a house. “We tend to think that small numbers are more precise,” he says, “so when we see a big number that is precise we instinctively assume it is less than it is.” If he's right the result is that you will pay more for expensive and non-round prices. Indeed his experiments seem to agree. In one, respondents viewed pictures of several houses and sales prices, some were round and some were larger and non-round (e.g., $490,000 and $492,332). On average subjects judged the precise one to be lower. As Bellos concludes on large numbers, “if you want to make money, don't end the price with a zero.”

Number Influence When Shopping

One of the ways to make a number seem more precise is by subtracting 1.

When we read a number, we are more influenced by the leftmost digit than by the rightmost, since that is the order in which we read, and process, them. The number 799 feels significantly less than 800 because we see the former as 7-something and the latter as 8-something, whereas 798 feels pretty much like 799. Since the nineteenth century, shopkeepers have taken advantage of this trick by choosing prices ending in a 9, to give the impression that a product is cheaper than it is. Surveys show that anything between a third and two-thirds of all retail prices now end in a 9.

Of course, we think that other people fall for this and surely not us, but that is not the case. Studies like this continue to be replicated over and over. Dropping the price one cent, say from $8 to $7.99 influences decisions dramatically.

Not only are prices ending in 9 harder to recall for price comparisons, we've also been conditioned to believe they are discounted and cheap. The practical implications of this are that if you're a high-end brand or selling an exclusive service, you want to avoid bargain aspect. You don't want a therapist who charges $99.99, any more than you want a high-end restaurant to list menu prices ending in $.99.

In fact, most of the time, it's best to avoid the $ altogether. Our response to this stimulus is pain.

The “$” reminds us of the pain of paying. Another clever menu strategy is to show the prices immediately after the description of each dish, rather than listing them in a column, since listing prices facilitates price comparison. You want to encourage diners to order what they want, whatever the price, rather than reminding them which dish is most expensive.

These are not the only nor most subtle ways that numbers influence us. The display of absurdly expensive items first creates an artificial benchmark. The real estate agent, who shows you a house way above your price range first, is really setting an artificial benchmark.

The $100,000 car in the showroom and the $10,000 pair of shoes in the shop window are there not because the manager thinks they will sell, but as decoys to make the also-expensive $50,000 car and $5,000 shoes look cheap. Supermarkets use similar strategies. We are surprisingly susceptible to number cues when it comes to making decisions, and not just when shopping.

We can all be swayed by irrelevant random numbers, which is why it's important to use a two-step framework when making decisions.

Numbers and Time

Time has always been counted.

We carved notches on sticks and daubed splotches on rocks to mark the passing of days. Our first calendars were tied to astronomical phenomena, such as the new moon, which meant that the number of days in each calendar cycle varied, in the case of the new moon between 29 and 30 days, since the exact length of a lunar cycle is 29.53 days. In the middle of the first millennium BCE, however, the Jews introduced a new system. They decreed that the Sabbath come every seven days ad infinitum, irrespective of planetary positions. The continuous seven-day cycle was a significant step forward for humanity. It emancipated us from consistent compliance with Nature, placing numerical regularity at the heart of religious practice and social organization, and since then the seven-day week has become the world’s longest-running uninterrupted calendrical tradition.

Why seven days in the week?

Seven was already the most mystical of numbers by the time the Jews declared that God took six days to make the world, and rested the day after. Earlier peoples had also used seven-day periods in their calendars, although never repeated in an endless loop. The most commonly accepted explanation for the predominance of seven in religious contexts is that the ancients observed seven planets in the sky: the Sun, the Moon, Venus, Mercury, Mars, Jupiter and Saturn. Indeed, the names Saturday, Sunday and Monday come from the planets, although the association of planets with days dates from Hellenic times, centuries after the seven-day week had been introduced.

The Egyptians used the human head to represent 7, which offers “another possible reason for the number’s symbolic importance.”

There are seven orifices in the head: the ears, eyes, nostrils and mouth. Human physiology provides other explanations too. Six days might be the optimal length of time to work before you need a day’s rest, or seven might be the most appropriate number for our working memory: the number of things the average person can hold in his or her head simultaneously is seven, plus or minus two.

Bellos isn't convinced. He thinks seven is special, not for the reasons mentioned above, but rather because of arithmetic.

Seven is unique among the first ten numbers because it is the only number that cannot be multiplied or divided within the group. When 1, 2, 3, 4 and 5 are doubled the answer is less than or equal to ten. The numbers 6, 8 and 10 can be halved and 9 is divisible by three. Of the numbers we can count on our fingers, only 7 stands alone: it neither produces nor is produced. Of course the number feels special. It is!

Favorite Numbers and Number Personalities

When people are asked to think of a digit off the top of their head, they are most likely to think of 7. When choosing a number below 20, the most probable response is 17. We'll come back to that in a second. But for now, let's talk about the meaning of numbers.

Numbers express quantities and we express qualities to them. Here are the results from a simple survey that paints a “coherent picture of number personalities.

From The Grapes of Math by Alex Borros
From The Grapes of Math by Alex Bellos

 

Interestingly, Bellos writes, “the association of one with male characteristics, and two with female ones, also remains deeply ingrained.”

When asked to pick favorite numbers, we follow clear patterns, as shown below in a heat map, in which the numbers from 1 to 100 are represented by squares. Bellos explains:

(The top row of each grid contains the numbers 1 to 10, the second row the numbers 11 to 20, and so on.) The numbers marked with black squares represent those that are “most liked” (the top twenty in the rankings), the white squares are the “least liked” (the bottom twenty) and the squares in shades of gray are the numbers ranked in between.

From the Grapes of Math by Alex Bellos
From the Grapes of Math by Alex Bellos

The heat map shows conspicuous patches of order. Black squares are mostly positioned at the top of the grid, showing on average that low numbers are liked best. The left-sloping diagonal through the center reveals that two-digit numbers where both digits are the same are also attractive. We like patterns. Most strikingly, however, four white columns display the unpopularity of numbers ending in 1, 3, 7 and 9.

Numbers are a part of our lives. We see them everywhere. They influence us, they guide us, and they help us solve problems. And yet, as The Grapes of Math: How Life Reflects Numbers and Numbers Reflect Life shows us, their history and patterns can also be a source of wonder.

The Colonel Blotto Game: How Underdogs Can Win

If you've ever wondered why underdogs win or how to improve your odds of winning when you're the underdog, this article on The Colonel Blotto Game is for you.

***

There is a rich tradition of celebrating wins by the weak—while forgetting those who lost—including the biblical Story of David vs. Goliath. It is notable, that “David shunned a traditional battle using a helmet and sword and chose instead to fight unconventionally with stones and a slingshot,” says Michael Mauboussin.

Luckily, David was around before Keynes said: “It is better to fail conventionally than to succeed unconventionally.” Turns out, if you're an underdog, David was onto something.

Despite the fact it is not as well known as the Prisoners' Dilemma, the Colonel Blotto Game can teach us a lot about strategic behavior and competition.

Underdogs can change the odds of winning simply by changing the basis of competition.

So what exactly is the Colonel Blotto Game and what can we learn from it?

In the Colonel Blotto game, two players concurrently allocate resources across n battlefields. The player with the greatest resources in each battlefield wins that battle and the player with the most overall wins is the victor.

An extremely simple version of this game would consist of two players, A and B, allocating 100 soldiers to three battlefields. Each player's goal is to create favorable mismatches versus his or her opponent.

According to Mauboussin, “The Colonel Blotto game is useful because by varying the game's two main parameters, giving one player more resources or changing the number of battlefields, you can gain insight into the likely winners of competitive encounters.”

To illustrate this point, Malcolm Gladwell tells the story of Vivek Ranadivé:

When Vivek Ranadivé decided to coach his daughter Anjali's basketball team, he settled on two principles. The first was that he would never raise his voice. This was National Junior Basketball—the Little League of basketball. The team was made up mostly of twelve-year-olds, and twelve-year-olds, he knew from experience, did not respond well to shouting. He would conduct business on the basketball court, he decided, the same way he conducted business at his software firm. He would speak calmly and softly, and convince the girls of the wisdom of his approach with appeals to reason and common sense.

The second principle was more important. Ranadivé was puzzled by the way Americans played basketball. He is from Mumbai. He grew up with cricket and soccer. He would never forget the first time he saw a basketball game. He thought it was mindless. Team A would score and then immediately retreat to its own end of the court. Team B would inbound the ball and dribble it into Team A's end, where Team A was patiently waiting. Then the process would reverse itself. A basketball court was ninety-four feet long. But most of the time a team defended only about twenty-four feet of that, conceding the other seventy feet.

Occasionally, teams would play a full-court press—that is, they would contest their opponent’s attempt to advance the ball up the court. But they would do it for only a few minutes at a time. It was as if there were a kind of conspiracy in the basketball world about the way the game ought to be played, and Ranadivé thought that that conspiracy had the effect of widening the gap between good teams and weak teams. Good teams, after all, had players who were tall and could dribble and shoot well; they could crisply execute their carefully prepared plays in their opponent’s end. Why, then, did weak teams play in a way that made it easy for good teams to do the very things that made them so good?

Basically, the more dimensions the game has the less certain the outcome becomes and the more likely underdogs are to win.

In other words, adding battlefields increases the number of interactions (dimensions) and improves the chances of an upset. When the basketball team cited by Malcolm Gladwell above started a full court press, it increased the number of dimensions and, in the process, substituted effort for skill.

The political scientist Ivan Arreguín-Toft recently looked at every war fought in the past two hundred years between strong and weak combatants in his book How the Weak Win Wars. The Goliaths, he found, won in 71.5 percent of the cases. That is a remarkable fact.

Arreguín-Toft was analyzing conflicts in which one side was at least ten times as powerful—in terms of armed might and population—as its opponent, and even in those lopsided contests, the underdog won almost a third of the time.

In the Biblical story of David and Goliath, David initially put on a coat of mail and a brass helmet and girded himself with a sword: he prepared to wage a conventional battle of swords against Goliath. But then he stopped. “I cannot walk in these, for I am unused to it,” he said (in Robert Alter’s translation), and picked up those five smooth stones.

Arreguín-Toft wondered, what happened when the underdogs likewise acknowledged their weakness and chose an unconventional strategy? He went back and re-analyzed his data. In those cases, David’s winning percentage went from 28.5 to 63.6. When underdogs choose not to play by Goliath’s rules, they win, Arreguín-Toft concluded, “even when everything we think we know about power says they shouldn’t.”

Arreguín-Toft discovered another interesting point: over the past two centuries the weaker players have been winning at a higher and higher rate. For instance, strong actors prevailed in 88 percent of the conflicts from 1800 to 1849, but the rate dropped very close to 50% from 1950 to 1999.

After reviewing and dismissing a number of possible explanations for these findings, Arreguín-Toft suggests that an analysis of strategic interaction best explains the results. Specifically, when the strong and weak actors go toe-to-toe (effectively, a low n), the weak actor loses roughly 80 percent of the time because “there is nothing to mediate or deflect a strong player‘s power advantage.”

In contrast, when the weak actors choose to compete on a different strategic basis (effectively increasing the size of n), they lose less than 40 percent of the time “because the weak refuse to engage where the strong actor has a power advantage.” Weak actors have been winning more conflicts over the years because they see and imitate the successful strategies of other actors and have come to the realization that refusing to fight on the strong actor’s terms improves their chances of victory. This might explain what's happening in the Gulf War.

In the Gulf War, the number of battlefields (dimensions) is high. Even though substantially outnumbered, the Taliban, have increased the odds of “winning,” by changing the base of competition, as they did previously against the superpower Russians. It also explains why the strategy employed by Ranadivé's basketball team, while not guaranteed to win, certainly increased the odds.

Mauboussin provides another great example:

A more concrete example comes from Division I college football. Texas Tech has adopted a strategy that has allowed it to win over 70 percent of its games in recent years despite playing a highly competitive schedule. The team’s success is particularly remarkable since few of the players were highly recruited or considered “first-rate material” by the professional scouts. Based on personnel alone, the team was weaker than many of its opponents.

Knowing that employing a traditional game plan would put his weaker team at a marked disadvantage, the coach offset the talent gap by introducing more complexity into the team’s offense via a large number of formations. These formations change the geometry of the game, forcing opponents to change their defensive strategies. It also creates new matchups (i.e., increasing n, the number of battlefields) that the stronger teams have difficulty winning. For example, defensive linemen have to drop back to cover receivers. The team’s coach explained that “defensive linemen really aren’t much good at covering receivers. They aren’t built to run around that much. And when they do, you have a bunch of people on the other team doing things they don’t have much experience doing.” This approach is considered unusual in the generally conservative game of college football.

While it's easy to recall all the examples of underdogs who found winning strategies by increasing the number of competition dimensions, it's not easy to recall all of those who, employing similar dimension enhancing strategies, have failed.

Another interesting point is why teams who are likely to lose use conventional strategies, which only increase the odds of failure?

According to Mauboussin:

What the analysis also reveals, however, is that nearly 80 percent of the losers in asymmetric conflicts never switch strategies. Part of the reason players don’t switch is that there is a cost: when personnel training and equipment are geared toward one strategy, it’s often costly to shift to another. New strategies are also stymied by leaders or organizational traditions. This type of inertia appears to be a consequential impediment to organizations embracing the strategic actions implied by the Colonel Blotto game.

Teams have an incentive to maintain a conventional strategy, even when it increases their odds of losing. Malcolm Gladwell explores:

The consistent failure of underdogs in professional sports to even try something new suggests, to me, that there is something fundamentally wrong with the incentive structure of the leagues. I think, for example, that the idea of ranking draft picks in reverse order of finish — as much as it sounds “fair” — does untold damage to the game. You simply cannot have a system that rewards anyone, ever, for losing. Economists worry about this all the time, when they talk about “moral hazard.” Moral hazard is the idea that if you insure someone against risk, you will make risky behavior more likely. So if you always bail out the banks when they take absurd risks and do stupid things, they are going to keep on taking absurd risks and doing stupid things. Bailouts create moral hazard. Moral hazard is also why your health insurance has a co-pay. If your insurer paid for everything, the theory goes, it would encourage you to go to the doctor when you really don't need to. No economist in his right mind would ever endorse the football and basketball drafts the way they are structured now. They are a moral hazard in spades. If you give me a lottery pick for being an atrocious GM, where's my incentive not to be an atrocious GM?

Key takeaways:

  • Underdogs improve their chances of winning by changing the basis for competition and, if possible, creating more dimensions.
  • We often fail to switch strategies because of a combination of biases, including social proof, status quo, commitment and consistency, and confirmation.

Malcolm Gladwell is a staff writer at the New Yorker and the author of The Tipping Point: How Little Things Make a Big Difference, Blink, Outliers and most recently, What the Dog Saw.

Michael Mauboussin is the author of More More Than You Know: Finding Financial Wisdom in Unconventional Places and more recently, Think Twice: Harnessing the Power of Counterintuition.

The Famous Game Show Problem

game show problem

Ahh, the famous game show problem (also known as The Monty Hall Problem).

This is a probability puzzle you've heard of:

Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what's behind the doors, opens another door, say #3, which has a goat. He says to you, “Do you want to pick door #2?” Is it to your advantage to switch your choice of doors?

Marilyn Vos Savant, one of the “smartest” people in the word, offers her answer:

When you switch, you win 2/3 of the time and lose 1/3, but when you don't switch, you only win 1/3 of the time and lose 2/3. You can try it yourself and see…..

The winning odds of 1/3 on the first choice can't go up to 1/2 just because the host opens a losing door. To illustrate this, let's say we play a shell game. You look away, and I put a pea under one of three shells. Then I ask you to put your finger on a shell. The odds that your choice contains a pea are 1/3, agreed? Then I simply lift up an empty shell from the remaining other two. As I can (and will) do this regardless of what you've chosen, we've learned nothing to allow us to revise the odds on the shell under your finger.

The benefits of switching are readily proven by playing through the six games that exhaust all the possibilities. For the first three games, you choose #1 and “switch” each time, for the second three games, you choose #1 and “stay” each time, and the host always opens a loser.

Warren Buffett and the Power of Nontransitive Dice

nontransitive diceIn a terrific book by William Poundstone, Fortune's Formula: The Untold Story of the Scientific Betting System that Beat the Casinos and Wall Street, is the story of a 1968 dinner meeting between mathematician Edward Thorp and fund manager Warren Buffett.

Poundstone casually mentions that Buffett and Thorpe discussed their shared interest in nontransitive dice.

“These are,” writes Poundstone, “a mathematical curiosity, a type of ‘trick' dice that confound most people's ideas about probability.”

A nice description of nontransitive dice by Ivars Peterson at the Mathematical Association of America's website:

The game involves four specially numbered dice. You let your opponent pick any one of the four dice. You choose one of the remaining three dice. Each player tosses his or her die, and the higher number wins the throw. Amazingly, in a game involving 10 or more throws, you will nearly always have more wins.

Here's what the dice look like:

The trick is to always let your opponent pick first, and then you pick the die to the left of his selection (if he picks the die with the four 4s, then circle round to the die with the three ones). It's just like playing Rock, Paper, Scissors — only you get to see what the other guy picks in advance.

With these dice, you always have a 2/3 probability of winning — what a great sucker's bet!

Free Book : The Mathematics of Gambling

Edward Thorp on the Mathematics of Gambling.

“The Mathematics of Gambling” is quite different from those other books. For instance, it does not focus on just one game like most of the others. In fact, it barely explains a game at all. Instead, it describes the mathematical methods that might be used to win at the game more consistently. Think of this book as a starting point to understanding gambling theories.

Part one | Part two | Part three | Part four