Pascal's Triangle

Last time I talked about the famous problem in probability known as the "problem of points" (also known as the "division of stakes").  Again, suppose that Bobby and Patty are playing a game of "Heads or Tails" by flipping a quarter (Patty wins with "Heads" while Bobby wins with "Tails").  They've decided that whoever is the first to win ten games will win the prize, in this case a dollar's worth of change that they found on the sidewalk.  Unfortunately, their parents call them in for dinner before they can finish the game.  Patty has 9 wins, while Bobby has only 7 wins.  What's the fairest way to divide up the dollar's worth of change now?  

I discussed how the mathematician Pierre de Fermat solved this problem, which while relatively straightforward could get very cumbersome.  Fermat's solution relied upon the fact that if one player needs r more rounds to win and the other needs s rounds to win, the game will have been won by someone after r + s - 1 rounds.  By the simple laws of mathematics, these rounds have 2^(r + s - 1) possible outcomes.  So, to use our example, Patty needs only 1 more win, while Bobby needs 3 more wins.  If the game had not been interrupted, there would have been a winner after three rounds (1+3-1=4-1=3), leading to 8 possible outcomes (2^3 = 2 x 2 x 2 = 8).  But what if the game had been interrupted when Patty needed 5 more wins and Bobby needed 7 wins?  We would have had to calculate the possible outcomes for (5+7-1) 11 more games, which would equate to 2^11 = 2,048 possible outcomes!  That would take forever to calculate the respective probabilities that Patty or Bobby eventually win the game.

Pascal's method of solving the "problem of points" definitely makes things a lot easier and introduces two concepts, specifically expected value and the arithmetic triangle that now bears his name (Pascal's Triangle).  Briefly, the expected value in a game of chance is the probability of that outcome multiplied by the value of the reward.  As an example, if Patty has a 25% chance of winning one thousand dollars, her expected value is 25% x $1,000 = $250.  Pascal suggested (and Fermat agreed with him) that the the division of stakes would be fair if the expected value to the player did not change.  

Pascal also further elaborated on the arithmetic triangle that now bears his name, though the concept was studied much earlier by mathematicians in India, Persia, and China.  Basically, the triangle is constructed by placing the number 1 in row zero (the top row).  Each entry of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as zero.  Here are the first eight rows of the triangle:

Notably, Pascal's triangle determines the coefficients which arise in any binomial expansion (i.e. when the a binomial such as x + y is raised to a positive integer power of n).  Importantly, the entry in the nth row and kth column is denoted by the following notation and determined by the rule below:

In general, according to the binomial theorem, when a binomial expression such as (x + y) is raised to a positive integer power of n, we have the following expression, where the coefficients a_k are precisely the numbers in row n of Pascal's Triangle!

So, let's go back to our original problem of points described in the previous post, where Patty needs to win 1 more round in order to win the game and Bobby needs to win 3 more rounds.  Using Fermat's equation (r + s - 1), we know that there will be a winner after 3 additional rounds.  We go to the 3rd row of Pascal's Triangle, which contains the following numbers 1, 3, 3, and 1.  Note that these numbers add up to 8, which is the number of possible outcomes that we determined above for the next 3 rounds of play.  The first number represents the outcome where Bobby wins (again, he needs 3 to win and Patty only needs 1), i.e. 1 chance out of 8 (which is the same as what we calculated above using Fermat's method).  The sum of the next 3 numbers (3, 3, and 1) add up to 7, which is the number of chances that Patty will win out of 8 (again, the same that we calculated using Fermat's method!).

Isn't mathematics fun!?!?  There are a number of online videos explaining how to use Pascal's Triangle to solve the "problem of points", but hopefully my explanation is sufficient to at least give you a rough idea of Pascal's method.  There's also a great book discussing the history of probability, as told through Pascal's and Fermat's correspondence about the "problem of points" by Keith Devlin called "The Unfinished Game".  

For the last 3 posts, I've been talking a lot about mathematics, specifically mentioning a few of the more famous problems in probability and statistics, namely the birthday problem, the Monty Hall problem, and the problem of points.  The question that I am sure all of you are asking is, "What does all of this have to do with leadership?"  The answer might surprise you - these problems have a lot to do with leadership!  These three famous mathematics problems provide a great foundation in the science of probability, which is needed in order to start talking about the science of game theory.  Game theory is a branch of economics that deals with strategic decision-making.  In other words, game theory can help your organization make better decisions!  So at least for the next few posts, I want to discuss a little bit more of the fascinating discipline of game theory, specifically as it pertains to making better decisions! 

The Problem of Points

Suppose that Bobby and Patty are playing a game of "Heads or Tails" by flipping a quarter (Patty wins with "Heads" while Bobby wins with "Tails").  They've decided that whoever is the first to win ten games will win the prize, in this case a dollar's worth of change that they found on the sidewalk.  Unfortunately, their parents call them in for dinner before they can finish the game.  Patty has 9 wins, while Bobby has only 7 wins.  What's the fairest way to divide up the dollar's worth of change now?  

Patty argues that because she is winning, she should take home all the money.  Bobby cries out "No fair!" and states that they should just split the money 50/50.  After a few more minutes of arguing, Patty suggests that because she has won 9 out of the 10 games needed to win, she should receive 9/10 of the dollar.  Bobby again cries out "No fair!" and states that using that argument, he should take home 7/10 of the dollar (and not 1/10 as Patty proposes).  Who's correct here?

Actually, mathematicians have been arguing about how to divide up the winnings going back as far as the 15th century.  It's called the "problem of points" or the "division of stakes", and it was first discussed by the mathematician Luca Pacioli (who was a contemporary of Leonardo da Vinci) in his 1494 book Summa de arithmetica, geometrica, proportioni et proportionalità.  Incidentally, this same book also contains the first published description of the double-entry system of accounting, so Pacioli is often referred to as the "father of accounting" (the book actually was plagiarized from the writings of  Piero della Francesca, but that is a topic for another day).  Pacioli solved the problem by dividing the stakes in proportion to the number of rounds won by each player (Patty would have been happy!).

Another Italian mathematician, Niccolò Fontana Tartaglia suggested that Pacioli's method would not be fair if the game was interrupted after the first round (in our example, Patty would have won the dollar after winning just the first game).  He proposed a different way of dividing the stakes based on the ratio between the size of the lead and the length of the game.  Unfortunately, mathematicians soon discovered that this wasn't a perfect solution either.

Here's where history gets even more interesting, at least for me.  The 17th century mathematicians Blaise Pascal and Pierre de Fermat (most famous for his eponymous last theorem) exchanged a series of letters in 1654 in which they discussed how to solve the "problem of points".  Pascal and Fermat started with the insight that the "division of stakes" shouldn't rely on what had already taken place during the interrupted game, but on the possible ways that the game could have finished if it had not been stopped prematurely.  For example, if Patty has a 9-7 lead over Bobby in a game to 10 versus a 19-17 lead in a game to 20, she has similar odds of winning in both games.  In other words, what matters is the number of rounds that each player still needs to win in order to win the game as opposed to the number of rounds he or she has won up to the point that the game is stopped.

Fermat's solution relied upon the fact that if one player needs r more rounds to win and the other needs s rounds to win, the game will have been won by someone after r + s - 1 rounds (work it out, he's absolutely correct!).  By the simple laws of mathematics, these rounds have 2^(r + s - 1) possible outcomes.   Fermat's solution then was to compute the odds for each player to win simply by writing down all of the possible outcomes and counting how many of them would lead to each player winning.  He would then divide the stakes accordingly.

So, to use our example, Patty needs only 1 more win, while Bobby needs 3 more wins.  If the game had not been interrupted, there would have been a winner after three rounds (1+3-1=4-1=3).  There would have been 8 possible outcomes (2^3 = 2 x 2 x 2 = 8), as follows:

In other words, Patty would win 7 out of the 8 possible outcomes (7/8), while Bobby would only win 1 out of the 8 possible outcomes (1/8).  Patty should therefore take home 7/8 of the prize, while Bobby should take home 1/8 of the prize.  But wait, that's kind of ridiculous too, isn't it?  For the first possible outcome in the Table above, why would they still continue to flip the coin since Patty already won?  Just to check that this still works, let's look at only the feasible outcomes:

Let's calculate the probability of Patty winning (remember, with a coin toss there is 1/2 chances of "Heads" and 1/2 chances "Tails"):

P(A) = 1/2

P(B) = 1/2 x 1/2 = 1/4

P(C) = 1/2 x 1/2 x 1/2 = 1/8

By the rules of probability, if you have mutually exclusive events, you add the probabilities.  Therefore, the probability of Patty winning, P(Patty) = 1/2 + 1/4 + 1/8 = 7/8, which is exactly what we found in the case above!

While Fermat's method of solving the "problem of points" works quite well, just imagine having to calculate the probabilities or listing out all of the possible outcomes when the number of rounds required to determine a winner becomes larger!  Here is where Pascal's solution makes things a little easier and involves something that is now called expected value and the arithmetic triangle that now bears his name (Pascal's Triangle).  We will cover Pascal's solution to the  "problem of points" next time.

Let's Make a Deal!

Our family started watching the 2008 movie "21" starring Kevin Spacey, Kate Bosworth, and Jim Sturgess the other night.  The movie is based upon the book Bringing Down the House and tells the story of how a group of MIT students won a lot of money playing Blackjack in Las Vegas.  We didn't make it to the end, but one of the opening scenes caught my attention.  The MIT mathematics professor played by Spacey challenges one of his students (played by Sturgess) with the classic "Monty Hall Problem", which is the subject of today's post.

The "Monty Hall Problem" is a brain teaser that is named in honor of Monty Hall, the host of the television game show "Let's Make a Deal".  The problem was originally submitted by Steve Selvin to the journal American Statistician in 1975, but it became famous from a letter sent by reader Craig Whitaker to Marilyn vos Savant (best known for having the highest known IQ reported by the Guinness Book of Records) for her weekly column "Ask Marilyn" in Parade magazine in 1990.  Here is the question as posed by Whitaker:

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 No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

Let's break this down using simple rules of probability.  At the beginning of the game, the contestant has a 1/3 chance of winning a car, which means that the probability of winning a goat is 2/3.  When the game show host reveals a goat behind one of the two remaining doors (not selected by the contestant), the contestant can increase the probability of winning a car to 2/3 if he or she switches to the other door!  

When vos Savant first explained this veridical paradox, approximately 10,000 readers, including 1,000 who had a PhD in mathematics, wrote back to the magazine to tell her that she was wrong.  The famous mathematician Paul Erdős (read about the so-called Erdős Number here) remained unconvinced until someone showed him a simulation of the problem.  If a simulation worked for Erdős, it will probably work for all of you too!

Let's look at all the different permutations of this game, which gives us a total of 9 different scenarios:

If the contestant (in this case, you) adopts the strategy of not switching, he or she wins 3 times out of 9 total chances, or 1/3.  However, if the contestant does indeed switch doors, he or she will increase the chance of winning a car to 6/9, or 2/3!

The "Monty Hall Problem" has made its way into pop culture.  For example, the problem was featured in a 2011 episode of the television show "MythBusters", in which the hosts confirmed that when posed with a version of the "Monty Hall Problem", most people will not switch their choices (when the better strategy, again confirmed by the hosts, would be to switch their choice).  The television show "Survivor" also used a version of the "Monty Hall Problem" in seasons 41 and 42.  Surprisingly, in both cases, the contestant chose not to switch and still won the game!  It just goes to show that playing the odds is not always a winning strategy!  Next time, I want to talk about one more famous problem in probability, and then I promise I will explain what all of this has to do with leadership!

Happy Birthday!

Have you ever shared a birthday with someone you know, either at school or work?  The chance that this occurs is a lot more common than we realize.  As a matter of fact, the probability of sharing a birthday in a group of only 30 people (which is around the size of an elementary school classroom) is over 70%!  

A few years ago, I posted about a concept by Douglas W. Hubbard known as the "rule of five", which he discussed in his book, How to Measure Anything.  At that time, I mentioned a famous problem in probability and statistics known simply as "the birthday problem", which seems counterintuitive until you actually take the time to work out the mathematics.  Let's take a closer look.

Let's first review a couple of rules of simple probability.  Remember that if you are rolling a standard die, the chance of coming up with the number "3" is 1 chance out of six, or 1/6.  Similarly, the chance of rolling an even number (2, 4, or 6) is 3 chances out of six, or 3/6 (which of course simplifies into 1/2 or 50%).  What happens if you roll two dice?  What is the chance that you will roll "3" with the first die and "4" with the second?  The simple rules of probability state that the chance of two independent events occurring is a product of their individual probabilities.  In other words, we multiply 1/6 (the chance of rolling a "3") by 1/6 (the chance of rolling a "4"): 1/6 x 1/6 = 0.03, or 3%.  What's the probability of not rolling a "3" on one die?  It's just 1 minus the probability of rolling a "3", or 1 - 1/6 = 5/6.  You now have all the tools to answer the birthday problem!

Suppose you are a student in a classroom with 29 other students.  What is the probability that you share a birthday with at least one other student in the class?  The best way to tackle this problem is to first calculate the probability (P) that these 30 individuals do NOT share a same birthday, P(A).  Therefore, according to the simple rules of probability discussed above, the probability that at least one student shares his or her birthday with someone else in the class is P(B) = 1 - P(A). 

Let's also make things a little more straightforward and assume that there aren't any twins in the class (twins would share the same birthday) and that there are no leap years involved, which means that there are 365 possible birthdays in a year.  Imagine that we ask all 30 students to form a line, and each student will go and mark his or her birthday on a calendar, one right after the other.  The probability that the first student will have a birthday that is not shared with anyone else is 365/365.  The probability for the second student not sharing a birthday with the first student is 364/365.  The probability for the third student not sharing a birthday with the first or second students is 363/365.  These are all independent events, so in order to calculate the probability of all 30 students not sharing a birthday, we have to multiply the probability of the individual students:

P(A) = 365/365 x 364/365 x 363/365 x ... 337/365 x 336/365 = 0.293684

The probability that at least two individuals share the same birthday, P(B), is therefore:

P(B) = 1 - P(A) = 1- 0.293684 = 0.706 or 70.6%!

If you look the "birthday problem" up online, you can find a graph of P(B) as a function of the number of people in a group that looks something like this:

The "birthday problem" is a great example of what is called a veridical paradox, which is something that appears to be false but is nonetheless true.  Next time, we will discuss another famous veridical paradox (and I promise I will explain what all of this has to do with leadership next time too).

Where is everybody?

Everyone so often, I come across an interesting article or hear about something that I want to write about, even if the article or bit of information has nothing to do with leadership.  As my father has always told me, it's good to be well-rounded and knowledgeable about a lot of different things.  With my father's sage advice firmly in mind, I wanted to explore a recent article I came across that involves the Drake Equation, the Fermi Paradox, and something called the Great Filter.  It's all in the spirit of being well-rounded!

I first heard about the so-called Drake Equation on Carl Sagan's 1980 television mini-series "Cosmos" (I remember getting the book that Sagan wrote as a companion to the mini-series that year for Christmas).  Frank Drake developed the equation bearing his name in 1961 to "estimate" the number of extraterrestrial civilizations that could theoretically exist in our Milky Way Galaxy.  Contrary to popular belief, the equation is more conceptual than numerical (in other words, the equation doesn't actually estimate the number of extraterrestrial lifeforms) and has been used to try to answer the questions "Is there life out there?" or "Are we alone?"  Here is the equation as originally formulated by Dr. Drake:

N = R* x fp x ne x f1 x fi x fc x L

where, 

N is the number of civilizations with which humans could communicate

R* is the rate of new star formation in the galaxy

fp is the fraction of those stars that have planets (like our solar system)

ne is the number of planets that could support life per star with planets

f1 is the fraction of life-supporting planets that actually develop life

fi is the fraction of planets with life where life develops intelligence

fc is the fraction of intelligent civilizations that develop technology to communicate with us

L is the length of time that civilizations can communicate

Again, the Drake Equation was never intended to actually come up with a quantifiable number of civilizations that which we could possibly communicate with, though several experts have provided estimates for each of these variables.  Drake originally estimated that N closely approximated L and concluded that there were probably between 1,000 and 100,000,000 planets with civilizations in the Milky Way Galaxy.  If that is indeed the case, then why haven't we been able to communicate with at least one of these civilizations?  

The great 20th century Italian physicist Enrico Fermi asked this very same question in the 1950's.  As the story goes, Fermi had a casual conversation in the summer of 1950 with fellow physicists Edward Teller, Herbert York, and Emil Konopinski in response to several reports of UFO sightings in the media.  While the exact quote is not known, Fermi is said to have blurted out, "But where is everybody?"  In other words, if there is intelligent life out there, why haven't we been able to communicate with them?  If, as revealed by the Drake Equation, that there are anywhere between 1,000 to 100,000,000 planets with civilizations in our own galaxy, why haven't we been able to contact them?  This, in essence, summarizes the Fermi Paradox.  

Here is where the article that I found comes in.  Scientists at NASA recently submitted a manuscript for publication that provides a possible explanation to the Fermi Paradox.  The manuscript has not been peer-reviewed (meaning that it hasn't been accepted for publication yet).  These scientists talk about something known as the Great Filter.  Robin Hanson, an economist at George Mason University argued in an essay he wrote in 1996 ("The Great Filter - Are We Almost Past It?") that our failure to locate extra-terrestrial life is due to some impenetrable barrier (or filter) that prevents intelligent life from developing to the point where it can communicate with us.  Hanson states that there are nine steps along the evolutionary path to intelligent life:

1. The right star system (potentially habitable planets that can support life - see the Drake Equation above)

2. Reproductive molecules (RNA, DNA, etc)

3. Simple (prokaryotic) single-cell life

4. Complex (eukaryotic) single-cell life

5. Sexual reproduction

6. Multi-cell life 

7. Tool-using animals with intelligence

8. A civilization advancing toward the potential for a colonization explosion

9. Colonization explosion (specifically, space colonization)

Hanson pointed out that life on planet Earth has made it successfully from step 1 through step 8, though we have yet to progress to the 9th and final step (so perhaps this is the ultimate barrier to finding out if life exists elsewhere).  He further argues that because we have yet to find evidence of extraterrestrial life, progression through these steps is very improbable, suggesting that there is a "Great Filter" along the path from the primordial soup to colonization.  He wrote then that "the fact that our universe seems basically dead suggests that it is very very hard for advanced, explosive, lasting life to arise."

Hanson also suggests the possibility of some sort of catastrophic social collapse, such as what happened to the dinosaurs here on planet Earth that effectively wipes out life's progression from steps 1-9.  Alternatively, intelligent life tends to destroy themselves before being able to make contact off-planet.  To that end, a group of investigators from NASA's Jet Propulsion Laboratory proposed several different ways that life on planet Earth could be wiped out, including nuclear war, pandemics, asteroid or comet impacts, climate change, and artificial intelligence.  I would argue that it doesn't take a creative genius to come up with a list of possible scenarios ending life as we know it, just watch any of a number of popular science fiction movies involving these apocalyptic scenarios.

There is one last possibility for why we haven't found evidence of intelligent extraterrestrial life (see my post "Weird Science" for a more detailed explanation).  Basically, this one comes straight from the science fiction movie "The Matrix" .  The Oxford philosopher Nick Bostrom proposed a few years ago  that at least one of the following statements is true - (1) the human species will become extinct before we progress to a post-human state (think artificial intelligence, virtual reality, robotics, and cyborgs!); (2) any post-human civilization is unlikely to actually run a computer simulation of real-world life and humans (basically because they either don't possess the ability or they are too ethical to do so); or (3) we are already living in a simulated world (which is to say that we all co-exist in the Matrix).  If we are living in a simulated world, then of course we wouldn't find evidence of life in outer space.  But then again, why wouldn't we?

All of this is very interesting to ponder.  And perhaps the takeaway lesson here is that we need to be careful about what we are doing to our planet.  It may be the only planet that we will ever have, and we need to be careful about not being the cause of our own destruction.

Happy Veterans Day 2022

On November 11, 1919, President Woodrow Wilson issued a message to all Americans on the one-year anniversary of Armistice Day, which marked the end of major hostilities in World War I at the 11th hour of the 11th day of the 11th month of 1918.  He issued a message to the people of the United States on that very first Armistice Day, in which he expressed what he felt the day meant to Americans:

“To us in America the reflections of Armistice Day will be filled with solemn pride in the heroism of those who died in the country’s service, and with gratitude for the victory, both because of the thing from which it has freed us and because of the opportunity it has given America to show her sympathy with peace and justice in the council of nations.”

Following that first anniversary of the end of World War I in 1919, Armistice Day was unofficially celebrated every year on November 11th, until Congress passed a resolution to officially honor and observe November 11th every year as Armistice Day in 1938.

Unfortunately, World War I didn't end up being "the war to end all wars" as everyone intended or maybe just hoped.  Soon, a little more than 20 years later, the world was again at war.  It was a veteran of that war (World War II) named Raymond Weeks who petitioned (then) General Dwight Eisenhower to expand Armistice Day to celebrate all veterans, not just those who died or served in World War I.  General Eisenhower, of course, supported Week's recommendation until he signed the bill that officially named November 11th as Veterans Day in 1954.  Incidentally, Weeks was later honored with a Presidential Medal of Freedom by President Ronald Reagan and was recognized as the “Father of Veterans Day" in 1982, just three years before his death.

Today, we celebrate Veterans Day, which honors all military veterans, those individuals who have served in one of the branches of the United States Armed Forces (Army, Air Force, Navy, Marine Corps, Coast Guard, and Space Force).  Veterans Day coincides with other holidays that are celebrated in other countries (Remembrance Day in the United Kingdom and Commonwealth countries, for example), though it is distinct from Memorial Day, a U.S. holiday honoring those who died while in military service and Armed Forces Day honoring those individuals who currently serve in the military.

While most calendars print November 11th as “Veteran’s Day” (with an apostrophe), the U.S. Department of Veterans Affairs states that the apostrophe is not necessary, “because it is not a day that ‘belongs’ to veterans, it is a day for honoring all veterans.”  The U.S. government also recommends that we honor all veterans at 2:11 PM Eastern Time with two minutes of silence.

I've always heard that during combat, soldiers fight primarily for each other.  Take a look at the following movie clip from the 2001 movie, Black Hawk Down.  The character "Hoot" (played by actor Eric Bana) is telling another character, First Sergeant Matt Eversmann (played by the actor Josh Hartnett) what motivates him to continue to serve.  Hoot, a Delta Force operator, says:

When I go home people'll ask me, "Hey Hoot, why do you do it man? What, you some kinda war junkie?" You know what I'll say? I won't say a goddamn word. Why? They won't understand. They won't understand why we do it. They won't understand that it's about the men next to you, and that's it. That's all it is.

I just recently finished the book Fairness and Freedom: A History of Two Open Societies, a comparative history of New Zealand and the United States by the historian and writer, David Hackett Fischer.  Fischer wrote about a study conducted by the nineteenth century French army officer Ardant du Picq, who interviewed veterans of the Algerian and Crimean wars and published his findings in the book Battle Studies.  He concluded that fear was nearly universal during battle, but the reason that soldiers overcame their fear and stayed to fight was the fear of letting down their comrades.  These findings were later confirmed and expanded upon in studies by the American journalist and military historian S.L.A. Marshall.  Following World War II, several studies led and conducted by the American sociologist Samuel A. Stouffer in the 1940's and 1950's confirmed that solderis fought primarily for each other, the soldiers fighting next to them.

The men and women who have served together in our nation's Armed Forces share a special bond, a bond shared even among those who have never been in combat.  But regardless of their motivations, there is still something truly special about their willingness to step forward, raise their hand, and potentially give their lives in the service of their fellow citizens and their country. 

If you are a veteran, today, we celebrate and honor you.  Thank you for your dedication and commitment to our country and thank you for your service!  

Advancing science one funeral at a time...

Do you remember when you were first taught the classic scientific method?  Believe it or not, we were all taught these foundational principles of science way back in elementary school.  Even if we didn't quite fully understand it back then, we were taught that scientific discovery is dependent upon careful observation, rigorous skepticism about what is observed, formulating hypotheses, experimentally testing these hypotheses, and then accepting or rejecting them based upon the analysis of our data.  Advances in science, in other words, occur through an iterative process based upon ideas developed and tested in prior experiments.  Consider what the English mathematician and scientist Isaac Newton once said, "If I have seen further, it is by standing on the shoulders of giants."

If Sir Isaac Newton claims that his discoveries built upon the discoveries of scientists that came before him, who am I to argue?  On the other hand, if advances in science depend upon a certain amount of skepticism, maybe we would be better off asking whether Newton's claim has ever been rigorously tested!  Consider for just one moment what another famous scientist once claimed.  The German physicist Max Planck, winner of the 1918 Nobel Prize for his early studies in quantum theory said, "A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it."

A group of investigators set out to test this exact question and published their findings online for the National Bureau of Economic Research ("Does science advance one funeral at a time?").  Specifically, they examined how the death of 452 eminent or "superstar" scientists (highly funded and highly cited scientists who were often members of the National Academy of Sciences or the Institute of Medicine) altered the number of publications and funded grants by their collaborators and competitors in the years after their death.  Perhaps surprisingly to Newton (but perhaps less surprising to Planck), the flow of articles by collaborators decreased precipitously (by an average of 40%) in the years following the death of a "superstar" scientist , while the number of articles published by competitors actually increased by 8%.    Moreover, these articles were disproportionately likely to be highly cited and often were authored by scientists who were not previously active in the deceased superstar's field.  These investigators concluded that "...outsiders are reluctant to challenge leadership within a field when the star is alive."

What exactly prevented the competitors, who were often younger scientists who were trying to start their careers from entering these fields while the "superstars" were still alive?  These investigators could not definitively answer this question, but they did pose a few possibilities.  Outsiders (particularly younger scientists at the beginning of their careers) may be more reluctant to challenge a luminary in the field.  The younger scientists simply choose not to try to compete with the superstars.  It's a well-known fact that scientists have to compete for limited funding resources, so it may be that the superstars' hold on grant funding represents a significant barrier to entry.  Rather than standing on the shoulders of giants, these younger scientists stand in their shadows.  Superstars have a number of established collaborations with other investigators in the field, which represents another barrier to entry.

So it seems we have more evidence to support Planck's claim rather than Newton's.  The question becomes then, what should we do about the fact that the presence of a superstar can "freeze out" new entrants, which can actually prevent a field from advancing beyond the widely accepted paradigms (which are usually based upon the superstar's research)?  That question is even more difficult to answer.  Maybe we should pay attention to what one superstar scientist, Carl Sagan, had to say about this, when he claimed, "If we can’t think for ourselves, if we’re unwilling to question authority, then we’re just putty in the hands of those in power."

Butterflies

The word chaos comes from the Greek khaos meaning "emptiness, vast void, or abyss."  It was first used in ancient Greek mythology to describe the state of the world before creation.  Chaos theory refers to the study of the irregular and unpredictable evolution of complex systems and is perhaps best illustrated by Edward Norton Lorenz and his famous "butterfly effect" (see also the famous poem, whose author is unknown, "For want of a nail").  Basically, Lorenz suggested that a butterfly flapping its wings in Brazil could set off a tornado in Texas.

Lorenz was studying local weather patterns using a computerized simulation, which included several different variables such as temperature, air pressure, and wind speed.  As the story goes, he had run the simulation earlier in the day and was repeating it to validate his initial results.  He briefly left the computer program running while he went to go for a cup of coffee.  Upon his return, he was shocked to find that the results had dramatically changed.  He had changed just one of the variables by rounding off a few decimal places (0.506127 was changed to 0.506).  However, to his amazement this small change yielded a much larger change in his results.  

Lorenz published his findings in a little known paper entitled, "Deterministic Nonperiodic Flow" in the Journal of the Atmospheric Sciences in 1963.  Very few scientists paid attention to this paper at first (the paper was cited just three times over the next 10 years).  However, over time, this little known paper was recognized as the first to describe one of the foundational principles in the field of chaos theory, which has been applied in fields as diverse as biology, geology, and physics.  

The "butterfly effect" has even made its way into pop culture.  Robert Redford's character in the 1990 movie, Havana claimed, "A butterfly can flutter its wings over a flower in China and cause a hurricane in the Caribbean."  Jeff Goldblum's character in Jurassic Park said, "A butterfly can flap its wings in Peking, and in Central Park, you get rain instead of sunshine."  

I recently came across a short story by Ray Bradbury that reminded me a lot of the "butterfly effect" called "A Sound of Thunder".  Bradbury's story first appeared in Collier's magazine in 1952, but nevertheless Lorenz is still credited for coining the term and concept.  After reading the short story, I was taken back to my childhood when I was fascinated with the concept of time travel.  

The story is set far into the future (in the year 2055, to be exact), and the main characters are talking about how relieved they are that someone named Keith beat someone else named Deutscher in the recent U.S. Presidential election.  The main character, Eckels is purchasing a ticket with a commercial safari company that takes their clients back in time to hunt dinosaurs.  The safari guide warns them that they shouldn't touch anything and that they should only kill designated dinosaurs (ones that the company followed closely in time to make sure that they were going to die shortly anyway).  The safari is even conducted upon a levitating path so that the clients don't touch and disturb the ground.  Apparently, even the slightest change could alter the course of history.  

Here's an excerpt from the story, when the safari guide is explaining what could happen if Eckels accidentally kills a mouse in the past.

"All right," Travis continued, "say we accidentally kill one mouse here.  That means all the future families of this one particular mouse are destroyed, right?"

"Right."

"And all the families of the families of the families of that one mouse!  With a stamp of your foot, you annihilate first one, then a dozen, then a thousand, a million, a billion possible mice!"

"So they're dead," said Eckel.  "So what?"

"So what?" Travis snorted quietly.  "Well, what about the foxes that'll need those mice to survive?  For want of ten mice, a fox dies.  For want of ten foxes, a lion starves.  For want of a lion, all manner of insects, vultures, infinite billions of life forms are thrown into chaos and destruction.  Eventually it all boils down to this: fifty-nine million years later, a caveman, one of a dozen on the entire world, goes hunting wild boar or saber-toothed tiger for food.  But you friend, have stepped on all the tigers in that region.  By stepping on one single mouse.  So the caveman starves.  And the caveman, please note, is not just any expendable man, no!  He is an entire future nation.  From his loins would have sprung ten sons.  From their loins, one hundred sons, and thus onward to a civilization."

You get the idea - for want of a nail, the kingdom was lost.  Well, you can probably guess what happens.  Eckels gets too scared to shoot the dinosaur and runs back to the time machine.  While running, he stumbles and unknowingly steps on a tiny butterfly.  They return back to their present day, 2055, to find things are just a little off compared to when they had left.  And they discover that President Deutscher has actually won the election!

My point here is not to talk about time travel (although that would be cool).  And I really don't want to talk specifically about butterflies either.  The take-home message here is that each and every one of us can have a tremendous impact on the world.  We may not ever see it, but we do.  

It's very easy right now to feel a little jaded about what is happening in our world.  It's very easy to be cynical, or worse maybe even hopeless.  But that would be the wrong response - because we can make a difference.  Each of us has a job to do, and no matter how small that job is, it can and will make a difference.  Somewhere.  Somehow.  It will make a difference.

Just consider a recent article written by Anton DiSclafani in the New York Times.  DiSclafani is a left-leaning Democrat who decided to vote in the most recent Alabama primary (Alabama has an open primary, so he voted in the Republican primary).  He voted for what he thought was the best choice for him, even though neither candidate came even close to representing his personal and political views and values (in this case, the best candidate was the least worst candidate).  DiSclafani's vote ended up being the deciding vote (his candidate won by one vote!).

This coming Tuesday is Election Day in the United States.  It's going to be an important one.  Remember, every vote counts, so go out and please vote!  Your vote matters.  Your contributions matter.  You are that butterfly in Brazil flapping his wings and causing a tornado in Texas!

"Everyone gets the best deal"

Have you seen the AT&T commercial for the new iPhone starring the professional basketball player LeBron James and saleswoman Lily Adams (played by the actress Milana Vayntrub)?  Well, if you haven't, it's because you've not watched any television during the last couple of weeks.  It seems that the commercial plays all the time on network television.  

The commercial opens with LeBron James sitting in an AT&T store talking with saleswoman Lily about the new iPhone 14 Pro.  Lily says that the phone is amazing and tells the basketball star, "You'll get our best deal."  Of course, LeBron argues that everyone should get the best deal, to which Lily replies that everyone does get the best deal, both new and old customers, on every iPhone.  LeBron then announces, "My work is done" and stands up to walk away.  He stares into the camera as he walks away and proudly announces that everyone gets the best deal on every iPhone.

The commercial hasn't been very popular (possibly because it's been too overplayed).  My initial gut reaction, to be 100% honest, was "Wow, LeBron James is taking credit for something that he didn't do."  If you check out the online comments, my initial reaction wasn't unique.  There is absolutely no question that LeBron James will go down as one, if not the, greatest basketball players of all time.  More importantly, his contributions outside basketball have been incredible.  I have nothing but admiration and respect for him as a player and as a person.  But come on.  He didn't get us a deal on every iPhone! 

I realize this is only a commercial, but I thought it was a great opportunity to talk about one of the more common cognitive biases.  Again, as a reminder, a cognitive bias is a systematic error in our thinking that occurs when we are trying to make sense of the world around us.  Our brains try to simplify all of the incoming information by taking a shortcut, if you will.  The shortcut might be the right one, but occasionally it is not, and we make an error.  There have been at least 180 different cognitive biases described in the literature.  

In my opinion, there are a couple of cognitive biases at play in the AT&T commercial.  First of all, we have the so-called "Illusion of Control" bias, the tendency for us to believe that we have more control over events than we really do.  We also have some elements of the "Self-Serving" bias, our tendency to take credit for for positive events or outcomes, coupled with a similar tendency to blame outside or external factors for negative ones.  Looking at this from a completely different angle, at its worst LeBron James is being far too arrogant and self-centered, and he is committing fraud by taking credit for someone else's work (in this case, Lily's).

Unfortunately, having a co-worker take credit for something that you did is far too common in the workplace, and an article in the Harvard Business Review offers some tips on how to handle this situation.  These situations can be infuriating, so the first tip is to take some time so that you can calm down and handle the situation calmly and professionally.  Next, assess the severity of the situation by asking yourself whether this was an intentional or accidental act.  Also, ask yourself whether it really matters long-term for your career to raise this issue.  Perhaps it's not as big of an issue as you think.  Admittedly, given the heavily matrixed and team-based organizations that we work in nowadays, it's often impossible to completely attribute credit solely to a specific individual or even a small group of individuals.  

If you still think it's important to address the issue (and if it happens more than once, you definitely should), first question why it happened.  Instead of making accusations, ask questions.  For example, you could start with "How did you feel the presentation went? Did you feel like you were able to hit all the main points?"  While this strategy may seem a little passive-aggressive, it will give your colleague an opportunity to recognize and potentially acknowledge his or her mistake.  If that doesn't occur, follow up with a statement and question such as, "I noticed that when you talked about the project you said “I” instead of “we.” Was that intentional? Why did you present it that way?" 

If the individual recognizes his or her mistake, talk about how the situation can be remedied.  Maybe the individual can send out a communication thanking you for your contributions to the success of the project.  If not, you should discuss the situation with your immediate supervisor, but frame it as a way to foster good teamwork rather than spreading blame or accusations.  

Finally, and perhaps most importantly, there are things that you can do to prevent this situation from happening in the future.  For example, prior to starting a project, sometimes it's good to discuss how credit will be allocated.  Who will present the project to senior leaders?  Who will field questions?  Who is the designated leader for the project (though that may not exist).  Make sure that all of these discussions are documented in writing, and be sure to take notes when you are working on the project so that you can quickly recall your contributions to its success.  Lastly, be sure to model good behavior by attributing credit where credit is due - if you are generous about sharing credit, studies show that others will reciprocate.

The last line in the AT&T commercial speaks volumes.  Lily asks LeBron if he is trying to steal her job.  Hopefully that will not occur.

Nothing but the best...

Thanks to the Apple TV series "Ted Lasso", I've started following the English Premier League.  I'm not quite sure how I chose "my team" (obviously, the fictional football team AFC Richmond was not a choice), but I've been following Everton FC, one of two professional teams from the city of Liverpool playing in the Premier League.  The Toffees (they are also known as "The Blues") had a difficult year last season and just barely escaped relegation.  Maybe that's why I started liking them - I suppose the fact that they are the favorite football team of Paul McCartney has something to do with it as well!

Everton FC was originally formed in 1878 and is one of the founding clubs for both the English Football League (1888) and the Premier League (1992).  The Toffees are the fourth most successful club in English professional football (behind their arch-rivals Liverpool, Manchester United, and Arsenal), having won nine league titles, five FA Cups and a European Cup Winners' Cup.  They've spent all but four seasons in the top division of the world's oldest professional football league - more than any other club, so it was quite a big deal that they barely escaped relegation last year.

The club's motto is "Nil Satis Nisi Optimum" which is Latin for "Nothing but the best is good enough."  I know I probably overdo it when it comes to using sports metaphors for leadership, but I can't think of a better motto for any organization than this one.  Winning is always great, but what should matter the most is that we always give it our best.  The 18th century Russian monarch Catherine the Great reportedly once said, "One does not always do the best there is.  One does the best one can."  Nothing else really matters.  If you are not giving something your best effort, maybe it’s time you ask yourself why not.