Tuesday, December 20, 2022

282: The Man With The Map

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Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school.   Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host.  And now, on to the math.

I was surprised to read about the recent passing of Maurice Karnaugh, a pioneering mathematician and engineer from the early days of computing.   Karnaugh originally earned his PhD from Yale, and went on to a distinguished career at Bell Labs and IBM, also becoming an IEEE Fellow in 1976.   My surprise came from the fact that he was still alive so recently:  he was born in 1924, and his key contributions were in the 1950s and 1960s, so I had assumed he died years ago.   In any case, to honor his memory, I thought it might be fun to look at one of his key contributions:  the Karnaugh Map, known to generations of engineering students as the K-map for short.

So, what is a K-map?   Basically, it’s a way of depicting the definition of a Boolean function, that is a function that takes a bunch of inputs and generates an output, with all inputs and the output being a Boolean value, that is either 0 or 1.    As you probably know, such functions are fundamental to the design of computer chips and related devices.   When trying to design an electronic circuit schematic that implements such a function, you usually want to try to find a minimum set of basic logic gates, primarily AND, OR, and NOT gates, that defines it.   

For example, suppose you have a function that takes 4 inputs, A, B, C, and D, and outputs a 1 only if both A and B are true, or both C and D are true.   You can basically implement this with 3 gates:   (A AND B) , (C AND D), and an OR gate to look at those two results, outputting a 1 if either succeeded.   But often when defining such a function, you’re initially given a truth table, a table that lists every possible combination of inputs and the resulting output.   With 4 variables, the truth table would have 2^4, or 16, rows, 7 of which show an output of 1.   A naive translation of such a truth table directly to a circuit would result in one or more gates for every row of the table, so by default you would generate a much larger circuit than necessary.   The cool thing about a K-map is that even though it’s mathematically trivial— it actually just rewrites the 2^n lines of the truth table in a 2^(n/2) x 2^(n^2) square format— it makes a major difference in enabling humans to draw efficient schematics.

So how did Karnaugh help here?   The key insight of the K-map is to define a different shape for the truth table, one that conveys the same information, but in a way that the human eye can easily find a near-minimal set of gates that would implement the desired circuit.   First, we make the table two-dimensional, by grouping half the variables for the rows, and half for the columns.   So there would be one row for AB = 00,  one row for AB = 01, etc, and a column for CD=00, another for CD=01, etc.   This doesn’t actually change the amount of information:  for each row in the original truth table, there is now a (row, column) pair, leading to a corresponding entry in the two-dimensional K-map.   Instead of 16 rows, we now have 4 rows and 4 columns, specifying the outputs for the same 16 input combinations.

The second clever trick is to order each set of rows and columns according to a Gray code— that is, an ordering such that each pair of inputs only differs from the previous pair in one bit.   So rather than the conventional numerical ordering of 00, 01, 10, 11, corresponding to our ordinary base-10 of 0, 1, 2, 3 in order, we sort the rows as 00, 01, 11, 10.   These are out of order, but the fact that only one bit is changing at a time makes the combinations more convenient to visually analyze.

One you have created this two-dimensional truth table with the gray code ordering, it has the very nice property that if you can spot rectangular patterns, they correspond to boolean expressions, or minterms, that enable an efficient representation of the function in terms of logic gates.  In our example, we would see that the row for AB=11 contains a 4x1 rectangle of 1s, and the column of CD=11 contains a 1x4 rectangle of 1s, leading us directly to the (A AND B) OR (C AND D) solution.    Of course, the details are a bit messy to convey in an audio podcast, but you can see more involved illustrations in online sources like the Wikipedia page in the show notes.   But the most important point is that in this two-dimensional truth table, you can now generate a minimal-gate representation by spotting rectangles of 1s, greatly enhancing the efficiency of your circuit designs.   

Over the years, as with many things in computer science, K-maps have faded in significance.  This is because the power of our electronic design software has grown exponentially:  these days, virtually nobody hand-draws a K-map to minimize a circuit.  Circuit synthesis software directly looks at high-level definitions like truth tables, and does a much better job at coming up with minimal gate implementations than any person could do by hand.   Some of the techniques used by this software relate to K-maps, but of course many more complex algorithms, most of which could not be effectively executed without a computer, have been developed in the intervening decades.   Despite this, Karnaugh’s contribution was a critical enabler in the early days of computer chip design, and the K-map is still remembered by generations of mathematics and computer science students.

And this has been your math mutation for today.



Sunday, October 30, 2022

281: Pascal Vs Mathematics

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Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school.   Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host.  And now, on to the math.

If you have enough interest in math to listen to this podcast, I’m pretty sure you’ll recognize the name of Blaise Pascal, the 17th-century French mathematician and physicist.   Among other achievements, he created Pascal’s Triangle, helped found probability theory, invented and manufactured the first major mechanical calculator, and made essential contributions to the development of fluid mechanics.   His name was eventually immortalized in the form of a computer language, a unit of pressure, a university in France, and an otter in the Animal Crossing videogame, among other things..   But did you know that in the final decade of his life, he essentially renounced the study of mathematics to concentrate on philosophy and theology?   

According to notes found after his death in 1662, Pascal had some kind of sudden religious experience in 1654, when he went into a trance for two hours, during which time he claims that God gave him some new insights on life.   At that point he dropped most of his friends and sold most of his possessions to give money to the poor, leaving himself barely able to afford food.   He also decided that comfort and happiness were immoral distractions, so started wearing an iron belt with interior spikes.   And, most disturbingly, he decided that math and science were no longer worthy of study, so he would devote all his time to religious philosophy.   He was still a very original and productive thinker, however, as in this period he wrote his great philosophical work known as the Pensees.   

There were a couple of reasons why he may have decided to give up on math and physics at this point.    Part of it was certainly just a change in emphasis:  he was concentrating on something else now, which he considered more important.   He also made comments about worldly studies being used to feed human egos, which means nothing in the eyes of God.   At one point he stated that he could barely remember what geometry was.

He never completely suppressed his earlier love of mathematics though.     Ironically, at several points in the Pensees he uses clearly mathematical ideas.    Most famously, you may have heard of “Pascal’s Wager”, where he discusses the expected returns of believing in God vs not believing, based on his ideas of probability theory.   You have two choices, to believe or not believe.   If you choose to believe, you may suffer a finite net loss from time spent on religion, if God doesn’t exist— but if he does, you have an infinite payoff.   Choosing not to believe offers at best the savings from that lifetime loss.   Thus, the rational choice to maximize your expected gain is to believe in and worship God.

As many have pointed out since then, there is at least one huge hole in Pascal’s argument:   what if you choose to believe in the wrong God, or worship him in the wrong way?   Many world religions consider heresy significantly worse than nonbelief.   He has an implicit assumption of Christianity, in the form he knows, being the only option other than agnosticism or atheism.   I think Homer Simpson once refuted Pascal’s Wager effectively, when trying to get out of going to church with his wife:  “But Marge, what if we chose the wrong religion? Each week we just make God madder and madder.”

Another surprising use of math in the Pensees is Pascal’s comments on why the study of math and science may be pointless in general.   He compares the finite knowledge that man may gain by these studies against the infinite knowledge of God:   “… what matters it that man should have a little more knowledge of the universe?  If he has it, he but gets a little higher.  Is he not always infinitely removed from the end…?   In comparison with these Infinites all finites are equal, and I see no reason for fixing our imagination on one more than on another.”   While he doesn’t write any equations here, the ideas clearly have a basis in his previous studies related to finite and infinite values.   We could even consider this a self-contradiction:  wouldn’t the fact that his math just gave him some theological insight mean that it was, in fact, worthy of study to get closer to God?

Pascal did also still engage a few times during this period in direct mathematical studies.   Most notably, in 1658, he started speculating on some properties of the cycloid, the curve traced by a point on a moving circle, to distract himself while suffering form a toothache.   When his ache got better, he took this as a sign from God to continue his work on this topic.   That excuse seems a bit thin to me:  clearly he never lost his inbuilt love for mathematics, even when he felt his theological speculations were pulling him in another direction.   

And this has been your math mutation for today.


Friday, August 26, 2022

280: Rubik's Resurgence

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Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school.   Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host.  And now, on to the math.

Do you remember the Rubik’s Cube, that 3x3x3 cube puzzle of multicolored squares, with each side able to rotate independently, that was a fad in the 1980s?   The goal was to take a cube that has been scrambled by someone else with a few rotations, and get it back to a configuration where all squares on each side match in color.   Originally developed by Hungarian architecture professor Erno Rubik in 1975, he had initially intended it to model how a structure can be designed with parts that move independently yet still hold together.   The legend is that after he demonstrated the independent motion of a few sides and had trouble rearranging it back to the original configuration, he realized he had an interesting puzzle.   In the early 1980s, it started winning various awards for best toy or puzzle, and quickly became the best selling toy of all time.   (A title which it apparently still holds.)   It was insanely difficult for the average person to solve, though typically with some trial and error you could get one or two sides done, the key to holding people’s interest.    To get a sense of how popular it was back then, there were cube-solving guidebooks that sold millions of copies, and even a Saturday morning cartoon series about a cube-shaped superhero.   But by 1983 or so sales were dropping off, and the fad was considered over.

Like everyone else who was alive in the early 80s, I spent some time messing around with cubes, but found it too frustrating after a while, and eventually solved it with the aid of a guide book.   I remember being impressed by a classmate who swore he hadn’t read any guidebooks, but could take a scrambled cube from me, go work on it in a corner of the room, and come back with it fully solved.   His secret was to remove the colored stickers from the squares and put them back in the right configuration, without rotating the sides at all.   But the non-cheating solutions in the guidebooks typically revolve around identifying sequences of moves that can move around known sets of squares while keeping others in their current configuration, then getting the desired cubes in place layer by layer.   These sequences are tricky in that they appear to be completely scrambling the cube before restoring various parts, which is a key reason why average cubers would fail to discover them— you need to mess up your cube on the way to completing the solution.    It’s actually been mathematically proven that any reachable configuration can be solved in 20 moves or fewer.

One aspect of the cube that is always fun to mock is its marketing campaign.   Typically the cube was sold and advertised with the phrase, “Over 3 Billion Combinations!”   But if you think about it for a minute, there are a lot more.   A cube has 8 corner pieces, so you could combine those in 8 factorial (8!) ways, which is 8 * 7 * 6 …  down to 1.   And since each of these combinations can have each corner piece in 3 different rotations, you need to multiply by 3 to the 8th power.   Similarly, the 12 side pieces can be arranged in 12 factorial (12!) ways, then you need to multiply by 2 to the 12th power.    So to find the total possibilities, we need to multiply 8! * 3^8 * 12! * 2^12.    It turns out that only 1/12 of these positions are actually reachable from a starting solved state, so we need to divide the result by 12.   But we still get a total a bit larger than 3 billion:  4.3 * 10^19.   So the marketing campaign was underestimating the possibilities by a ridiculous amount— in fact, if you square the 3 billion that they gave, you still don’t quite reach the true number of cube configurations.   Perhaps they were afraid that the terms for larger numbers, such as the quintillions needed for the true number, would confuse the average customer.

I was surprised to read recently that Rubik’s Cube actually has actually gained in popularity again in the modern era, fueled by daring YouTubers who speed-solve cubes, solve them with their feet, and perform similar feats.     Lots of enthusiasts take their cubing very seriously, and there are Rubik’s Cube speed-solving championship events held regularly.   If you’re a professional-class cuber, you can buy specially made Rubik’s lubricants to enable you to rotate your sides faster.  The latest championship, this year in Toronto, included standard cube solving plus events on varying size cubes (up to 7x7x7), blindfolded cube solving, and one-handed cube solving.   (Apparently they eliminated the foot-only solving, though, concerned that it was unsanitary.) Champion Matty Inaba fully solved a 3x3x3 cube in 4.27 seconds, which sounds like about the time it usually takes me to rotate a side or two.     Author A.J. Jacobs, in “The Puzzler”, also points out that if you’re too intimidated to compete yourself, there are Fantasy Cubing leagues, similar to Fantasy Football, where you can bet on your favorite combination of winners.

So, what does the future hold for the sport of Rubik’s Cubing?   Well, even though the big leagues are only competing up to the 7x7x7 level this year, Jacobs tracked down a French inventor who has put together a 33x33x33 one.   As you would guess, the larger cubes are pretty challenging to build— this one involved over 6000 moving parts and ended up the size of a medicine ball.   Experienced cubers do say, however, that the basic algorithms for solving are fundamentally the same for all cube sizes.   Due to the home manufacturing capabilities enabled by modern 3-D printing, one of Jacobs’ interviewees points out that we are in a “golden age of weird twisty puzzles”.    Hobbyists have invented many Rubik’s like variants that are not perfect cubes, and have numerous asymmetric parts, to create some extra challenge.   Personally I’m not sure I would ever have the patience to deal with anything beyond a basic 3x3x3 cube, though maybe I’ll look into joining a fantasy league sometime.  

And this has been your math mutation for today.


Saturday, July 2, 2022

279: Improbable Envelopes

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Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school.   Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host.  And now, on to the math.

Today we’re going to talk about a well-known paradox that a co-worker recently reminded me about, the Two Envelopes Paradox.   It’s similar to some others we have discussed in past episodes, such as the Monty Hall paradox, in that a slightly incorrect use of the laws of probability gives an apparent result that isn’t quite correct.

Here’s how the basic paradox goes.   You are shown two envelopes, and told that one contains twice as much money as the other one, with the choice of envelopes for each amount having been pre-determined by a secret coin flip.  No information is given as to the exact amount of money at stake.   You need to choose which envelope you want.   After you initially point at one, you are told the amount of money in it, and asked, “Would you like to switch to the other one?”   Since you have been presented no new information about which envelope has more money, it should be obvious that switching makes no difference at this point, as either way you have a 50% chance of having guessed the right one.

But let’s calculate the expected value of switching envelopes.   Say the envelope you chose contains D dollars.   If you stick with your current envelope, the expected amount you will get is simply D.  The other one contains either 2D or D/2 dollars, with 1/2 probability for each.   The expected value if you switch is then 1/2*2D + 1/2*D/2, which adds up to (5/4)D.    Thus your expected winnings if you switch are greater than the D you would gain from your first choice, and you should always switch!   But this doesn’t make much sense, if you had no additional information.   Can you spot the flaw in this reasoning?

The key is to recognize that you’re combining two different dollar amounts in your expected value calculation:  the D that exists in the case where you initially chose the smaller envelope is different from the D if you chose the larger one.   The easiest way to see this is if you define another variable, T, the total money in the combination of two envelopes.   In this case, the larger one contains (2/3)T, and the smaller has (1/3)T.   Now your expected winnings become the same as the expected value from switching, (1/2)*(2/3)T + (1/2)*(1/3)T, or simply T/2.  Alternatively, you could have come to the same conclusion by modifying our original reasoning using Bayes’ Theorem, replacing our reuse of D with correct calculations for the conditional values in each envelope.

But weirdly enough, brilliant.org seems to point out an odd strategy that will let you choose whether to switch with a greater than 50% chance of getting the higher envelope.    Here’s how it works.   After you choose your first envelope, choose a random value N using a normal probability distribution.   This is the common “bell curve”, with the important property that any number on the number line has a probability of being chosen, as the ‘tail’ of the bell asymptotically approaches and never reaches 0.   So if the center of the normal distribution is at 100, you have the highest probability of choosing a number near 100, but still a small chance of choosing 1000, and a really tiny chance of choosing 10 billion.   Therefore, if you’ve chosen a number N using this distribution, there is some probability, P, that N is between the dollar values in the two envelopes.   Now assume the envelope you didn’t choose contains that number N, and choose to switch on that basis:  if N is greater than the amount in the envelope you chose, you switch, and otherwise keep your original envelope.

Why does using this random number help?   Well, if your random number N was smaller than both envelopes, then you will always keep your first  choice, and there is no change to the overall 50-50 chance of winning.   If it was larger than both, you’ll always switch, and again no change.   But what if you got lucky, and N was between the two values, which we know can happen with nonzero probability P?   Then you will make the right choice:  you will switch if and only if your initial envelope was the smaller one!    Thus, with probability P, the chance of N being in this lucky interval, you will be guaranteed a win, while the rest of the time you have the original odds.   So the overall chance of winning is (1/2)*(1-P) + 1*P, which is slightly greater than 1/2.   

Something seems fishy about this reasoning, but I can’t spot an obvious error.   I remember also hearing about this solution from a professor back in grad school, and periodically tried searching the web for a refutation, but so far haven’t found one.   Of course, with no information about the actual value of P, this edge can be unpredictably small, so is probably of no real value in practical cases.    There also seems to be a philosophical challenge here:  How meaningful is an unpredictable bonus to your odds of an unknowable amount?   I’ll be interested to hear if any of you out there have some more insight into the problem, or pointers to further discussions of this bizarre probability trick.

And this has been your math mutation for today.



Sunday, May 22, 2022

278: Bicycle Repair Man

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With the arrival of summer weather in many parts of the U.S., it’s time to think again about various outdoor activities.    Watching a few bicycles pass by my house the other day brought to mind a famous anecdote about pioneering mathematician and computer scientist Alan Turing.   As you may recall, Turing was the famous British thinker who not only founded theoretical computer science, but also was the primarily visionary in the project to crack the German Enigma code, a key contribution to the Allied victory in World War II.   As often happens with such geniuses, his personal life was very odd, though he usually had reasons for whatever he did.   For example, he was mocked as ridiculously paranoid for chaining his favorite coffee mug to a radiator.  But there are rumors that years later, a pile of smashed coffee mugs was found near his old office, apparently thrown away by a disgruntled co-worker.    Another crazy story involves Turing’s efforts to repair his bicycle.

As the story goes, Turing noticed that every once in a while, the chain on his bike would come loose, and he would have to stop and kick it a certain way to get it back on its track again.   After a while, he noticed that the intervals at which this was happening seemed kind of regular, so decided to check that theory rigorously.   He attached a mechanical counter, and started measuring the exact interval at which this problem was occurring.    It turned out he was right— the intervals were regular.   The number of clicks between failures was proportional to the product of the number of spokes in his wheel, the number of links in the chain, and the cogs in the bicycle gears.   Once he discovered that, he took a close look at the components, and soon discovered the root cause:   there was a particular spoke that was slightly bent, and when this got too close to a particular link in the chain that was also damaged, the chain would be pulled off.   Armed with this knowledge, he was able to correctly fix the bent spoke and resolve the problem.    It’s been said that any competent bike repairman would have spotted the issue in a few minutes without bothering with counters and intervals, but the mathematics of that would be pretty boring.

Naturally, as with any anecdote about someone famous, there are some alternate versions of this story.   My favorite is the one that changes the ending slightly:  once Turing figured out the formula for when the chain would jump off, he started carefully calculating the intervals as he rode the bike, and stopping to kick it at the exact right times he calculated.   That’s a fun one, and certainly fits into the stereotypes of Turing’s eccentricity.   But I do find it a bit hard to believe.  When riding a bike outdoors, there are lots of variables involved to interrupt your concentration:  road obstacles, changing inclines, approaching cars, etc.   Could someone safely riding a bicycle successfully keep a running count of the wheel and chain rotations, over a continuous ride of several miles?   And in Turing’s case, it was further complicated by the fact that he always wore a gas mask as he rode, to prevent triggering his allergies.   But the alarm clock he was known to wear around his waist might have helped.

In honor of this story, there was actually a proposal back in 2015 to name a bicycle bridge in Cambridge, England after Turing.   I didn’t see any further references to this online, so it looks like it didn’t pass.   But there’s plenty of non-bicycle-related stuff named after him in the computer science world.    If you want someone to propose a bicycle bridge in your name, next time your bike breaks down, think about the clever mathematical tricks you might use to diagnose the issue.   Also, remember to found a branch of mathematics and win a world war.    Or forget about the bridge, and just take your bike to a competent repair shop.

And this has been your math mutation for today.


Wednesday, March 30, 2022

277: Bad Career Advice

 Audio Link

Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school.   Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host.  And now, on to the math.

As I mentioned in the last episode, I’ve recently reread Nassim Nicholas Taleb’s  classic book “The Black Swan” , about the disproportionate role of unlikely extreme events, the Black Swans, in shaping our lives and our history.    Today I’d like to discuss another of the intriguing ideas he discusses in the book:  scalable and unscalable jobs, and which you should choose if starting out your career.   

Back when he was in college, Taleb received some advice from a business student:  choose a scalable, rather than a nonscalable, profession in order to become rich.    It’s a pretty simple concept:  a scalable job is one where you are paid for ideas, not hourly labor, and thus can affect many people with a small amount of work.   This contrasts with nonscalable jobs, where you are directly paid for the labor you perform.    An example of a scalable job is a corporate CEO, a derivatives trader, or an author:   in any of these professions, a small amount of work can impact a massive number of people.    On the opposite end of the spectrum, you can think of cases like a dentist or a chef:   your services are inherently delivered one-on-one, and your output is essentially dependent on the time you spend.     Some like to refer to scalable professions as “idea” professions, and nonscalable ones as “labor” professions.

You can see why scalable work has the potential to earn much more money, since it’s a matter of simple math.   If your actions affect one or a small number of people, there is fundamentally less money to go around, since whatever you earn must ultimately come from the people you serve.    If your actions affect millions of people, then there is the capacity to draw in money from many directions; as Taleb phrases it, this can “add zeroes to your output… at little or no extra effort.”   An idea person does the same amount of work no matter how many people it affects.   The CEO gives his orders once, the derivatives trader presses the same button regardless of the quantity traded, and the author writes the book once.     Taleb also glosses over the fact that some professions are kind of in-between:  for example, a chip design engineer at Intel impacts many millions of customers, though that impact is shared with about 100,000 other co-workers, creating a pretty good income potential, though unlikely to make someone super-rich unless they progress high on the even-more-scalable management ladder.  

So, was the advice correct, to choose a scalable rather than a nonscalable profession?   Well, it is true that if you go around looking at super-rich people, almost all did it through being in the scalable world.  But be careful:   if you make this kind of observation, you are making a fundamental logical fallacy, confusing “A implies B” for “B implies A”.    If someone is rich, they probably got that way through a scalable profession— but does that mean that if an arbitrary person chooses a scalable profession, they are likely to become rich?   The answer to that is a definite NO.   For every J.K Rowling or Nassim Nicholas Taleb, there are thousands of aspiring authors who barely sell a copy of their book.   (I won’t comment on how the Math Mutation book fits into this discussion.)   The scalable professions are massively profitable for the small number of people who are successful, but what’s a lot less visible are the corresponding masses of unsuccessful aspirants to these careers who failed miserably.     

Thus, Taleb points out that the advice he got was not very good, even though he happened to follow it and succeed himself.    In a nonscalable profession, the average worker makes a decent living— in some, like skilled tradesmen, engineers, or doctors, they can be pretty sure of heading towards the upper middle class if they do a good job, even though they are not likely to become rich.   Choosing a scalable profession is like entering a lottery, while nonscalable or mid-range jobs give you much better odds of earning a solid living.   And of course, there is usually the possibility of later moving into management of whatever career path you’re on, if you decide later that you want to gamble on the scalable route, while having a solid profession to fall back on if you don’t win that lottery.  

Another interesting point Taleb makes is that the nature of some professions changes over time.   If you look to the 19th century or earlier, being a singer was a nonscalable, labor-type position:  you had to be physically present before a relatively small audience, and repeat that activity every time you wanted to earn money for your music.     Thus many singers across the world could make a living producing music for eager audiences.   Then came the 20th-century revolution in recorded sound.   Once that happened, you could see superstars like Elvis, Pavarotti, or the Beatles become household names— and the majority of the money that in previous years would have been distributed among thousands of local performers ended up in their pockets.   For those who think this situation was unfair to the local singers, think about the effect of the printing press on monks, or of the invention of the alphabet on traveling storytellers.     The truth is, making a job more scalable almost always benefits huge numbers of people who consume the goods or services being produced, while creating an inconvenience for those who had previously profited off the nonscalability of their jobs.

Another interesting point Taleb makes is to look at this concept at a national level.  In the case of the United States, it seems to have been much more economically successful in the past century than the intellectual European nations of, as Taleb puts it, “museumgoers and equation solvers”.    He attributes this to the US’s much greater tolerance for creative tinkering and trial-and-error, which result in the development of new concepts and ideas.   Because the economic benefits of concepts and ideas are scalable, this has resulted in a large multiplier on the money that can be made by US companies in general.    The much-lamented loss of US manufacturing jobs is just a reflection of this shift of focus.   Nike can design a shoe, or Boeing can design an airplane, with a relatively small number of ideas, and subcontract the grunt work to foreign companies.   

I wish I could say that I pondered all these insights when initially setting out on my career, but like most of us, I just blundered my way around until I settled into something that seemed good.   It seems to have worked out pretty well for me.    But if you’re at an earlier stage of your life, Taleb’s ideas are worth strong consideration. 

And this has been your math mutation for today.



Sunday, February 20, 2022

276: Don't Believe the Math

Audio Link

Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school.   Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host.  And now, on to the math.

The last episode’s discussion of randomness brought to mind the classic book “The Black Swan” by economist-philosopher Nassim Nicholas Taleb.    His books discuss the disproportionate role of unlikely extreme events, the Black Swans, in shaping our lives and our history.    Noticing online that there is a 2nd edition now, I decided to reread Taleb’s book, and got many intriguing new ideas for podcast episodes.   Today we will talk about the “Ludic Fallacy”, the incorrect use of mathematical models and games to predict real-life events.   To understand this better, let’s look at one of his key examples.

Suppose we do an experiment with two observers in a room, a professor and a gambler.   We present them the following mathematical puzzle:   I have a fair coin that I plan to flip 100 times, with everyone watching.   The first 99 flips are all heads.   The two observers are asked to estimate the probability that the next flip will turn up heads.    The professor confidently answers, “Since you said it’s a fair coin, previous flips have no influence on future flips.   So the chance is the same as always, exactly 50%.”   On the other hand, the gambler answers, with equal confidence, “If you got 99 heads, I’m almost certain that the coin is biased in some way, regardless of whether you said it’s fair.   So I’ll estimate a 99% chance that the next flip is heads.”    Naturally, in a purely mathematical sense, the professor was right, according to the information we provided.   But if this were a real-life situation, and you had to bet money on the outcome of the next flip, which answer would you go with?   The gambler probably has a point.

And this is Taleb’s key insight that forms the Ludic Fallacy:   while abstract mathematical models may provide some insight into possibilities, you cannot consider them reliable models of real life.     Issues or events that are outside your simple model may have a huge effect.   Taleb criticizes a lot of professionals who spend their lives creating complex mathematical models, and claim that they deserve large salaries or become media darlings for using them to make intricate predictions about the future, which then turn out to have little more accuracy than random chance.   Economists are some of the most notorious in this regard.    You may recall that back in the 1990s, a large hedge fund called Long Term Capital Management (or LTCM) was built around some insights from supposedly genius economists who had Nobel Prizes.   But when its “mathematically proven” strategy led to buying massive numbers of Russian bonds with borrowed money, which then defaulted, LTCM failed so badly that it needed a multi-billion dollar bailout to avoid crashing the world economy.      

There are plenty of other examples like this, and it’s not just experts who fall for this kind of fallacy.   Taleb is a bit critical of the modern software, such as features in Microsoft’s Excel spreadsheets, that make it easy for even ordinary workers to mathematically extend existing data into future extrapolations, which are very rarely accurate in the face of unpredictable real-life events.   In effect, computers allow anyone to transform themselves into an incompetent economist with high self-confidence.

I think my favorite example that Taleb cites is the story of a casino he consulted with in Las Vegas.  They had very meticulously modeled all the ways that a gambler could cheat, or that low-probability events in the games might threaten their cash flow, and had invested massive amounts of money in gambling theory, security, high-tech surveillance, and insurance to guard against these events.   So what did the four largest money-losing incidents in their casino turn out to be?    

1. The irreplaceable loss of their headline performer when he was maimed by one of this trained tigers.

2. A disgruntled worker, who had been injured on the job, attempted to blow up the casino.

3. An incompetent employee had been putting some required IRS forms in a drawer and failing to send them in, resulting in massive fines.

4. The owner’s daughter was kidnapped, and he illegally took money from the casino in order to ransom  her.

Now of course, it would have been very hard for any of these to be predicted by the models the casino was using.   That’s Taleb’s point:  no mathematical modeling could cover every conceivable low-probability event.

This is also an important reason why Taleb opposes centrally-planned economies.   One of the few Nobel-winning economists who Taleb respects is F.A.Hayek, whose 1974 Nobel speech offered a harsh critique of his fellow economists who fall back on math due to their physics envy,  and try to claim that their equations model the world just like the hard sciences.  No matter how many measurable elements they factor into their equations, the real world is much too complicated to model accurately and make exact predictions.   Modern free economies are largely successful because millions of individuals make small-scale decisions based on local information, and are free to take educated risks with occasional huge payoffs for society in general.   In his conclusion Hayek wrote, “The recognition of the insuperable limits to his knowledge ought indeed to teach the student of society a lesson of humility which should guard him against becoming an accomplice in men’s fatal striving to control society – a striving which makes him not only a tyrant over his fellows, but which may well make him the destroyer of a civilization which no brain has designed but which has grown from the free efforts of millions of individuals.”

We should mention, however, that Taleb and Hayek are not showing that mathematical models are totally useless— we just need to recognize their limitations.   They can be powerful in finding possibilities of what might happen, and opening our eyes to potential consequences of our basic assumptions.   For example, let’s look again at the coin flipping case.    Suppose instead of 99 heads, our example had shown a variety of results, including a run of 5 heads in a row somewhere in the middle.   The gambler might spot that and initially have a gut feeling that this is an indication of bias.   But the professor could then walk him through a calculation, based on the ideal fair coin, that if you flip a coin 100 times, there is over an 80% chance of seeing a run of length 5 at some point.   So using the insight from his modeling, the gambler can determine that this run is not evidence of bias, and make a more educated guess, considering that the initial promise of a fair coin has not yet been proven false.    Remember, however, that the gambler still needs to consider the possibilities of external factors that are not covered by the modeling— maybe as he is making his final bet, that disgruntled employee will return to the casino with an angry tiger.

So, in short, you can continue to use mathematical models to gain limited insight, but they are not confident sources for practical predictions.   Don’t get overconfident and fool yourself into making big bets that your model will guarantee discovery of all real-life risks.

And this has been your math mutation for today.