Last year, Professor Steven Strogatz of Cornell University wrote a series of op-eds for the New York Times that discussed the presence of mathematics in unlikely places. I discussed one of these columns here. Now, either those articles were well-received, or Professor Strogatz is well-connected, because this year he’s back in the Times with a much more ambitious series of articles. This time around, Strogatz is attempting to “[write] about the elements of mathematics, from preschool to grad school, for anyone out there who’d like to have a second chance at the subject.”
Preschool to grad school is a significant amount of ground to cover, but thus far Strogatz has used his articles to assault this goal with gusto. To date, he has tackled counting, patterns in addition, negative numbers, division, and basic high school algebra. This doesn’t really do justice to his content, though. Along the way he gives the reader some Sesame Street, and discusses a number of tangential topics, including the inability of Verizon employees to do math, the half-your-age-plus-seven rule, and pre-WWI European history. The latter comes about in a discussion of that old adage which is familiar to anyone who saw the first Alien vs. Predator movie: the enemy of my enemy is my friend.
Predators must be awesome at math.
While some of Professor Strogatz’s explanations are a bit hand wavy (in particular, his explanation of why (-1) x (-1) = 1 is a lacking), on the whole they are quite good. In particular, he offers a nice explanation of what it is for a mathematical argument to be “elegant.” But even more impressive than his writing is its location – to have a discussion of mathematics with as wide an audience as the New York Times readership is commendable. Even if people are not inspired to learn more mathematics after reading these pieces, hopefully they will have at least learned something. As with exercise, a little mathematics is better than no mathematics at all.
Moreover, these articles highlight aspects of math not usually seen in popular discourse. Much like Paul Lockhart’s A Mathematician’s Lament (which Strogatz references), these snack-size essays are focused on simple mathematical ideas, and the beautiful (and sometimes unexpected) results that follow. Nowhere here does Professor Strogatz multiply two really big numbers together; in fact, he’s quite sympathetic to the fact that for many people, there is nothing more tedious than calculation. By leading the conversation in this way, he’s hopefully able to give a taste of what makes math beautiful to an audience for whom such a statement might otherwise be labeled heresy.
I don’t know where this series of articles is headed, but I look forward to finding out, and hope you do to. Professor Strogatz’s articles are grouped together here.
(Hat tip to dad for sending me a few of these articles.)
Some time ago, I wrote an article on the optimal way to select a mate, assuming you know how many eligible partners exist, and that once you’ve dated someone, you can’t go back and date them again (sorry, Drew Barrymore and that dude from the Apple commercials). This is less romantically known as the secretary problem. Let me briefly recall the problem and its solution: suppose you have n candidates, from which you want to pick the best one. This applies to a variety of situations, from hiring a secretary to finding a girlfriend to apartment hunting. In either case, the outcome is the same: you should look at roughly the first n/e of them (yes, thate), and then select the first one after those n/e which is better than all that you have seen so far. While this strategy won’t guarantee you get the best choice, it will give you the best choice around 37% of the time.
The major problem with this model is that in many situations, the value of n is unknown. There are ways to circumvent this problem, which I will not discuss here. Instead, in the context of finding a mate, I offer the following method to calculate the number of partners you could reasonably expect to find in your area. This method recently gained some attention when Peter Backus, a Ph.D. candidate from the University of Warwick wrote a paper titled “Why I Don’t Have a Girlfriend.”
The basic technique involves modifying the Drake Equation, an equation used to estimate the number of potential extraterrestrial civilizations in our galaxy. For those who have never been introduced to this equation, it asserts the following:
N = R · fp · ne · fl · fi · fc · L.
According to Wikipedia, these variables represent the following quantities:
R = the average rate of star formation per year in our galaxy, fp = the fraction of those stars that have planets, ne = the average number of planets that can potentially support life per star that has planets, fℓ = the fraction of the above that actually go on to develop life at some point, fi = the fraction of the above that actually go on to develop intelligent life, fc = the fraction of civilizations that develop a technology that releases detectable signs of their existence into space, L = the length of time such civilizations release detectable signals into space.
Given estimates for all of these parameters, one could then estimate the number of civilizations in our galaxy. Since we don’t know the values of any of these parameters, however, this is more of a thought experiment than anything else.
Nevertheless, the idea can be easily modified to try and find the number of eligible mates in a given area. Peter Backus’ approach is fairly specific to him, but he links to a more general approach discussed here, in which the following equation is presented:
n = P · ft · fo · fc · A · R
In this case, the parameters are given by
P = the population. This could be the population of your university, your city, or your country, depending on how ambitious you are. ft = the fraction of that population which you would want to mate with, in broad terms. If you’re a straight male, this would be the fraction of females. If you’re a gay male, it would be the fraction of males, and so on. fo = the fraction of the population you’d want to mate with which wants to mate with you. For example, if you’re a straight male who wants to mate with females, this will compensate for the fact that some females will be lesbian and therefore unwilling to mate with you. fc = author Raymond Francis labels this the out fraction, and describes it as the answer to the question “Of the people in your target gender and orientation, how many of them are open enough about their sexuality to engage in a relationship of the sort you’re hoping for?” If you are straight, this value is likely 1, or very close to it. If not, things can be a little bit fuzzier. A = the fraction of those remaining who fall within your desired age range. This is, of course, personal to you – if you’d like a socially acceptable age range, you could follow the “half your age plus seven” rule. R = factors for any remaining filters you wish. Do you want your partner to have a certain level of education, or a certain income? Do you need a non-smoker, or demand a euphonium player? Here’s where you can fold that into the mix.
Yes, Wikipedia has a graph illustrating the half your age plus seven rule. Amazing.
With all these parameters accounted for, N will give you the number of potential mates in your area.
Let’s take this equation for a spin, shall we? Suppose you are a straight male living in Los Angeles, and looking for a girl to date in Los Angeles. According to Wikipedia, the estimated population of LA as of 2008 was 3,833,995. Of those, let’s say that 51% are female, and of the females, let’s posit that 90% are straight or bisexual. fc should be high in this case – to be conservative, let’s put it at 95%.
To estimate the age filter, one can obtain some data from this site. Suppose you are a 25 year old man – then, absent any personal preference, the socially acceptable age range of women for you to date is between 19.5 and 36. According to census data, in 2000 there were 3,694,820 people in Los Angeles, and of them, 974,004 were between the ages of 20 and 34. Additionally, there were 251,632 people between the ages of 15 and 19, and 584,036 people between the ages of 35 and 44. If we make the assumption that ages are roughly uniformly distributed within these brackets, this gives us an additionlal 141,970 people either between the ages of 19.5 and 20, or between the ages of 35 and 36. Combining this gives a total of 1,115,974 people between the ages of 19.5 and 36 in Los Angeles in 2000, or roughly 30% of the population. Let’s use this for our value A.
Assuming you have no other restrictions (i.e. taking R = 1), this gives us n = 3,833,995 · .51 · .9 · .95 · .3 = 501,544. That’s a lot of ladies out there for the taking. Of course, taking R = 1 is probably unrealistic. It’s unlikely you want to date women who are married, for example, and everyone has their own personal taste that will decrease the pool even further. Once you’ve calculated your personal value for R, however, you then know how many eligible mates will be in your area. Given that, you’ll know how large n/e is, and then you’ll know how many people you should date before you think about settling down.
Although Peter Backus has received a fair amount of buzz for the short paper he has written on this idea, he readily admits that he is not the first to think of applying Drake’s equation in this situation. I’ve discussed mostly Raymond Francis’ approach here, but Backus has links to many other people that have discussed the idea on his website. In particular, here’s an exchange from CBS’s The Big Bang Theory (don’t worry, there’s a laugh track so you know when things are supposed to be funny).
In summary, not only can the Drake Equation be used to consider the existence of extraterrestrial life, it can also be used to consider potential mates right here on Earth. The next step, of course, is obvious: we must combine these two equations to calculate the number of potential extraterrestrial mates. Undoubtedly the number will be small, but one should never underestimate the power of love.
Late last year, a study was published in Proceedings of the National Academy of Sciences which tried to pin down origins for the gender gap in mathematics education. As I’ve discussed before, the gender gap in math education is shrinking, and has been shown to be less about biology and more about culture – in cultures where gender equality is weaker, the gender gap is stronger. Nevertheless, even in American culture, the gender gap still persists, and this study by Sian Beilock and others has tried to figure out how, if the gender gap is culturally based, it comes about in young students. The original study can be found here, while a discussion of the study that was featured in the news can be found here.
Professor Beilock and her colleagues tried to correlate young students’ math anxiety with the math anxiety of their teachers. In particular, they looked at 1st and 2nd grade students, of whom a vast majority (over 90%) have teachers who are female. The study assessed the math anxiety of the teachers and measured the math achievement of the students at the beginning and end of the year. Here are the results, taken from the introduction to the paper:
There was no relation between a teacher’s math anxiety and her students’ math achievement at the beginning of the school year. By the school year’s end, however, the more anxious teachers were about math, the more likely girls (but not boys) were to endorse the commonly held stereotype that “boys are good at math, and girls are good at reading” and the lower these girls’ math achievement. Indeed, by the end of the school year, girls who endorsed this stereotype had significantly worse math achievement than girls who did not and than boys overall.
These findings make intuitive sense, and lend further support for the need to better our mathematics education at all levels, or at the very least require primary educators to study mathematics more seriously. Teaching mathematics with confidence is not something that comes automatically, even for those who may have been good at math in their early years.
It’s interesting that boys weren’t more likely to endorse the view that boys are good at math and girls are good at reading if their teacher had math anxiety. I’d be curious to see what the case is in a classroom led by a male teacher, both with and without math anxiety. Given the dearth of male primary educators, however, this type of data may be harder to acquire. In any event, the lesson here is clear: if you want your daughter to not fear math, it wouldn’t hurt to demand that her teachers not fear it either. Or at the very least, demand that any math fear be exhibited only by male teachers. That may be a cheaper solution.
I’d also be interested in knowing whether this trend can be reversed by a suitably competent teacher. If a group of 2nd grade girls is taught math by a woman who is unqualified, but in 6th grade is taught by a woman who is exceptional, can this help undo the damage that the 2nd grade teacher has done? I would hope so.
In the aftermath of the Super Bowl, some of you fans may be dreading the next six months. To kick off this football drought, I’d like to highlight this article, which was featured on Yahoo yesterday. The article says that Saints quarterback Drew Brees should hope to lose the coin toss at the start of the game, because in the past 43 Super Bowls, the team that won the coin toss had only won 20 times.
An unlucky coin? Unlikely.
Um…what? Who cares? While 20/43 is slightly less than the expected 50%, this difference is not even close to being statistically significant. Actually, the fact that this ratio is only 1 1/2 games shy of the mean is pretty good. Matt Springer has posted an article that discusses why we shouldn’t really care about this difference.
Of course, the sample size is naturally restricted by the small number of Super Bowls, but if the author (Mark Pesavento) had really been interested in the question of whether or not the coin toss is correlated with the winner in a football game, he could’ve easily collected data over a couple of seasons and obtained an answer to the question. At the very least, he could’ve owned up to the fact that his analysis is worthless, but instead, to the critics he offers only the following rebuttal: “because of the small sample size, some statisticians argue that the win-loss record of coin-toss winners is statistically insignificant.”
This is completely disingenuous, because it suggests that there would be a debate among statisticians about the significance in the data Pesavento uses, when no such debate exists. Anyone with even a rudimentary background in statistics would understand that the sample size here would be too small to draw the conclusion he draws.
Moreover, Pesavento falls for one of the most common traps in statistics: mistaking correlation for causation. Even if the data was much stronger in indicating that the coin toss winner is at a disadvantage, this would not imply that Brees should hope to lose the toss. A correlation between these two effects does not imply a causal relationship between the two. I feel like I’ve discussed this before, but just in case, here’s a thorough discussion of this misconception.
Here this point is moot, since we don’t even have a correlation. I thought no one would need to point out that “No correlation does not imply causation,” but apparently we do.
Thankfully, most of the comments on Pesavento’s post are scathing in regards to his methods. But that’s cold comfort in light of the fact that the article was deemed fit for posting on the front page of Yahoo.
Big ups to Liz Landau for bringing attention to one of the most important unsolved math problems of our time, the Riemann Hypothesis. Over at the CNN SciTechBlog, she has written a nice article on the problem aimed at a general audience.
This year marks the 150th anniversary of the publication of Riemann’s manuscript, where he proposed the now famous conjecture on the zeros of the Riemann-zeta function, and November was the month in which it was published. However, as Landau points out, the exact date of publication isn’t known, which makes having a birthday celebration a little tricky. The American Institute of Mathematics picked today to celebrate, and in honor of Riemann talks were held all around the world.
The Riemann Hypothesis has held the attention of the mathematical community for a century and a half, but it’s also made occasional forays into the realm of popular culture. For starters, there are quite a few books on the conjecture that are aimed for a general audience (one of the more recent ones is by Dan Rockmore). It has also made cameos in television shows such as Law and Order and Numb3rs. The Numb3rs episode is particularly notable because in it, the Riemann Hypothesis is solved by none other than Neil Patrick Harris. While he may not be a major contender to solve the problem in real life, it may be a bit too early to make a final verdict.
This is what NPH looks like when he thinks about mathematics.
While the Riemann Hypothesis may not be as simple to state to a general audience as, say, the statement of Fermat’s Last Theorem, there is ample room for its inclusion in popular discourse. This is important not just because the conjecture excites mathematicians, but also because it shows that the work mathematicians do involves more than multiplying large numbers and being awkward in social situations.
In an ideal world, we would celebrate the Riemann Hypothesis every November 18th, just as we celebrate Pi Day on March 14th. The difference would be that 11/18 would commemorate something substantive within mathematics. For that reason, the idea probably won’t take hold. But at least a man can dream. Thanks again, Liz, for doing your part!
Last month marked the release of Superfreakonomics, a sequel by economist Steven Levitt and journalist Stephen Dubner to the 2005 bestseller Freakonomics. The fanfare surrounding this prefix-enhanced release has been marred, however, by controversy surrounding a chapter on global warming. Starting with this entry on ClimateProgress.org, the debate has drawn a fewresponses on the Freakonomics blog, but nothing has seemed to blunt the allegations that Dubner and Levitt wrote the chapter from a contrarian perspective without understanding even the fundamental principles of climate science, and as a result, what they’ve written is garbage.
Much of the writing back and forth has been quite heated, and being a student of mathematics I am averse to conflict. However, one response resonated with me a great deal, and as a case study of the arguments that can be made using only simple calculations, it’s quite effective. The response in question comes from RealClimate.org, and is titled “An Open Letter to Steve Levitt.”
Written by fellow University of Chicago Professor Raymond T. Pierrehumbert, the letter takes Steve Levitt to task by harnessing the power of mathematics. After some opening remarks, Pierrehumbert sets the stage in the following way:
By now there have been many detailed dissections of everything that is wrong with the treatment of climate in Superfreakonomics , but what has been lost amidst all that extensive discussion is how really simple it would have been to get this stuff right. The problem wasn’t necessarily that you talked to the wrong experts or talked to too few of them. The problem was that you failed to do the most elementary thinking needed to see if what they were saying (or what you thought they were saying) in fact made any sense. If you were stupid, it wouldn’t be so bad to have messed up such elementary reasoning, but I don’t by any means think you are stupid. That makes the failure to do the thinking all the more disappointing. I will take Nathan Myhrvold’s claim about solar cells, which you quoted prominently in your book, as an example.
Myhrvold’s claim in this context is essentially that using solar cells to fight global warming is not a good idea, because solar cells must be dark in order to absorb solar energy. However, only a fraction of that energy is converted into electricity, while the rest simply becomes waste heat that in turn will heat up the atmosphere.
Using nothing more than simple arithmetic, Pierrehumbert then tries to reason his way through such an argument to see if it makes any sense. As expected, it does not. My favorite part of the argument is the graphic that shows how many solar panels would be required to supply the world’s electricity:
That black square in Saudi Arabia certainly doesn’t look like it should make any significant contribution to the planet’s heating, and indeed, Pierrehumbert uses mathematics to argue quite effectively that it wouldn’t.
The letter is worth a read, not just for the strength of Pierrehumber’s argument, but for the simple mathematics that gives his argument such strong support. Levitt offered a meek response in the comments (#47, I believe), which was then quickly rebutted. Since then, all’s been quiet.
Of course, Levitt and Dubner may think that the math is on their side – since I haven’t read the chapter in question, I can’t comment. But the arguments put forth by Pierrehumbert are quite compelling, due in no small part to the simple calculations he performs. There’s no doubt that mathematics can be used both for good and for evil, but Pierrehumbert, like Spider Man, seems to understand that with great power comes great responsibility.
As you may have heard, last week Martin Gardner celebrated his 95th birthday. Gardner, who authored the “Mathematical Games” column in Scientific American for a quarter of a century, is often credited for introducing generations of young students to the beauty and charm inherent in mathematics. My favorite quote in this vein comes from professor Ron Graham, who is quoted in a recent New York Times article on Gardner as saying that “Martin has turned thousands of children into mathematicians, and thousands of mathematicians into children.”
A warm brain is the key to mathematical dexterity.
Both Scientific American and Wired ran articles on Gardner last week, and each one used a different expression to represent his age. Scientific American congratulated him on reaching an age of 25 x 3 – 1, while Wired proclaimed that Gardner had turned 5! – 25. Upon reflection I think I prefer the latter expression over the former, since the exclamation point in the factorial makes his birthday feel just a little more exciting. No matter your preference, however, I hope this encourages others to write their ages using mathematical expressions – more complicated expressions would be especially useful for ladies who don’t wish to reveal their age.
In any event, I’d like to take the opportunity to celebrate Martin Gardner’s birthday in my own way, by discussing a problem credited to him. There are a few variations on this problem, but all fall under the general name of the Truel. I first encountered this problem during a job interview, and the variant I heard then is the variant I’ll discuss now.
Suppose there are three men: Mr. White, Mr. Gray, and Mr. Black. These three men have a score to settle, and so they decide to engage in a three way duel (or a “Truel,” if you want to be cute). Mr. White is a rather poor marksman who hits his target only 1/3 of the time, In contrast, Mr. Gray is successful 2/3 of the time, and Mr. Black is an expert marksman who never misses. Because of this disparity, they agree that Mr. White shall shoot first, followed by Mr. Gray, followed by Mr. Black, and this sequential rotation shall continue until only one man remains standing.
Setup of the truel.
Although Mr. White isn’t so good with a gun, he is an expert strategist, and surmises that Mr. Black will dispose of Mr. Gray first, since Mr. Gray poses more of a threat than Mr. White. By the same reasoning, Mr. White assumes that Mr. Gray will shoot at Mr. Black before shooting at Mr. White.
Assuming that this analysis holds true, what should Mr. White do on his first turn to maximize the chance that he’ll win?
I’ll discuss the answer below, but you should try this out for yourself if you’ve never seen this problem before. So that you don’t peek at the answer accidentally, let’s pause for a moment and look at some drawings courtesy of M.C. Escher.
Relativity
Drawing Hands
Circle Limit III
So, what should Mr. White do? Should he shoot first at Mr. Gray or at Mr. Black?
Phrased this way, it’s a bit of a trick question, since the answer is that Mr. White shouldn’t shoot at either one. Instead, he should intentionally miss!
Intuitively, this seems reasonable if you reflect on it. The reason is that it’s too risky for Mr. White to shoot at either of his opponents in the first round, because if he actually hits one of them, the remaining opponent will immediately turn and shoot at Mr. White. For example, if Mr. White shoots and hits Mr. Grey, he’s done for, since Mr. Black will shoot at Mr. White and is guaranteed to hit. Similarly, if Mr. White hits Mr. Black, Mr. Grey will turn to shoot Mr. White. If, however, Mr. White shoots neither one, then either Mr. Black or Mr. Grey will be eliminated by the other one, leaving Mr. White with only one opponent to worry about.
For those looking for a more quantitative argument, we can use the odds given in the problem to calculate the probability that Mr. White will win if he a) shoots first at Mr. Black, b) shoots first at Mr. Gray, or c) shoots first at neither one.
As part of the analysis, we’ll need to determine the odds that Mr. White will win when he’s paired one-on-one against either one of his adversaries. For Mr. Black this is easy: if Mr. White faces off against Mr. Black, he has a 1/3 chance of winning, since if he misses Mr. Black will win with certainty.
In a face-off between Mr. White and Mr. Gray, however, things are more complicated. This is because there are many ways for Mr. White to win. He could win by hitting Mr. Gray on his first shot, he could win if both men miss and he hits Mr. Gray on his second shot, he could win if both men miss twice and he hits Mr. Gray on his third shot, and so on. In general, since the probability that both men will miss in a given round is 2/3 x 1/3 = 2/9, the probability that Mr. White wins in the first round is 1/3, the probability that he wins in the second round is (2/9) x 1/3, the probability that he wins in the third round is (2/9)2 x 1/3, and so on.
In general, we find that the probability Mr. White wins when facing Mr. Gray is 1/3 x (1 + 2/9 + (2/9)2 + (2/9)3 + … ). The value of the geometric series is 9/7, so the probability is just 1/3 x 9/7 = 3/7. That is, when facing off against Mr. Gray directly, Mr. White has a 3/7 chance of winning (assuming Mr. White shoots first).
How are these calculations relevant to the matter at hand? Well, we can analyze the outcome of the truel depending on Mr. White’s initial action by using some tree diagrams. Let’s first see what happens in the simplest case when Mr. White intentionally misses. In this case, Mr. Gray will shoot, and either he will hit Mr. Black or he will miss: Here, GHB and GMB stand for “Gray hits Black” and “Gray Misses Black,” respectively.
We know that if Gray hits Black, we will now be in a face-off between Gray and White. Similarly, if Gray misses Black, Black will in turn kill Gray, putting us in a face-off between Black and White. Since we calculated the probability of White winning in either one of these face-offs, and since Gray has a 2/3 chance of hitting Black, this diagram shows that the probability White will win must be 2/3 x 3/7 + 1/3 x 1/3 = 25/63, which is roughly 39.7%.
What if White shoots at Gray instead of shooting at Black? In that case, the picture gets a bit more complicated: In this case, if White hits Gray the truel will end, since Black will hit White with certainty. Therefore, the only way White can win is if White misses Gray – but in this case, the situation is just as it was before, when white shot to miss. Since White has a 2/3 chance of missing, and from above we know that White has a 25/63 chance of winning if he misses, this shows that the probability of white winning is 2/3 x 25/63 = 50/189 in this case, or roughly 26.5%.
Finally, if White shoots at Black, we get an even more complicated tree diagram: Notice that the probabilities in the bottom half of this diagram are the same as in the earlier diagram, and therefore it suffices to consider what happens in the top half. Here we see that White can only win if he hits Black, if Gray misses White, and then White wins the face-off against Gray. Following the probabilities on the tree, we see that the odds of this happening are 1/3 x 1/3 x 3/7 = 1/21. Therefore the total probability that White will win if he shoots at black is 50/189 + 1/21 = 59/189, which is roughly 31.2%.
From this analysis, we see that White’s odds are significantly increased by shooting to miss: 39.7%, versus 31.2% if White shoots at Black and a paltry 26.5% if White shoots at Gray.
Another interesting feature of this problem is that by shooting to miss, not only does White maximize his chances of winning, but he also becomes the most likely person to win. Using similar arguments, one can show when White shoots to miss in the first round, the probability that Gray will win drops to 38.1%, and the probability that Black will win goes down to just 22.2%. In other words, using this strategy not only makes White the most likely to win, it makes Black the least likely to win, even though Black is the strongest marksman!
As discussed in the Wikipedia entry on truels, there are other variants one can consider as well. One could change the order of the shooters, for example, or toy with the probabilities involved. One could also consider more realistic situations in which the players have only a finite number of bullets, or think about what happens if the players fire simultaneously rather than sequentially. These variants will lead to generalizations of the phenomena observed here.
As with much of mathematics, this recreational problem shows that once you start digging, there’s practically no limit to the questions you can ask and answer with a bit of mathematics. Here’s to Martin Gardner, for providing us with such a delicious taste of the bounty mathematics has to offer.
I’m not sure, but this seems like a good candidate for a new bar. According to a recent study out of the University of Washington, as many as half of the population may fail to understand simple probability statements, in the context of weather forecasts.
Here’s the summary:
If, for example, a forecast calls for a 20 percent chance of rain, many people think it means that it will rain over 20 percent of the area covered by the forecast. Others think it will rain for 20 percent of the time, said Susan Joslyn, a cognitive psychologist at the University of Washington who conducted the study.
Coming out of Washington, one would think that the participants would have a better than average understanding of rain forecasts, but now I certainly hope that’s not the case.
That’s American math education for you. Maybe everyone should just move to LA – at least here, the forecast is the same every day.
For those who don’t believe we can actually use math to fight crime, the story of Harry Markopolos, the man who blew the whistle on Bernie Madoff, shows that a dream of using math to catch criminals need not be untenable. In a recent interview for 60 Minutes, Mr. Markopolos describes how he harnessed the power of mathematics to discover that whatever Mr. Madoff was doing, it had to be illegal.
Bernie’s luck was bound to run out sooner or later, as he must’ve known. His seeming success was really nothing more than a giant Ponzi scheme – in other words, he was able to pay his investors amazing returns by taking money from new investors, rather than by creating new wealth. It doesn’t take a mathematician to realize that such a plan is unsustainable, since the more successful your scheme becomes, the more new investors you require in order to keep the scheme successful. Eventually, the pool of new investors will become too small, and the scheme will collapse. Bernie Madoff must have known this, and this inevitability is perhaps part of the reason why he decided to confess.
If you’d like to put some numbers to such a wordy explanation, you’re more than welcome to. In fact, Professor Marc Artzrouni of the University of Pau in France has attempted to do just that, with a recent paper titled “The Mathematics of Ponzi Schemes.” In it, Professor Artzrouni models the amount in a fund taking into account a collection of 7 parameters, including the rate of return promised to investors (rp), as well as the actual interest rate at which the money is invested (rn). Notice that in the case where rp is larger than rn, investors are promised a rate of return that is less than the actual rate, as in the case of a Ponzi scheme.
Charles Ponzi, founder of the Ponzi scheme. The lesson here: never trust a man who uses that much Pomade.
Professor Artzrouni models the asymptotic behavior of the amount of money in a particular fund subject to different restrictions on the initial conditions. In particular, his models produce three types of funds: those that remain solvent (the amount of money in the fund is always positive), those that collapse, and those that collapse, but could regain solvency with a bailout.
Interestingly, he is able to produce examples of what would be considered Ponzi schemes that nevertheless remain solvent. This can happen in certain situations if the fund manager is able to supply enough outside capital to the initial investment – this initial input on behalf of the fund manager must be enough to offset the fact that the promised rate of return is larger than the actual rate of return. Professor Artzrouni discusses the existence of these so-called “philanthropic” Ponzi schemes in the context of Social Security (so named because the initial capital put in on behalf of the manager goes towards the solvency of the fund rather than being considered an investment on behalf of the manager), which has sometimes been criticized as being nothing more than a government sponsored Ponzi scheme.
Unfortunately, it doesn’t look like Madoff’s fund falls into such a category. What’s even worse, from the 60 minutes article it’s clear that Bernie Madoff was not the only one who knew that he was running a scam. Mr. Markopolos knew as well, and called shenanigans on Madoff to the SEC nearly 9 years ago. Sadly, it seems that the SECs own poor background in financial mathematics blindsided them to Madoff’s antics for nearly a decade, even as Markopolos continued to submit reports detailing Madoff’s fraudulent practices.
The conclusion here is that mathematics can only be used to fight crime when the people fighting it have a strong enough background in mathematics. Or, failing that, every investigative unit should have one go-to math guy, à la Charlie Eppes.
“A Ponzi scheme, you say? Quickly – to the faculty lounge!”
Either way, it’s unfortunate that nobody was able to put Madoff away before he screwed so many people out of so much money. One hopes that next time we will be able to act more quickly when the mathematical evidence so strongly suggests that something bogus is happening.
I made my reservations fairly clear regarding the doubledose of math holidays last month. Despite my objections, I remained confident that the headlines they gathered would quickly fade away, and I wouldn’t have to worry about these faux math headlines for the next 12 months. In this way, I was able to sleep peacefully at night.
Unfortunately, it seems there are those who wish to disturb my slumber.
Dan Vergano over at USA Today recently wrote a brief article which highlighted the fact that this year there are a whopping 2 “square days,” one of which is today, 4/01/2009. The day is called a square day because if you read the date as a number, the number turns out to be square. In this case, 4,012,009 = 2003 * 2003.
The article attempts to be relevant by making a tenuous link between this sort of mathematical wizardry and the latest film excursion into numerology – Nicolas Cage’s most recent triumph, Knowing. Mr. Vergano was also kind enough to link to my article on Square Root Day, although based on the tone of his article, I’m not sure he appreciated the point I was trying to make. Perhaps he intended to address my concerns, but in the process of writing he got lost in Nic Cage’s eyes. Lord knows it can happen to the best of us.
What’s that, Mr. Cage? Sorry, I got a little distracted.
So, Mr. Vergano, if you’re reading this, I beg of you: use your powers for good. With a readership as large as I’m assuming yours must be, you have a venue to help dispel stereotypes about people who study mathematics. Of course, those stereotypes include, but are not limited to, the idea that mathematicians spend their days looking for significance in arbitrary dates.
To his credit, Mr. Vergano does point out the insignificance of these types of diversions. But if there isn’t any significance, what’s the point in writing about it? Does this happen in other fields besides mathematics?
I get that advances in math may not seem as sexy to the lay person as certain advances in the sciences, and sometimes the ideas can be difficult to communicate. But there are opportunities for those willing to look. Here’s one: why not write an article celebrating the contributions of Mikhail Gromov, recent recipient of the Abel prize? This, it seems to me, would be a much more worthy topic for a writer with such exposure.
I don’t think I’m alone in this, either. A look at the comments to Mr. Vergano’s article reveals many others who fail to see the importance of today as a square day.
Perhaps one day the USA Today blog will discuss some real mathematics. And on that day, I shall declare a legitimate math holiday. Until then, I will remain here and nervously await Pi Day 2010.
