Math in the News: Elephants are Smarter than your Babies

I missed the memo on this one, but apparently worms aren’t the only animals capable of doing math. A recent experiment coming out of the University of Tokyo suggests that Asian elephants have an unexpected aptitude for arithmetic. While many animals have a rudimentary counting ability, and are able to distinguish between sets with only a few elements, it seems that elephants are able to take things a step further, and can consistently differentiate between larger numbers such as 5 and 6.

Is this difference significant? Within the animal kingdom, it would seem so. Here’s how it breaks down, courtesy of this article:

A theory held by some is that humans and other animals share a basic neural system called an “accumulator” that can clearly distinguish numbers of objects less than three or four but that cannot reliably discriminate between bigger numbers. This accumulator is active in animals and, perhaps, in human infants, the theory contends. Higher-order number abilities require the collaboration of other, more highly developed brain systems found only in humans.

An ability to consistently distinguish between larger number (by larger, I mean larger than four) may therefore indicate a more advanced accumulator system than is found among the general kingdom’s populace.

What does this mean for the noble elephant? While it’s certainly a bit premature to start hiring them as our accountants or financial advisers (although, given the current economic conditions, perhaps giving elephants access to our finances isn’t such a bad idea), it certainly does highlight what those active in elephant research already know: these majestic creatures aren’t all looks. Each one has a head on its shoulders as well.

This elephant is no doubt pondering some very deep mathematics. The picture is taken from a collection that accompanies an excellent National Geographic article on these mathematical savants.
It is natural to ask what sort of evolutionary process would lead to the elephant’s surprising counting aptitude (aside from the obvious benefit of being able to impress the ladies). An article from the London Times suggests the following alternative hypothesis:
Speculation among scientists over why the elephant should have developed its limited but nonetheless impressive mathematical ability centres on the way in which the lumbering creatures move in herds. A basic counting ability, say experts, might act as a guarantee that no calf is left behind.

Is the acquisition of mathematics knowledge driven by evolution? Perhaps in the animal kingdom, although if you ask graduate students in mathematics, I doubt they will say that an aptitude in math has really helped them to propagate the species. Those days are coming, my friends, but they are not here yet. For now, let us find solace in the fact that when it comes to defending the belief that mathematics is of fundamental importance, we will have a mighty ally in the Asian elephant.

The results of the experiment came as no surprise to Babar, whose sharp intellect not only allowed him to become king of the elephant empire, but also blessed him with a keen eye for fashion.
November 20, 2008 | 1 Comment »

Math in the News: Are Mathematicians the Reason Why You’re Broke?

As you may have heard, the economy is in a bit of trouble. People continue to debate the root cause of the current crisis: some blame so-called predatory lenders for pushing mortgages on people who couldn’t afford them, some blame the borrowers themselves for recklessly taking on loans to try and live beyond their means. And of course, as with any problem, there are those who try to shift the blame to mathematicians.

Why mathematicians? Proponents of this theory assert that our current financial collapse is the fault of the math whiz kids hired to work on Wall Street or manage hedge funds. It is no secret that investment banks have been hiring bright mathematical minds for years, then squeezing that brainpower into models for trading.

The use of mathematics in this case isn’t the problem (indeed, when could using mathematics ever be a problem?). The problem, the critics cry, is that nobody understood the models that were developed – not the economists, not the credit ratings bureau, and certainly not your average Joe Sixpack. Nevertheless, the models worked, at least for a time, and made many people very rich. And so the models were used, without a complete understanding of how they worked, until everything began to go horribly wrong.

This general idea has found its way to most media outlets. Below is a story from 60 minutes that briefly touches upon the issue in an interview with former US Deputy Treasury Secretary Roger Altman. The BBC jumped on this bandwagon even earlier, with an article on “quants” (a.k.a. quantitative analysts, math junkies hired for finance jobs) that appeared nearly a year ago.


Jump to the 7:00 mark to learn how mathematicians RUINED YOUR LIFE!!!

It certainly is convenient to blame mathematicians for this crisis. The more pressing question, of course, is whether these assertions are merited.

I think not. After all, mathematicians never told investors to use their models irresponsibly. If the investors didn’t understand what they were doing, they should have spoken up, or at least brushed up on their mathematics. Why would you gamble with billions of dollars if you didn’t understand the game you were playing?

Some will argue that perhaps these mathematicians and physicists built their models on false assumptions, thereby creating simulations that were bound to fail. This is a weak copout at best. These people were hired by these institutions to model the markets, but in the end, the investors are the ones to pull the triggers. It is their responsibility to ensure that the models they are working rest on sound hypotheses, and it should go without saying that nobody should act on what the model is saying without understanding it.

What about the criticism from the BBC article mentioned above? In the article, a quant by the name of Paul Wilmott throws himself under the bus by explaining why he believes quants should share some of the blame for the economic downturn:


“The way in which quants are compensated encourages them to use the same strategies as everyone else.”

He claims that many quants calculate that if they lose money as a result of following a novel strategy they will be fired.

However, if they lose money as a result of following the same strategy as everyone else, they will not get the blame.

“The problem with this,” says Mr Wilmott, “is that if something bad happens, it happens across the board.”

If true, it’s certainly the case that mathematicians could have been more forward. Perhaps they were too attached to the lavish lifestyle that a job in finance provided. Even still, it’s difficult to imagine that a group of mathematicians could have actively built a culture that stifles creative problem solving. Certainly it’s in the investors’ best interests to not lose money, but any true mathematician should follow the data, rather than making data fit with expectations. Is it the fault of mathematicians that the culture of the industry was so resistant to the intuitive idea that you can’t always make your investments pay off?

If mathematicians are to blame for the evaporation of your retirement savings, I have yet to find a compelling argument for why (unless of course, you were robbed by a mathematician, in which case I offer you my sympathies). This seems like just another case of passing the buck – we blame the passive mathematician because we know he won’t fight back.

The moral here is one that will be familiar to anyone who has seen Harold and Kumar Go to White Castle. On the fateful night that Harold and Kumar set out on their pilgrimage to the aforementioned eatery, Harold had been screwed over by some of his coworkers, who talked him into staying late and finishing their work for them so that they could go out, party, and be generally irresponsible. The audience is led to believe that Harold has been doing their work for some time, but he never gets the credit.

During the course of the film, owing to some hijinx and a bit of tomfoolery, Harold loses all the work he has done, and when he serendipitously meets up with his coworkers later, they tell him that there will be hell to pay come Monday morning because of it.

Harold (left) takes his mathematics very seriously.

It is at this point that we witness the transformative power of Harold’s journey to White Castle. Whereas before he may have shied away from confrontation, in this climactic scene he confronts his aggressors with strength and confidence. He makes it clear that he will not allow them to walk all over him and blame him for their own shortcomings, then suggests that they both have gonohrrea.

While this last step may be a bit overboard, it certainly couldn’t hurt for mathematicians to say a little something to stop investors for laying the blame on their shoulders. A little blame, maybe. But with so much to go around, there’s no need to burden us unnecessarily.

November 13, 2008 | 2 Comments »

Math Gets Around: Politics, Part 3

As promised, in this thrilling final installment to the relationship between math and voting (the first two parts can be found here and here), we will look at what many people see as the holy grail of voting systems: Range voting.

The concept of range voting is simple. Given a set of candidates, in a range voting system you simply put a score next to each name that reflects how strongly you support that candidate. Of course, this is quite different from our current voting system, where we only get to vote for one candidate, but more importantly, it differs significantly from other voting systems where you are just asked to rank candidates in order of preference, because a ranking gives no information about the degree to which your support varies from candidate to candidate.

For example, if Anna, Bob, and Charlie are all running for President, you and I may both prefer Anna to Bob, and Bob to Charlie. However, I may LOVE Anna and HATE Charlie, while you may be relatively indifferent, with only a slight preference for one over the other. In a ranked voting system, both our preferences would be recorded as A > B > C. However, using range voting, our preferences may look something more like: A = 99, B = 50, C = 0 for me, and A = 51, B = 50, C = 49 for you.

This example shows that range voting allows us to capture more information about voter preferences than the other voting systems discussed. Therefore, one might heuristically expect that because it captures more information, it leads to better results, i.e. a more accurate representation of the will of the people. The question, of course, is whether this is actually the case.

The answer may depend on your definition of “better,” but by most measures the range voting system comes out on top of the others. One important feature of this system is that it is not subject to the constraints of Arrow’s Impossibility Theorem (discussed in Part 1). In other words, range voting has basically every property you would like a voting system to have. Range voting doesn’t contradict Arrow’s theorem because Arrow’s theorem deals only with voting systems that only rank preferences.

There are many other benefits to range voting as well, and these benefits are no doubt well known by any range voting advocate. The website Rangevoting.org gives a list of reasons why range voting is so great – I won’t list them all here, but I will highlight some of the more interesting ones.

  • Range voting encourages honest voting, rather than strategic voting. There is never an incentive for you to score a candidate you support more lower than a candidate you support less.
  • Range voting allows for a larger range of political parties to flourish. Because you are not restricted to one vote, people from third parties can feel free to support their candidate without fear of “wasting their vote.” This is also good for independents who may not feel particularly strongly about any major party candidate.
  • (Perhaps the cutest result) Range voting maximizes the number of “pleasantly surprised” voters, i.e. the number of voters for whom the winner of the election is better (scored higher) than they thought it would be.

As with any other idea, though, range voting is not without its share of criticism. However, these criticisms pale in comparison to the critiques that can be made about our current voting system. The main critique with range voting has to do with strategic voters, and comes in two forms:

  • Why doesn’t this just degenerate into the system we already have? For example, if supporters for one candidate feel strongly enough, they will simply give that candidate the highest score and every other candidate a 0. Doesn’t this benefit dishonest voters, and hurt candidates whose supporters are honest and may not give their candidate the full score, or score everyone else with a 0?

This criticism is a little suspect, because while there are certainly people who may vote in this way, it’s certainly hard to believe that everyone will, or that even a disproportionate number of supporters of one candidate will. It’s more likely that roughly the number of people for each major candidate will feel strongly enough to vote in this way, so that in the end it should all balance out. There certainly are extremes of political belief, but this is true of both the left and the right, with a wider swath of moderates somewhere in the middle.

  • Doesn’t the system inflate support for third party candidates? For example, people will be more likely to throw support to a candidate they believe has no chance of winning – this will amplify support for lesser known candidates, and dampen support for well known candidates.

This seems plausible. One way to combat this is to require that a candidate receive a minimum number of scores in order to be viable. For example, we could say that in order for a candidate to be declared the winner, at least 10% of the population must have voted for the candidate. The percentage should be high enough to be significant, but not so high that it’s possible it couldn’t be obtained if enough voters strategically abstain from voting for a particular candidate.

However, it’s still far more likely that a major candidate with a large base of support will win over an independent candidate with a small base of fervent supporters. Overall, range voting will certainly reflect preferences better than the current system, so it’s hard to argue that this is much of a valid criticism when compared to the current system, where your support is for all intents and purposes meaningless unless it is for a major party candidate.

In summary, from a mathematical point of view, there really is no argument: range voting certainly trounces are current voting system, and it looks like it beats everything else as well. The question then becomes: why don’t we use it? I’m not sure there’s a good answer.

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Range voting is not used by any democratic nation, but examples of it can be found all over the internet. For a concrete example you are probably familiar with, you need look no further than the Internet Movie Database. On this site, you are free to vote on any movie you like, giving it a rating between 1 and 10. These votes are then compiled into IMDB’s rankings of the top 250 movies of all time, which can be found here. What’s even better, the votes are compiled using a true Bayesian estimate, the formula for which can be found at the bottom of the page. If you have any doubts about the validity of range voting, you need only go view this list and see all the awesome movies on it to conclude that indeed, this system has it going on.

Of course, you may find that this list of movies is terrible, but in this case, don’t worry. It doesn’t mean that range voting doesn’t work, it just means you have bad taste in movies.

*

In conclusion, our voting system is horribly broken. There are solutions out there, but getting from where we currently are to a new voting system is a problem that goes beyond the scope of a pop culture math blog. For now, we’ll have to deal with things as is, and of course, that includes voting tomorrow, November 4th. So make sure you go out and do it. We can fight the larger fight of voting systems another day.

November 3, 2008 | 3 Comments »