Non sequitur of the month: Very amusing, completely unimportant

This time it's a line from a TV commercial. It's so funny (in a non sequitur kind of way) that I just couldn't help but quote it. Some company is advertising its product by giving away free samples. One of the customers participating in the commercial says

They're giving it away for free? It must be good!

As Paul Krugman is fond of saying, not only is it not true, it's the exact opposite of truth.

An idea that changed the world. Can you name one?

At some point in a very interesting NOVA PBS show, a scholar says something we hear quite often: monotheism was a revolutionary idea that completely changed the world.

Did it really change the world, though? If so, how much?

Technically, we have no way of knowing. I may be wrong, of course, but it seems to me that we often misjudge the impact of ideas and historical events because we have an intuitive notion of causality that's just wrong. If we have some event A that happened at time T, we think of its causal impact on the history of the world as the difference between what the world looked like before and after time T. But that's incorrect. The world after any event is different than it was before it, so by this metric, everything would be "revolutionary" and "world-changing."

The true causal impact of A is the difference between the state of our world after time T and the state of a hypothetical world in which A did not happen at T (also after time T). In other words, we tend to forget that causal inference is necessarily about counterfactuals.

So the impact of monotheism is the difference between our world after monotheism appeared, and a counterfactual world in which it did not appear. I have no idea how to even begin to estimate this difference, but in my completely subjective opinion it is quite small.

We like ladders better than trees

Sports fans are usually obsessed with rankings (be they player or team rankings); soccer fans are no different. FIFA (the international soccer organization) has its own rankings of national teams, but given how flawed those are, many soccer statisticians are trying to come up with their own. The best ones seem to be Voros McCracken's (his soccer blog is also very interesting) and Nate Silver's (the political scientist that runs the famous FiveThirtyEeight.com website).

The problem with rankings is that they imply a linear order that sometimes just doesn't exist in reality. For example, the usefulness of ranking soccer teams is presumably that if we see that, say, Spain is ranked no. 2 in the world whereas Denmark is ranked no. 12, then Spain is the more likely winner in a Spain vs. Denmark game. The problem is that linear rankings require transitivity (i.e., transitivity is a necessary condition for a relation to be a linear order.) The transitive property is easy to explain. Suppose Spain is a better soccer team than Denmark, and Denmark is better than Venezuela. Then it seems common sense that Spain is also better than Venezuela. If this is indeed the case and if, in fact, this property is true for any three teams we choose, then the relation "better soccer team than" is transitive. If there's no transitivity, we cannot rank soccer teams from best to worst. And in reality it isn't there. For example, all major soccer rankings put Mexico ahead of the U.S. But, at least in the recent eight years, the U.S. has a better head-to-head record against Mexico. It seems as though Mexico's record is superior to that of the U.S.--except when those two teams play each other. But this means there's no transitivity: Denmark is ahead of the U.S., Mexico is ahead of Denmark, and the U.S. is ahead of Mexico. The linearity of the order breaks down.

This is not to say that statistical methods of measuring team strength are useless. Far from it. Both Silver's and McCracken's systems, for example, are capable of producing the odds for essentially any given game. That's extremely useful--but very different than providing a consistent linear ordering of all teams in the world. The latter just does not exist in reality.

Coincidentally, there are many more examples of us trying to impose a linear order where there is none. It's often the case that we see a total order in situations where the true order is partial--i.e. we're trying to put things on a ladder where the true underlying structure is that of a branching tree. (Note: unlike the soccer rankings situation, a partial order is still transitive. It's just not linear.) For example, you sometimes hear the question: "If humans evolved from chimps, why are there still chimps?" The confusion from which this question arises is that of treating the relation "evolved from" as a linear order, whereas in reality it's a partial one (evolution is not a ladder but a tree; chimps are not our "parents" but "cousins:" our "parents" and chimps' "parents" were "siblings"). For another example, the way we teach grammar in middle schools is based on an implicit assumption that grammatically correct sentences can be derived from rules if sentences are treated as strings of words. This assumption is incorrect: as shown by Noam Chomsky, in order for us to be able to provide algorithmic rules for generating syntactically correct sentences of any human language, those sentences necessarily have to be treated as trees, not strings, of words and phrases.

Most important concepts we don't teach in the courtroom: signaling

Imagine you're a college math teacher. This coming semester, you're supposed to teach Calculus 1. As an only prerequisite for your course, you list a grade of at least A- in a pre-calc course. However, the pre-calc course offered students a choice between taking it for a grade or as "pass-fail." A student named Danny signed up for your class. He passed the pre-calc course, but opted out of taking it for a grade. You don't want to let him in your class, reasoning that no A-student would choose to hide his grade. This, however, makes Danny's dad very angry. Danny's dad wants to speak with you. Even worse, the very President of the college you teach at insists on being present during your this conversation.

Danny's dad: Do you hold the fact that Danny opted out of taking pre-calc for a grade against him?

You: This fact tells me something.

Danny's dad: It shouldn't tell you anything.

You: But him opting out of a grade signals me something.

College President: You have to ignore the fact that Danny didn't want a grade. It doesn't signal anything.

You: But it does.

President: If you're selected as Danny's teacher, can you ignore the fact that he chose not to take a grade?

You: I could try, but subconsciously I know why he's doing that.

President: Will you try your best not to be prejudiced?

You: I'll try my best, but I can't control my subconscious thought that he's not taking a grade.

Danny's dad: You're excused from teaching Calculus 1.

Sounds absurd, right? I mean, you're clearly right--the very fact that someone chose a "pass-fail" option over a grade option says that they're most likely not an A-student. This fact signals something, whether anyone likes it or not. And yet, the above conversation (as well as the flawed decision-making resulting from it) is commonplace in American courtrooms. Don't believe me? See this.

Two thoughts

Substantively, they are completely unrelated. It's just that I think both are great observations, and wish they were mine.

First, Steven Landsburg writes

The Intelligent Design folk tell you that complexity requires a designer. The Richard Dawkins crowd tell you that complexity must evolve from simplicity. I claim they're both wrong, because the natural numbers, together with the operations of arithmetic, are fantastically complex, but were neither created nor evolved.

(By "fantastically complex" he means the unsolvability of the halting problem: it's logically impossible to write an algorithm that could tell you whether or not any given arithmetic problem is solvable in finite time.)

The second thought is Eric Falkenstein's:

... 150 years ago it would be proper to be a racist and think that the 'upper class' was a thoroughly different beast altogether, whereas today smarmy college freshmen at top universities think every human grouping possible has equal genetic ability and interest in every meaningful human dimension. People never get rid of their prejudices, they just change them.

It's always a good idea to look at the data

Before making a sweeping statement, no matter how commonsensical the statement may seem to you.

"Preventive medicine saves money" seems to be something virtually everyone believes. All mainstream politicians and political commentators certainly do. In fact, they all treat this statement as if it were simply self-evident, not as something that would need empirical verification. How can it be possible for preventive medicine to increase costs of healthcare?

Easy. Imagine you run a health insurance company and you cover a large population of people all of which are susceptible to a certain genetic disease. In a single patient, the disease is extremely expensive to treat if detected in its late stages, but fairly cheap to deal with when detected early. You can either pay to screen your customers and pay for their early treatment, or not screen them and just pay for late-stage treatment of whoever ends up getting sick. Holding costs constant, it is clearly possible that the latter choice costs less money, provided the contraction rate of the disease is low enough.

A 2008 metastudy from The New England Journal of Medicine claims that, on aggregate, preventive medicine does in fact increase the costs of healthcare.

Two things need to be remembered. First, preventive care increases aggregate costs; this means that most preventive measures increase costs, but not that all of them do. Some preventive measures do actually save money. More importantly, I was only considering the question of whether or not prevention saves money. Saying that something increases costs is different that saying it's a bad deal; even though prevention adds to the healthcare bill, it may still be cost-effective (if it improves the overall quality of life by an amount that offsets additional costs).

(HT: Healthcare Economist).

Most important concepts we don't teach in school: opportunity cost

Thomas Friedman's recent NYT column is another good example of a fallacy I've written about previously: focusing solely on probabilities and ignoring utilities when doing cost-benefit analysis. Essentially, Friedman thinks that the reasoning behind Cheney's "One Percent Doctrine" is sound, and applies it to the climate change debate. He writes

When I see a problem that has even a 1 percent chance of occurring and is "irreversible" and potentially "catastrophic," I buy insurance.

Really? You just "buy insurance?" Without even asking how much said insurance costs?

The root cause of Friedman's remarkably thorough confusion is the following: he thinks that insurance against the effects of global warming costs less than it actually does, because he is unfamiliar with the concept of opportunity cost. Friedman writes

If we prepare for climate change by building a clean-power economy, but climate change turns out to be a hoax, what would be the result? Well, during a transition period, we would have higher energy prices. But gradually we would be driving battery-powered electric cars and powering more and more of our homes and factories with wind, solar, nuclear and second-generation biofuels. ... In short, as a country, we would be stronger, more innovative and more energy independent.

Every single action we take has a cost in forgone opportunities--the value of the best thing we could have done instead. Producing electric cars, wind, solar and nuclear plants, and second-generation biofuels is no exception. So in the case of climate change, is this opportunity cost worth bearing? It depends on two things: the probability that global warming is real (which is extremely high) and the costs of it if we do nothing about it (which are extremely uncertain). The problem is that Firedman's reasoning simply ignores some of that crucial information.

Most important concepts we don't teach in school: expected utility

Imagine you're talking to an insurance salesperson. She tells you that DNA testing reveals you have a 1% chance of developing a certain form of cancer which, when untreated, is always fatal. She says there is a cure, but it's so expensive no one can afford it without insurance. She hands you an insurance contract to sign; if you do, the costs of treatment, should you need it, will be fully covered so even if you do get sick you will still live. You stare at the dotted line holding a pen, then ask what your premium would be. "Oh, we don't know that yet," she says. "But we'll get back to you on that real soon." Would you still sign the contract?

Of course not; only a complete idiot would. Whether or not it's worth it to insure yourself against a 1% chance of dying depends on the premium you'd have to pay. And yet we are offered exactly the same absurd sales pitch as described above when we're being sold public policy. I'm referring to what is now known (perhaps misleadingly) as Dick Cheney's "One Percent Doctrine." The name comes from the following line from the former Vice-President:

If there's a 1% chance that Pakistani scientists are helping al-Quaeda build or develop a nuclear weapon, we have to treat it as certainty in terms of our response.

This quote contains the very same error in reasoning as the insurance deal above; and both could be remedied if more attention was paid to the concept of expected utility.

Suppose you're trying to decide whether or not to carry out some action A. If you don't do A, there's an X-percent chance you will lose a Y amount of money. If you do A, you won't lose Y; however, doing A is costly too (say it costs Z). Expected utility says you should do A if and only if Z<X*Y. In other words, you need to compare the costs of action to the costs of inaction times the probability of bad things happening due to inaction. Cheney's doctrine focuses exclusively on the likelihood of bad outcome of inaction, without trying to balance it against the costs of action.

The piano smells like a bomb

In my previous post I wrote about a problem with many public policies: the fact that we often focus only on their benefits while ignoring their costs. Many times we see government agencies implementing such policies being rewarded for their benefits while not absorbing any of their costs. TSA is a perfect example of such an agency. TSA gets rewarded for coming up with screening procedures that deter terrorists--but are not punished if those procedures are too costly for the non-terrorist passengers. TSA doesn't care if the lowered risk of terrorist acts due to their procedures warrants the huge amount of time wasted at the security gates, or the inconvenience caused to passengers diverted from flights or put on no-fly lists because of TSA's oversensitivity.

Or about the inconvenience caused to a world famous classical pianist by seizing and destroying his piano at the airport because it smelled funny. This happened to a Polish pianist Krystian Zimerman shortly after September 11 at New York's JFK airport. Apparently Zimerman alters his instruments by hand, and always travels to concerts with his own customized Steinway. On one of such trips, TSA confiscated his piano and then subsequently destroyed it because, as they said, the glue in it smelled like explosives. This has prompted Zimerman to 1) start traveling with his piano dismantled into little pieces that he would later reassemble on his own before concerts; 2) embark on a weird rant during one of his shows about how the U.S. military wants to control the entire world and 3) announce that he would cease to play in the U.S. altogether.

Shame on you, TSA, for the ridiculous "take off your shoes and belts" routine at the security gates, and for destroying Zimerman's piano. And here is how this piano sounds: Zimerman playing Chopin's Ballade No. 4 in F Minor (the most beautiful piece of music I have heard so far).

What we should be ashamed of

We like to feel morally superior to our ancestors. When we look back at our history, we like to feel outraged about grave sins committed by our great-grandparents (such as nazism, racism, etc.), and think that we, being more civilized than they were, are no longer capable of similar depravity or confusion. This is dead wrong, because we are doing things today that future generations will look at with horror. We are doing them primarily because most people either don't see anything wrong about them, or else don't even notice that they're being done.

What are those things then, that we should be ashamed of? I don't claim to know I'm right, but here are my candidates.

1) The fact that we are not equally compassionate to all groups of people who are suffering. Our compassion seems to depend on politics, ideology and religion. We are outraged at our great-grandparents that they did almost nothing to stop Holocaust; but we are doing almost nothing to help Palestinians suffering from the Israeli government, Chechens suffering at the hands of the Russian government, or the people of North Korea who are being terribly oppressed by the communist government. (Before you click, be warned: reading texts linked above is likely to make you sick to your stomach.)

2) Our inability to recognize that every policy has costs as well as benefits, and that they have to be weighed against each other. In many cases (especially those that involve "moral panic"), we ignore the costs altogether, which leads us to implement cures that are much worse than the disease (such as the drug war).

3) Tribalism. The tribal instinct is very powerful, and it can lead to morally unacceptable attitudes. We recognize the immorality of some of those attitudes (e.g. racism) and try to curb them; but there are many outlets in which we let tribal prejudice run rampant. For example, saying things like "Atheists cannot be moral people" is ethically equivalent to racism--but no one who says them (including politicians) faces any type of social punishment. Similarly, public discourse is full of arguments supporting tightening immigration laws or implementing trade protection measures, on the grounds that it would prevent American jobs from being taken over by foreigners. All such arguments rest on an implicit moral assumption that foreigners are less human than Americans--and yet no one who uses them seems to be ashamed of this fact. In fact, the postulate that immigrants should be treated as people is completely absent from mainstream American politics; as far as I know, only libertarian right and anarchist left support it.

4) The fact that we rely on moral intuitions to settle ethical dilemmas. We seem entirely unaware of the fact that our moral intuitions are often wrong, because they evolved to facilitate efficiency in a small band of hunter-gatherers, not to minimize suffering in a society as complex as ours. For example, according to our moral intuition, we tend to judge actions by the intent of those who act, instead of by the consequences of those actions, ignoring the fact that acting on selfish motives can have good consequences, or that acting on benevolent motives can have terrible consequences indeed.

Battle of the sexes

If you are not convinced that nature is a cruel joke, take a look at the image above. What you see there is the penis of a beetle.

Any time you think life is rough for you, thank your lucky stars you're not a female beetle.

(HT: Pharyngula).

Does soccer reward cheating?

French striker Thiery Henry's recent dribbling the ball with his hand instead of his foot has lots of soccer fans asking this question. British economist Tim Harford thinks it does not; here's his reasoning:

Henry has been selfless. The rewards of his cheating go largely to his team-mates, who get to go to the World Cup with their names unblemished, and to fans of French football, once they get over the embarrassment – which they will. Henry himself faced all the risks. He might have been cautioned or sent off, but surely the far greater risk was what happened: only the TV cameras noticed the handball and a great striker’s reputation was tarnished. His subsequent pronouncements of guilt, shame and remorse have hardly put matters right. So, what would an economist have done? The answer is absolutely clear: economists would never cheat in front of the camera.

In other words, Harford thinks that when soccer players cheat it's because of an uncontrolled impulse rather than a deliberate response to incentives. From a single player's perspective, it doesn't pay to cheat, but sometimes the hand is just quicker than the head (or foot).

When it comes to Thiery Henry, Harford is probably right. There's no way the French striker could have though his blatant handball wouldn't be exposed on cameras, so it was probably just a reflex (that went unnoticed by the referee). But in general I think this is wrong; you don't always get caught on camera when you cheat. If your cheating is subtle enough, you'll have some room for plausible deniability despite the fact that the cameras will register what you did. Some blatantly dishonest plays don't look all that bad in replays.

Ignorance or confusion?

Dan Drezner talks about a recent Pew Research poll with a general theme of foreign policy. According to the poll, 44% of the population view China as "top global economic power" (only 27% think so about the U.S.) Drezner wonders (and I with him) how anyone could think that; there is absolutely no reasonable measure of economic power by which China can be regarded as the very top one. China's GDP is about two times smaller than the U.S. GDP; its output per capita is roughly eight times smaller.

It could be ignorance. Some respondents who see China as the world's biggest economic power may simply have no idea what the world's economies look like right now. Or it could be confusion: perhaps there are other respondents who aren't careful enough to distinguish between stock and flow. After all, for about a decade now, we keep hearing about China's amazingly fast GDP growth rate. That rate is in fact much higher than in the U.S. (for example, in the last three years, the average annual output growth rate in the U.S. was under three percent, whereas in China it was over eleven). But income growth rate and wealth are two completely different things.

No numbers, no formulas

A few months back, Mark Chu-Carroll complained (and rightfully so) about some idiot on NPR who said that, since sudoku puzzles don't have to use numbers but can use any symbols, they're not mathematical puzzles at all.

It's a common misconception, I think: If something doesn't have lots of numbers or formulas in it, it's not math. In reality, math isn't about either of those things. In fact, we could in principle rewrite the entire body of mathematical knowledge so that it wouldn't contain one single formula. It would be extremely difficult and utterly pointless, of course, but nonetheless possible; we use formulas only because they're convenient, not because they're necessary. Scroll through Gottlob Frege's Foundations of Arithmetic; it's mostly just plain text, not a whole lot of formulas there. And yet it's definitely mathematics, and important mathematics at that; it contains the first logically correct definition of the concept of number.

Here's another example of this misconception at work. Back when I was in college, a friend of mine who was a psychology undergrad asked me to participate in an experiment he was conducting for his thesis (why he called something that had no control group an "experiment" I'll never know, but let's leave that for now). He gave me a sheet of paper with integers 1 through 10 randomly scattered about the page, interspersed with (also randomly scattered) letters A through J. He asked me to connect the integers in ascending order as quickly as I could, and timed me. Then he gave me another sheet just like the first one, and asked me to connect the letters in alphabetical order; this whole thing was then repeated a few times. When, after it was finished, I asked him what he was trying to get at he said he was interested in relative speeds with which brain hemispheres process information, and that this would allow him to do it because the task of connecting numbers was managed by the right hemisphere whereas the task of connecting letters was managed by the left one. I asked him how he knew that. He said: "Because the right hemisphere is responsible for mathematical reasoning so it must control connecting numbers, and the left hemisphere is responsible for verbal skills so it must control connecting letters." In other words, to him, the first task was mathematical (because it was about numbers), and the second one was verbal (because it was about letters), even though, quite clearly, conceptually it was the same damn task!

So now that we know that math is not about numbers or formulas, can we say what it is about? It's about a certain type of reasoning, I guess. What type of reasoning? Who knows, really. In this respect math is like porn: very hard to define, but very easy to recognize when you see an example.

A great employee with a bad hobby

If you are, like me, a fan of Arsenal F.C., you probably hate national team soccer. Things of the sort that has just happened to Arsenal's brilliant Dutch striker Robin van Persie happen to soccer clubs all over the world, and make their managers furious. In mid November, van Persie got called for national duty to play for the Netherlands in a friendly (i.e. sparring) game against Italy. In that game, he suffered an injury that rendered him unable to play for the rest of the 2009/2010 season. And this in effect means that Arsenal is no longer able to mount a serious campaign to win the English Premier League.

Think about this from a club manager's perspective. You buy players for your team, pay their wages, as well as invest money in coaching necessary to make sure their skills improve. In exchange, those players do their best to help your team win trophies. Of course, every once in a while during your trophy-winning campaign, some of your players will get injured. Soccer is very physical, sometimes downright brutal, so this is unavoidable. If this happens to one of your players important enough that their absence significantly diminishes team value, you'll probably whine a lot and curse your bad luck. You will not think it's unfair, however; after all, every player in every team faces a positive risk of injury in about every game they play. But then there are national teams, whose managers can call your players and use their services in their own competitions, thus exposing them to an additional risk of injury and, what's worse, when they do call them you cannot refuse (clubs that won't release their players for national duty face severe sanctions from soccer governing authorities such as FIFA or UEFA, ranging from fines to suspending a player or even revoking his license altogether).

To me, this situation is blatantly unfair to the clubs. What's also interesting is that most soccer fans I talk to do not see this fundamental unfairness, and tend to just dismiss club managers' complaints without even trying to provide an argument. In fact, there is only one rational argument in favor of the status quo that I've ever heard; it comes from a sports statistician Voros McCracken, and goes something like this. The very best soccer players are also more likely to want to play for their national teams (for reasons of prestige or whatever); therefore if you, as a club manager, wanted to be able to forbid them from doing so, you'd face a situation in which those very best wouldn't want to play for your club so you'd have to settle for choosing your employees from a weaker pool. You can think of the best soccer players as of great potential employees that have a dangerous hobby (namely, playing in national team competitions); since players that don't have that hobby tend to be weaker, you can't really complain about how unfair it is when one of your best employees gets hurt while exercising his dangerous hobby.

Rational as it may be, this argument is still wrong; here's why. The relevant question is not whether or not better soccer players are more likely to want to play for their national teams than weaker players do. The relevant question is: how important is the best players' desire to play for their national team as compared to the commitment to the club that currently employs them, in a situation when those two goals are in conflict? The problem is that, given the current set up, we can't know the answer, because whenever a player gets called for national team duty, it's not just his club that can't refuse; he himself can't, either. The price of such refusal is long suspension or perhaps even the end of career, and that's just prohibitive. In a perfect world, clubs would be free to draw contracts that could either let players partake in their national teams or forbid them to do so, and then players would be free to choose which type of contract they'd want to sign. If Voros were right, we'd see forbidding contracts paying out higher wages than the non-forbidding ones (holding player quality constant); if he were wrong, there'd be no such difference; and if he were really right, the monetary price that clubs would have to pay to make a player get rid of his hobby would be just too high, so there'd be no forbidding contracts at all. The thing is, as of now, neither the clubs nor the players have such freedom, so Voros' claim is perfectly unfalsifiable.

Interestingly, Voros' "dangerous hobby" analogy helps defeat his own argument. Great soccer players also are, on average, more likely to have personalities that make them seek thrills in dangerous activities such as motorcycle racing, mountain climbing, sky diving, bar fighting etc. Some clubs do draw contracts that actually contain clauses explicitly forbidding players to, say, ride bikes or climb mountains, on pain of being fined or even fired, and some players do choose to sign such contracts. Until players are allowed to exercise the same amount of choice over national team duty versus club duty, we can't claim to know which is more important to them.