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Unlike the first bottle, the second must be profound. In selecting, you are obliged to demonstrate sophistication without descending into foolishness. Nobody wants to hear your reflections on the beads of dew you observed on the leaves of Pinot Noir vines when bicycling through the Côte d’Or in ’92. Save your ruminations on imaginary nymphs frolicking in the fields. And for goodness’ sake, do not talk like some early-twentieth-century English fop, describing the bottles you decide against as guttersnipes. Never mention Bacchus. Ever.
That’s from Alan Richman’s tips for ordering wine at dinner.
- Roger Federer is William Tell
- New album from The Bad Plus. And here are the places I need to be this fall.
- Basque wine.
- Charles Keating’s pornography research
- Punk math.
The data on first vs. second serve win frequency cannot be taken at face value because of selection problems that bias against second serves. The general idea is that first serves always happen but second serves happen only when first serves miss. The fact that the first serve missed is information that at this moment serving is harder than usual. In practice this can be true for a number of reasons: windy conditions, it is late in the match, or the server is just having a bad streak. In light of this, we can’t conclude from the raw data that professional tennis players are using sub-optimal strategy on second serves.
To get a better comparison we need an identification strategy: some random condition that determines whether the next serve will be a first or second serve. We would restrict our data set to those random selections. Sounds hopeless?
When a first serve hits the net and goes in it is a “let” and the next serve is again a first serve. But if it goes out then it is a fault. The impact with the net introduces the desired randomness, especially when the ball hits the tape and bounces up. Conditional on hitting the tape, whether it lands in or out can be considered statistically independent of the server’s current mental state, the wind conditions, and the stage of the game. These are the ingredients for a “natural experiment.”
Here is Sandeep’s post on the data discussed in the New York Times about winning percentages on first and second serves in tennis.There are a few players who win with higher frequency on either the first or second serve and this is a puzzle. Daniel Khaneman even gets drawn into it. (To be precise, we are calculating the probability she wins on the first serve and comparing that to the probability she wins conditional on getting to her second serve. At least that is the relevant comparison, this is not made clear in the article. Also I agree with Sandeep that the opponent must be taken into consideration but there is a lot we can say about the individual decision problem. See also Eilon Solan.)
And the question persists: would players have a better chance of winning the point, even after factoring in the sure rise in double faults, by going for it again on the second serve — in essence, hitting two first serves?
But this is the wrong way of phrasing the question and in fact by theory alone, without any data (and definitely no psychology), we can prove that most players do not want to hit two first serves.
One thing is crystal clear, your second serve should be your very best. To formalize this, let’s model the variety of serves in a given player’s arsenal. For our purposes it is enough to describe a serve by two numbers. Let 
Your second serve should be the one that has the highest
But it’s a jumped-to conclusion that this means your second serve should be as good as your first serve. Because your first serve should typically be worse!
On your first serve it’s not just
So, you want in your arsenal a serve which has a lower
But at the margin, if you can reduce
So when Vanderbilt tennis coach Bill Tym says
“It’s an insidious disease of backing off the second serve after they miss the first serve,” said Tym, who thinks that players should simply make a tiny adjustment in their serves after missing rather than perform an alternate service motion meant mostly to get the ball in play. “They are at the mercy of their own making.”
he might be just thinking about it backwards. The second serve is their best serve, but nevertheless it is a “backing-off” from their first serve because their first serve is (intentionally) excessively risky.
Statistically, the implications of this strategy are
- The winning percentage on first serves should be lower than on second serves.
- First serves go in less often than second serves.
- Conditional on a serve going in, the winning percentage on the first serve should be higher than on second serves.
The second and third are certainly true in practice. And these refute the idea that the second serve should use the same technique as the first serve as suggested by the Vanderbilt coach. The first is true for most servers sampled in the NY Times piece.
Twenty years ago, David Kaplan of the Case Western Reserve University had a manuscript rejected, and with it came what he calls a “ridiculous” comment. “The comment was essentially that I should do an x-ray crystallography of the molecule before my study could be published,” he recalls, but the study was not about structure. The x-ray crystallography results, therefore, “had nothing to do with that,” he says. To him, the reviewer was making a completely unreasonable request to find an excuse to reject the paper.
The article surveys the usual problems with peer review and is worth a read. I recognize all of the problems and they are real but I am less bothered than most. We don’t really need journals for dissemination anymore so the only function they serve is peer-review. The slowness of the process is not so big a deal anymore. (Unless you are up for tenure this year of course.)
Also, it’s true that reviewers are often just looking for excuses to reject a paper. But that is mainly because they feel obliged to give some reason to justify their decision. In many cases bad papers are like pornography. You know them when you see them. Few referees are willing to write “I reject this paper because it’s not a good paper,” so they have to write something else. To the extent that this is a failure it’s because the effort in reading through the paper looking for an excuse could be more productively spent elsewhere.
Hard Hat Heave: Lance Fortnow.
Abdelbasset al-Megrahi. He was the Lockerbie bomber who was released from a Scottish prison because he had just a few weeks to live. As it turns out, he lived for a year and may live for many more.
The public outrage has an uncomfortable edge to it.
Clinton yesterday implied strongly that Megrahi’s release, and his continued survival long beyond the three months predicted by Scottish ministers, meant justice for the families of the dead had been denied.
Let’s get this straight. We agree on the following ranking of outcomes, from best to worst.
- Alive and in prison
- Alive 1 year after being released from prison.
- Dead 1 week after being released from prison.
But its easy to get confused because, conditional on learning that he was going to be alive 1 year later, we see that releasing him was a mistake. Because we assumed that he was going to die in a week so that 1 and 2 were not feasible. Now that we see that 1 was actually feasible we are outraged that he is alive. When we feel outraged that he is alive we start to think that means that we wish he was dead. But alive/dead is just the signal that we made a mistake. What we really wish is that we hadn’t made the mistake of setting him free.
Parallel paths meet and end on one astounding episode of The Price is Right. Beautiful writing.
Ted says that when he went back for the afternoon taping, the producers moved him to a part of the studio where the contestants couldn’t see him. He says that he has heard “through unofficial channels” that he has been banned from the Bob Barker Studio, the way casinos have started asking Terry not to play blackjack inside their walls again. That Kathy Greco gave him a “Sicilian death stare” after the show, and that nobody ever needs a three-digit PIN. That according to his database, the Big Green Egg had appeared on the show only twice — before Terry and Linda began recording it — and that it was $900 before it was $1,175. That so many contestants — not just Terry — had won that day because they had listened to him. That his only mistake came when Terry played Switch?, because Ted didn’t realize there were two bikes, and he thought that a terabyte sounded like a lot of memory. That he was edited out of the show when it aired, that he can be seen only once, shaking his head when the prize is a Burberry coat, a prize that had never before appeared on the show. Otherwise, he would have known how much it was worth, the way he knew that a Berkline Contemporary Rock-a-Lounger was worth $599, and Brandon’s Ducane gas grill was worth $1,554, and Sharon’s car was worth $18,546. And for all that knowledge, for all his devotion, Bob Barker had called him a Loyal Friend and True, and Drew Carey called him that guy in the audience.
Deerstalker display: The Browser.
In golf:
How big a deal is luck on the golf course? On average, tournament winners are the beneficiaries of 9.6 strokes of good luck. Tiger Woods’ superior putting, you’ll recall, gives him a three-stroke advantage per tournament. Good luck is potentially three times more important. When Connolly and Rendleman looked at the tournament results, they found that (with extremely few exceptions) the top 20 finishers benefitted from some degree of luck. They played better than predicted. So, in order for a golfer to win, he has to both play well and get lucky.
I don’t understand the statistics enough to evaluate this paper. Apparently they call “luck” the residual in some estimate of “skill.” What I don’t understand is how such luck can be distinguished from unexpectedly good performance. If a schlub wins a big tournament and then returns to being a schlub, is that automatically luck?
Imagine the game: you and your partner are holding opposite ends of a rope which has a ribbon hanging from the middle of it. Your goal is to keep the ribbon dangling above a certain point marked on the ground.
This game is the Tug of Peace. Unlike a tug of war, you do not want to pull harder than your partner. In fact you want to pull exactly as hard as she pulls.
That shouldn’t be too difficult. But what if you feel that she is starting to tug a little harder than at first and the ribbon starts to move away from you. You will tug back to get it back in line.
But now she feels you tugging. If she responds, it could easily escalate into an equilibrium in which each of you tugs hard in order to counteract the other’s hard tugging.
This is metaphor for many relationship dysfunctions. For no reason other than strategic uncertainty you get locked into a tug of peace in which each party is working hard to keep the relationship in balance.
There is an even starker game-theoretic metaphor. Suppose that you choose simultaneously how hard you will tug and your choice is irreversible once the tugging begins. You never know how cooperative your partner is, and so suppose there is a tiny chance that she wants the ribbon just a little bit on her side of the mark.
Ideally you would both like to tug with minimal effort just to keep the ribbon elevated. But since there is a small probability she will tug harder than that you will tug just a little harder than that too to get the ribbon centered “on average.” Now, she knows this. And whether or not she is cooperative she will anticipate your adjustment and tug a little harder herself. But then you will tug all the harder. And so on.
This little bit of incomplete information causes you both to tug as hard as you can.
Doctors are skeptical. But they should be. When I approach a doctor with symptoms he has no way of knowing how many other doctors have already examined me. On average that number is larger than zero. So the doctor should factor in the wisdom of at least that fractional number of doctors who have already given me a clean bill of health. My symptoms have to be especially bad to compensate for that automatic positive signal.
I don’t mind rejection but a lot of people do.
When I want to ask someone to join me for a coffee or lunch I always send email. And its all about rejection even though I don’t mind rejection so much. The reason is that nobody can know for sure how I deal with rejection. And almost everybody hates to reject someone who might have their feelings hurt.
If I ask face-to-face I put my friend in an awkward position. Because every second she pauses to think about it is an incremental rejection. Because while the delay could be just because she is thinking about scheduling, it also could be that she is searching for excuses to get out of it.
These considerations combined with her good graces mean that she feels pressure to say yes and to say yes quickly. Even if she really does want to go. I would rather she have the opportunity to consider it fully and I would rather not make her feel uncomfortable.
Email adds some welcome noise to the transaction. It is never common knowledge exactly when she is able to read my email invitation. If she gets it right away she can comfortably consider the offer and her schedule and get back to me on her own time. And she knows that I know that… that I have no way of knowing how much time it took her to decide.
Game theorists can’t stop trashing email as a coordination device but that’s because we always think that common knowledge is desirable. But when psychology is involved it is more often that we want to destroy common knowledge.
Locavores advocate paying attention to the distance your food traveled before it reached your table. They want you to be aware of the social cost of the the tomatoes you buy. Steven Landsburg ridicules this kind of micro-mindedness as follows:
You should. You should care about all those costs. And here are some other things you should care about: How many grapes were sacrificed by growing that California tomato in a place where there might have been a vineyard? How many morning commutes are increased, and by how much, because that New York greenhouse displaces a conveniently located housing development? What useful tasks could those California workers perform if they weren’t busy growing tomatoes? What about the New York workers? What alternative uses were there for the fertilizers and the farming equipment — or better yet, the resources that went into producing those fertilizers and farming equipment — in each location?
And he helpfully points out that accounting for all of this is unnecessary because these costs are already all summarized by the price of the tomato.
But even if markets are perfectly competitive, the marginal social cost of a tomato equals the price plus the under-priced cost of the environmental damage from the fuels used in transportation. Any given locavore has his own private belief about the size of that gap. So a locavore wants to know how much energy was used in order to calculate the total gap. And locavore advocates are doing precisely the right thing by presenting that information.
On the other hand, this guy, the guy who Landsburg was actually ridiculing, is spot on when he points out they often present distorted information.
I think all sides should compromise. We should build a very tiny mosque. And it should have wings. The Ground Zero Mosquito. But instead of sucking your blood it would enrich your blood with the dual nectars of Retribution and Tolerance. We will all make pilgrimages to the site and our children will line up to be kissed by it. And one by one they will announce their mosquito-induced blessings like “I want to vanquish all of those who do Evil to our homeland but still I Love them all the same.” Thank you Ground Zero Mosquito!
Mark Kleiman proposes making it legal to grow, share, and consume cannabis, but not to sell it.
To the consumer, developing a bad habit is bad news. To the marketing executive, it’s the whole point of the exercise. For any potentially addictive commodity or activity, the minority that gets stuck with a bad habit consumes the majority of the product. So the entire marketing effort is devoted to cultivating and maintaining the people whose use is a problem to them and a gold mine to the industry.
Take alcohol, for example. Divide the population into deciles by annual drinking volume. The top decile starts at four drinks a day, averaged year-round. That group consumes half of all the alcohol sold. The next decile does from two to four drinks a day. Those folks sop up the next thirty percent. Casual drinkers – people who have two drinks a day or less – take up only 20% of the total volume. The booze companies cannot afford to have their customers “drink in moderation.”
Some questions come to mind:
- Would it be legal to sell the seeds? If not, how could anybody get them? If so, has the problem (assuming there is one) really been solved?
- The economics of the problem boil down to which market structure minimizes the private incentive to boost demand, say via marketing. If we outlawed the sale of tobacco but allowed the sale of grow-your-own-tobacco kits would we see more or less marketing? The less competitive the market the more each individual seller gains from a boost in generic demand. Should we expect the market for growing kits to be more competitive than the market for the final product?
- If we are going to ban something, why not advertising? Seems more direct and arguably a Pareto improvement if advertising acts as rent-seeking between producers and creates “artificial” demand as the premise of the policy seems to be.
Subjects in an experiment were shown pictures of people of the opposite sex and were asked to rate their attractiveness. Some subjects were first subliminally flashed pictures of their opposite-sex parent. For other subjects the pictures they were rating were actually a composite image made from pictures of the subject himself. Both of these induced higher attractiveness ratings.
Finally, in a third treatment the subjects were falsely (!) told that the images they were seeing were partly morphed from pictures of themselves. This reduced the attractiveness ratings.
Here is a link to the paper.
First watch the video below. The dark haired guy, Booth, has just made a big bet. He is claiming to have three-of-a-kind (fours). If he does he would win the hand, but he might be bluffing. The other guy, Lingren, has to decide whether to call the bet and he does something unexpected: he asks Booth to show him one of his cards:
The strategic subtext is this: if Booth has the third four then he wants Lingren to call. If not, he wants him to fold. Implicitly, Lingren is offering the following mechanism: if you do have the third four then you won’t want me to know it because I would then fold. So show me a card, and if it’s not a four I will call you.
What is left unsaid is what Lingren would do if Booth declined to show a card. The spirit of the mechanism is that showing a card is the price Booth has to pay to have his bet called. So the suggestion is that Lingren would fold if Booth is not forthcoming because that would signal that he is hiding his strong hand.
But in fact this can’t be part of the deal because it would imply exactly the opposite of Lingren’s expectations. Booth, knowing that it would get Lingren to fold, would in fact hide his cards when he is bluffing and show a card when he actually has the three-or-a-kind (because then he gets a 50% chance of having his bet called rather than a 100% chance of Lingren folding.)
So what exactly should happen in this situation? And did Booth really play like a genius? Leave your analysis in the comments.
Visor visit: the ever-durable Presh Talwalker.
Here is the abstract from a paper by Matthew Pearson and Burkhard Schipper:
In an experiment using two-bidder first-price sealed bid auctions with symmetric independent private values, we collected information on the female participants’ menstrual cycles. We find that women bid significantly higher than men in their menstrual and premenstrual phase but do not bid significantly different in other phases of the menstrual cycle. We suggest an evolutionary hypothesis according to which women are genetically predisposed by hormones to generally behave more riskily during their fertile phase of their menstrual cycle in order to increase the probability of conception, quality of offspring, and genetic variety.
Believe it or not, this contributes to a growing literature.
Tyler links to this paper which asks the question “Do Minimum Parking Requirements Force Developers to Provide More Parking than Privately Optimal?” and answers in the affirmative.
I would think that is exactly the purpose of minimum parking requirements, especially in suburbs. Take as given that residents will resist the installation of parking meters on their streets or other parking restrictions. Then parking spaces in strip malls have positive externalities for homeowners because they reduce spillover parking in the neighborhoods. The privately optimal number of parking spaces is therefore too low and mandatory minimums are aimed at correcting that. The question we are interested in is whether mandatory minimums increase parking space beyond what is socially optimal.
This guy thinks so.
There it is: Zynga’s dirty technique for making its $500 million. It ropes players into the game with the promise that absolutely anyone can play. It will even float you coins the first time you run out, not unlike the casino that gives a high-roller luxury accommodations in anticipation of making back the house’s stake. It dangles the prospect of a bigger, prettier, better farm; as the game loads, you’re faced with idyllic images of well-off farms, not unlike the glossy ads for high-end residences. But it’s nearly impossible to get some of those goods without ponying up a buck or two here and there. When Zynga’s got a user base of 61 million digital farmers, it’s easy enough to make ends meet, to say the least.
The argument seems to be: it sucks your time, it gets your friends hooked too, it’s stupid. Like TV, reading blogs, talking on the phone, etc. I do recommend reading the article though because its a fine case study in how to try and substitute a logical argument in place of “Why do people have tastes that are so different from mine?” (Full disclosure: some people very close to me play Zynga games. I think Zynga games are really stupid.)
Ten Gallon Greeting: GeekPress.
Even as parking meter technology improves to handle credit cards and flexible pricing, one relic remains: pay-in-advance. You have to commit to a period of time and pay the meter at the beginning of that time interval. You are fined if you don’t move before your pre-specified time expires.
Wouldn’t it be better if you paid only at the end and only for the amount of time you actually occupied the parking space? This is easily implemented in smart parking meters: you swipe your card when you park and you swipe it again when to pay when you are done. To prevent scofflaws, if you never swipe the second time the rest of the day’s parking is on you (sorta like the lost ticket fee in parking lots), and if you are parked in a meter that wasn’t swiped the first time you get a ticket.
Holding fixed the meter rate and demand for parking such a switch would lower your total parking bill and hence revenues. To show this you could try writing down a model where the parker has a probability distribution over waiting times and chooses optimally his parking time, but there is a simpler argument. With pay-in-advance two things can happen: either you pay for more time than you actually use or you get a ticket and pay a fine!
But switching to pay-as-you-leave parking must surely increase overall demand for parking and may even increase individual parking spells so the net effect on revenue is not so clear.
While we are on the subject I liked Tyler Cowen’s bit about free parking. But I think that there is a second-best logic that justifies mandatory free parking in the suburbs. Homeowners want their streets free of cars (but still the option to park on the street when necessary) and they don’t want parking meters on their sidewalks. This makes strip-mall parking a public good. The private incentive of commercial developers is to provide too little parking.
- There is diversity in sexual preference (same-sex or opposite-sex) but most people are at corner solutions.
- The ideal temperature in San Diego is 66F. You will worry in the morning when its cloudy but once the marine layer burns off the sun by itself will keep you warm. The air will keep you cool.
- It is not possible to define ‘fitness’ in a way that doesn’t make “survival of the fittest” into a tautology.
- People who argue that tenure should disappear should begin by saying what they think the market failure is that prevents that from happening by itself.
Here is a theory of why placebos work. I don’t claim that it is original, it seems natural enough that I am surely not the first to suggest it. But I don’t think I have heard it before.
Getting better requires an investment by the body, by the immune system say. The investment is costly: it diverts resources in the body, and it is risky: it can succeed or fail. But the investment is complementary with externally induced conditions, i.e. medicine. Meaning that the probability of success is higher when the medicine is present.
Now the body has evolved to have a sense of when the risk is worth the cost, and only then does it undertake the investment. Being sick means either that the investment was tried and it failed or that the body decided it wasn’t worth taking the risk (yet! the body has evolved to understand option value.)
Giving a placebo tricks the body into thinking that conditions have changed such that the investment is now worth it. This is of course bad in that conditions have not changed and the body is tricked into taking an unfavorable gamble. Still, the gamble succeeds with positive probability (just too low a probability for it to be profitable on average) and in that case the patient gets better due to the placebo effect.
The empirical implication is that patients who receive placebos do get better with positive probability, but they also get worse with positive probability and they are worse off on average than patients who received no treatment at all (didn’t see any doctor, weren’t part of the study.) I don’t know if these types of controls are present in typical trials.
I liked this post from Dan Reeves.
Remember, if it’s in the news don’t worry about it. The very definition of news is “something that almost never happens.” When something is so common that it’s no longer news — car crashes, domestic violence — that’s when you should worry about it.
The truth of that[1] hit home recently when I saw a news feature on the abduction of a four-year-old girl from her front yard in Missouri. Candlelight vigils, nation-wide amber alert, police blockades where every single car was stopped and questioned, FBI agents swarming the house. I think the expected reaction from parents is “oh my god, I need to be so vigilant, even in my own front yard!” My reaction was the opposite: Wow, this sort of thing really does essentially never happen. Let the kids run free!
While I generally have similar reactions, it is easy to take this too far. The weather is the prime counter-example. Valuable minutes of every nightly newscast in Southern California are devoted to repeating the same weather-mantra “Late Night and Early Morning Low Clouds and Fog” which indeed happens every single day.
Also, your interpretation of news depends on your implicit model of competition between news sources. A perfectly plausible model would be consistent with abductions happening every day but being reported only once in a while because we have a taste for variety in our stories of tragedy.
In San Francisco no less
San Francisco has been working on making parking “smarter” for quite a while now, and it’s just recently taken another big step in that direction by starting to replace over 5,000 older parking meters with the snazzy new model pictured above. Those will not only let you pay with a credit or debit card (and soon a special SFMTA card), but automatically adjust parking rates based on supply and demand, which means you could pay anywhere from $0.25 to $6.00 an hour depending on how many free spaces there are. Those rates are determined with the aid of some sensors that keep a constant watch on parking spaces, which also means you’ll be able to check for free spaces in an area on your phone or your computer before you even leave the house.
fedora flip: rob jefferson.
How to allocate an indivisible object to one of three children, it’s a parent’s daily mechanism design problem. Today I used the first-response mechanism. “Who wants X?” And whoever says “me!” first gets it.
This dimension of screening, response time, is absent from most theoretical treatments. While in principle it can be modeled, it won’t arise in conventional models because “rational” agents take no time to decide what they want.
But the idea behind using it in practice is that the quicker you can commit yourself the more likely it is you value it a lot. Of course it doesn’t work with “who wants ice cream?” But it does make sense when its “We’ve got 3 popsicles, who wants the blue one?” We are aiming at efficiency here since fairness is either moot (because any allocation is going to leave two out in the cold) or a part of a long-run scheme whereby each child wins with equal frequency asymptotically.
It’s not without its problems.
- Free disposal is hard to prevent. Eventually the precocious child figures out to shout first and think later, reneging if she realizes she doesn’t want it.
- There’s also ex-post negotiation. You might think that this can only lead to Pareto improvements but not so fast. Child #1 can “strongly encourage” child #2 to hand over the goodies. A trade of goods for “security” is not necessarily Pareto improving when the incentives are fully accounted for.
- It prevents efficient combinatorial allocation when there are externalities and/or complementarities. Such as, “who’s going in Mommy’s car?” A too-quick “me!” invites a version of the exposure problem if child #3 follows suit.
Still, it has its place in a parent’s repertoire of mechanisms.
- The art of 1-star Amazon reviews. I have been looking for a new hobby…
- Stanley Milgram, the TV show, starring William Shatner.
- History of the laugh track.
- Massaging your oppossum properly.
This is in fact an excellent introduction to game theory full stop. It covers strategic and extensive games, complete and incomplete information, sequential rationality, etc. Very nicely. And then on page 64 it gets really interesting, applying evolutionary game theory to pragmatics, a field in linguistics concerned with the contextual meaning of language.
After a disaster happens the post-mortem investigation invariably turns up evidence of early warning signs that weren’t acted upon. There is a natural tendency for an observer to “second-guess,” to project his knowledge of what happened ex post into the information of the decision-maker ex ante. The effects are studied in this paper by Kristof Madarasz.
To illustrate the consequences of such exaggeration, consider a medical example. A radiologist recommends a treatment based on a noisy radiograph. Suppose radiologists differ in ability; the best ones hardly ever miss a tumor when its visible on the X-ray, bad ones often do. After the treatment is adopted, an evaluator reviews the case to learn about the radiologist’s competence. By observing outcomes, evaluators naturally have access to information that was not available ex-ante; in that interim medical outcomes are realized and new X-rays might have been ordered. A biased evaluator thinks as if such ex-post information had also been available ex-ante. A small tumor is typically difficult to spot on an initial X-ray, but once the location of a major tumor is known, all radiologists have a much betterchance of finding the small one on the original X-ray. In this manner, by projecting information, the evaluator becomes too surprised observing a failure and interprets success too much to be the norm. It follows that she underestimates the radiologistís competence on average.
The paper studies how a decision-maker who anticipates this effect practices “defensive” information production ex ante, for example being too quick to carry out additional tests that substitute for the evaluator’s information (a biopsy in the medical example) and too reluctant to carry out tests that magnify it.
A tip of the boss of the plains to Nageeb Ali.
I thank Presh Talwalker for the pointer. Pretty soon I won’t have to do any teaching, I’ll just play YouTube clips for 90 minutes, pass out the chocolate and send them on their way.
Cheap Talk 