Some Trick: Thirteen Stories

My Heart Belongs to Bertie[1]

Let’s take 2 people, A and B. A is a heroin addict. B’s idea of a narcotic is Earl Grey tea.

We take a randomly selected infant, toss a coin, and allocate it according to the result of the coin toss: Heads A, Tails B.

We repeat the procedure.

In 10 trials, the likeliest number of Heads is 5. If we run sequences of 10 trials, though, we shall sometimes have fewer than 5, sometimes more, and the distribution of Heads will follow the familiar Gaussian curve:

In 20 trials, the likeliest number of “successes” is 10 — that is, the Gaussian curve shifts to the right:

The two PDFs lodged awkwardly upside down at Peter’s edge of the table, jamming up against napkin dispenser, sugar pourer, salt & pepper shakers, red plastic bottle of ketchup, yellow plastic bottle of mustard.

When one gives a lecture or seminar, one does not have to do battle with condiments. He had not prepared for the contingency.

If we repeat the procedure on a daily basis, said Peter, ineffectually shifting the PDFs to give Jim a better view, the infant’s exposure to misallocation will tend to be rectified with relative frequency — though, on the other hand, the infant will never be guaranteed enjoyment of a good draw for very long. If the procedure is conducted weekly, more hangs on the result; if monthly, quarterly, yearly, more still.

He pressed the palms of his hands to his eyes.

Peter had written a book of robot tales with a happy beginning which had made, as it turned out, what seemed a lot of money, and yet not enough money to mitigate contractual relations with persons who had professed to love it yet sought to remove references to e^iπ. He had explained that discovering e^iπ at the age of 9 was the only thing that had kept him from suicide and been brushed indulgently aside.

The money had appeared to make it worth the while of Jim (said to be a “hot shot literary agent”) to represent Peter’s interests on a second book of robot tales. It had proved more complicated than had first appeared.

Jim did not know the secret of a happy beginning: My parents died when I was born.

Peter had made a special trip to New York to explain the binomial distribution in person to Jim. He wanted to reduce the likelihood of contractual obligation to persons like the persons he knew too well.

Now they sat in a booth in a diner.

Peter had suggested meeting in Jim’s flat, on the assumption that this would minimise the number of people in the vicinity mistakable for Jim.

Jim had held out for the diner.

Peter had suggested that Jim wear a yellow sweater for ease of identification.

Jim had not taken up the suggestion.

Jim was wearing a brown pullover and brown trousers. His hair and eyes were brown. For the whole of the walk from office to diner Peter had been terrified of losing sight of Jim in the crowd and then failing to recognise him again. The worry now was that if, for instance, Jim went to the men’s room and someone else came and sat down, Peter might fail to notice the difference. That was one worry, and another worry was that the name “Jim” might slip his mind, as names generally did.

The surface of the table was taken up with all sorts of paraphernalia superfluous, not to say impedimental, to ratiocination. Peter had done what he could to maximise the surface area available for display of illustrative materials; he had waived the offer of lunch. Jim had ordered something or other. He had exchanged badinage with the waitress.

We posit daily reassignment, Peter went doggedly on. As our number of trials approaches 100, the number of days of infant allocation to A comes to cluster, of course, around 50:

He had been up all night running variations in R on

par(mfrow=c(3,1))

barplot(dbinom(0:100,10,.5))

barplot(dbinom(0:100,20,.5))

barplot(dbinom(0:100,30,.5))

—all this before catching the 5 am AirBus to Gatwick from Gloucester Green, Oxford’s answer to Port Authority. In his haste he had, he realised, forgotten to label the x axis.

He would have liked, at this point, to throw in the towel, or rather retreat to his hotel and address the issue of the unlabelled x axis. But they were here. One must make the most of it.

Jim squirted ketchup placidly on crinkled fries. It seemed unlikely that a properly labelled x axis would have made the thing usefully clearer.

Perhaps Jim was one of the lucky ones. If he was an orphan he probably had all kinds of rosy notions.

But in fact, of course, persevered Peter — he had really been talking, understand, for only a minute or so, in a lecture he could have talked uninterrupted for an hour, with hand-outs, a blackboard — 50% of adults are not heroin addicts, so a model with a 50:50 chance of drawing one isn’t a very good fit. Suppose we try to get a feel for uneven odds.

You see, I suppose, that we can imagine a pool of 10 parents with 1 heroin addict and 9 tea-drinkers, or a pool with 9 heroin addicts and 1 tea-drinker. Instead of tossing a coin we draw, perhaps, a ball from an urn (containing, as it might be, 1 black ball 9 white or 9 black balls 1 white) and replace it before the next draw. In the first instance, over one hundred trials, the number of As — that is, draws of a heroin addict — would cluster around 10,

whereas in the second instance it would cluster around 90.

So the point is, on our model of daily draws, the likelihood of drawing a heroin addict 90 days out of 100 arises only with a pool of 9 heroin addicts and 1 tea-drinker: it does not arise when only 1 in 10 possible parents is a heroin addict, and by extension would be even less likely in a population where only 1 in 100,000 was an addict. (According to Heyman, in a recent national, that is to say American, survey, there were in fact about 3.4 opiate addicts per thousand persons and about 10.8 nonaddicted heavy users.) Whereas, under the current system, even if only 1 in 100,000 is an addict, the fact that all depends on a single draw, the accident of birth, means that a child born to such a person is assigned 365 days out of 365 until the age of majority.

The point is simply, said Peter, that the family is a barbarous institution. One is, for the most part, stuck with the luck of a single draw.

Oh, families, said Jim. I know, I know, I know.

He took a healthy swallow from his glass of Diet Coke and set it down.

Look, said Jim. This is fascinating, but it’s way over my head. I don’t really get it all, but I don’t need to get it.

Jim put his plate unhurriedly to one side, rested his forearms on the table and laced his fingers together, with a quiet mastery of the space that — that is, if Jim had been the one who had wanted to present PDFs displaying the binomial distribution, an army of ketchup bottles would not have stood in his way.

You’re a very brilliant guy, said Jim. You’re the genius. You found a way to capture the imagination of a lot of kids who would not normally go for this stuff. You captured the imagination of a lot of adults who wouldn’t normally read books for kids. So if you want to talk about the odds, maybe I’m in a better position to know the kinds of odds you were beating. I’m in a better position to know why this is very exciting to a lot of other people who understand the kind of odds you were beating. What I can say is that a lot of people are very excited by your work; I know a lot of editors who would love to see a new book. So this would be a very good time to send something out, have an auction. Bottom line, if we don’t get a significant six-figure deal it’s time for me to take up knitting, and if we play our cards right we could be talking low seven.

Jim had already explained, by e-mail, that the option on Peter’s second book, held by the lucky publisher of Peter’s first book (advance: £5,000; sales: 500,000), was not an obstacle. The book must be submitted first to the lucky publisher, but if their offer was unsatisfactory Peter (or, rather, Peter’s agent on his behalf) was entitled to submit the book elsewhere.

This was, obviously, an improvement on our barbarous domestic arrangements: a parent does not have an option on a child, and the terms of the relationship do not come up for renegotiation. Peter’s position — and the reason for this ill-starred trip to New York — was that the objection to the lucky publisher was not financial. The objection was that it had done its best to dilute elements of the book likely to appeal to the underserved numerate, and to put off the innumerate who were already, one might have thought, amply provided for, an example being the hideous war of attrition it had waged over inclusion of e^iπ.

The fact that Jim could unashamedly admit to finding a perfectly simple explanation of the binomial distribution over his head, that he could unblushingly dismiss it as the province of genius, only went to show how deep-seated innumeracy actually is in our benighted culture. (If an agent, a ‘hot shot’, who notionally represents a client’s financial interests, can be functionally innumerate — !!!) But how could he possibly do battle with ignorance if he himself —

By the time a boy is 10 he has spent 3,652 days under governance of the allocation of a single draw. There’s nothing to be done about it. All the more reason not to enter into contractual relations lightly.

It’s unreasonable, perhaps, to expect someone like Jim to understand the full horror.

Exactly, said Peter. (Meet the man on his own ground.) Exactly. This is the whole beauty of business relations: we leave barbarity behind. Let us suppose I know about Merovingian kings; I wish either to work with someone with comparable knowledge of the Merovingians or, perhaps, to work with someone whose knowledge of the Carolingians supplements my relative ignorance of the period. We see at once that I should be highly unlikely to find a match leaving the matter to chance, but the invisible hand is my friend: I can pay for the information or, aliter visum, the value of the match enables all concerned to maximise profits regardless of whether money changes hands. Let us say no one with relevant knowledge can be found. Perhaps someone happens to know about the Dutch Tulip Bubble, and I discover in myself a hitherto unguessed-at interest in the Dutch Tulip Bubble. I can order my preferences, you know, in a way which is wholly out of the question in a family setting. As it happens, I have written a second book on robots and would like an editor with relevant expertise; if none can be found with expertise relevant to the book in hand, I would happily write a book relevant to such expertise as can be had. I rely on you to brief me so that I can make a rational decision.

Jim said he didn’t work that way. Look, he said, we could waste a lot of time talking about editors. We’re only interested in the ones who are willing to buy the book we have to sell. Once we have a list of serious contenders we can definitely talk about who would be best for the book.

As a child Peter had not been unduly, he wouldn’t have said, troubled by the shortcomings of his parents per se. The thing that had bothered him was the fact that all other adults colluded in placing him in the largely unchecked power of these individuals. All adults, even the apparently decent ones, were in collusion with evil.

He had worked it out when he was, perhaps, 7 and never forgotten. That was why he was able to write for children. He was 35, a bad age.

Peter said, Please.

He tried to think of the sort of thing Americans say.

He said, It would mean a lot to me to work with someone who admired Bertrand Russell.

He said, It would really mean a lot to me.

The statement seemed, if not meaningless, then uselessly imprecise.

(The first book had made all this money. Why could he not use the money to buy what he wanted? Was that not the general point of having money in the first place?)

He said, I’d be happy to switch the percentages round if that would help. You’d be very welcome to take an 85% commission.

This was undoubtedly precise but was perhaps not the sort of thing Americans say. Jim said he was happy with the normal 15% commission.

Peter pressed the palms of his hands to his eyes.

Russell, he said presently.

Russell was born in 1872. His father, Viscount Amberley, was an atheist and Utilitarian; he asked John Stuart Mill to be godfather to the child. Russell’s mother died when he was 2, his father when he was 4. His father’s will appointed two atheists to be guardians to Russell and his brother, and stipulated that the children be raised as agnostics. Russell’s grandmother, the Countess Russell, overturned the will and won guardianship of the children. She raised Russell on strictly religious principles. At the age of 11 he was introduced to geometry by his brother Frank; he said he had not imagined there was anything so beautiful in the world. He said later that only the desire to know more about mathematics restrained him from suicide.

I do understand, Peter said wearily, that we can’t reasonably expect to find an editor of children’s books with mathematical, scientific, or even philosophical training. But Russell, after all, was a great populariser; it’s surely not beyond the realm of probability that a general reader should be familiar with his popular work, work written for a general audience. The thing that matters is not, ultimately, an understanding of number theory, or the structure of the atom, or the semantic tradition, but an unswerving commitment to the pursuit of truth. I should be happy to forgo 70% of the revenue from a book to avoid entrusting it to a person to whom this is perfectly indifferent; one has to be particularly scrupulous in these matters when writing for children. That is the overriding interest which I hope to persuade you to represent. As ours is a business relationship, a financial incentive cannot, it seems to me, be offensive as it would be among mathematicians, scientists or philosophers. It is entirely reasonable for me to determine my own ends and offer financial compensation to you for the inconvenience of promoting them.

Jim made a number of friendly American remarks. It was by no means clear that the offer of a financial inducement within the context of a business relationship had not been perceived as offensive. It was, unfortunately, painfully obvious that he had not warmed to the style of social interaction which is a robot’s principal source of appeal. The business of adapting a robot to American manners is, of course, an engineering feat of considerable ambition.

Peter said, Will you excuse me a minute?

He stood up and left the diner.

In Oxford he was able to smoke at the Union, to which his father had given him a life membership in his freshman year.

He drifted to the lee of a wall. He took out a packet of Dunhills and lit one.

As a child he had had five imaginary robots as friends.

The robots had stopped talking to him somewhere during the protracted battles over e^iπ. It was a happy accident that a second book had been finished before they stopped talking.

It seemed unlikely that intervention from Jim would bring the robots back.

Perhaps Jim was an orphan.

Peter stubbed out his Dunhill on the sole of his shoe. He slipped it into a pocket. He pressed the palms of his hands to his eyes.

At his shoulder he heard a voice he had not heard for some time.

The robot pointed out pleasantly, dispassionately, that it would, certainly, be political suicide for a legislator to attempt to introduce an aleatory element into the allocation of minors, but that this was far from exhausting possible solutions. Research had already demonstrated that autistic children responded well to robot companions. It was only a matter of time before a robot companion was seen as an effective developmental adjunct for every child. From this it was a short step to recognition of the fact that each child would best be served by a complement of robotic aides. Robots could be exchanged, upgraded, used in different combinations with none of the social constraints affecting human subjects.

That’s true, said Peter.

He propped his right shoulder against the wall, right leg loosely bent, left leg perhaps 20 degrees from the perpendicular, left hand shoved in a pocket.

The robot observed that, as all children would be guaranteed a minimum of one rational companion, exposure to the variable rationality of the human beings in whose charge a child found itself would be significantly limited in the damage it could do.

Peter did not say anything, for he was listening attentively. His gaze had fallen to the ground as the least troublesome place for it.

The robot continued to speak. It was restful in the way that robots are restful.

When a human being develops an argument, when a human being attempts not only to think but to speak with precision, he or she is often made to feel that this is a mark of social inadequacy and that there is something comical about it. The younger the human being, the more humorous it becomes. So that humans whose inclination it is to think and speak in this way become self-conscious from an early age, and a kind of minstrelisation creeps in.

It was once the case that so-called minstrel shows were put on for the amusement of white audiences, and for these blacks would black up, put on black-face (!!!!!!!), exaggerating what whites perceived as comically grotesque features of negroid appearance, exchanging dialogue exaggerating what whites perceived as the ignorance and stupidity of the inferior race. In a similar way the rationalist is socialised to mug for the camera, trotting out recondite facts, objecting to logical fallacies, using polysyllabic words in sentences with a high number of dependent clauses, with the quizzical air of one who knows he is amusing the interlocutor by conforming to a fondly held stereotype.

A robot lacks this self-consciousness; one becomes aware of one’s own seeing the robot’s lack. A robot is, in any case, a machine: if it were conscious it would know that machines are not socially stigmatised for sounding like machines. A man who is accustomed to social stigma tends, curiously enough, to be repelled by persons like himself who are tarred with the same social stigma; it is a comfort to talk to a robot, in which rationality carries no stigma.

Peter lit another Dunhill while the robot talked persuasively on.

Ah! the revivifying properties of tobacco! Peter remembered suddenly that in his hour of need, viz. 3 am Greenwich Mean Time, he had fired off an e-mail to Andrew Gelman, Director of Applied Statistics at Columbia University, entreating assistance. If he had not been relatively new to R he would undoubtedly have known what to do; if he had not been catching the AirBus at 5 am he could undoubtedly have worked it out; these are not the circumstances in which one goes to the R Help Forum and exposes one’s ignorance to Brian Ripley, Duncan Murdoch, Gabor Grothendieck, Uwe Ligges, Peter Dalgaard, Deepayan Sarkar and other R supremos. Professor Gelman, on the other hand, had a son who had liked the first robot book; at 3 am the circumstance had seemed to extenuate.

At this very moment a reply to his e-mail might be waiting! He had spotted a Staples and a Kinko’s on 6th Avenue; either, surely, would permit him to see if anything useful had come in. Or for that matter — His laptop was in his hotel room. And his hotel (chosen for proximity to a bar at which one was permitted to smoke) was just off 8th Avenue, which was, he rather thought, just around the corner. (No longer a penniless academic, he had been easily able to afford the rates of the Gansevoort.)

Peter strode purposefully off — nothing like talking to a robot for clearing the head!

Restored, presently, to the comforts of the Gansevoort, he opened his trusty laptop, and what to his wondering eyes should appear but an e-mail from Andrew Gelman, his friend.

n <- c (10, 20, 50, 90, 100)

n.graphs <- length (n)

par (mfrow=c(n.graphs,1), mar=c(2,2,0,1), mgp=c(1.5,.5,0), tck=-.01)

total <- 70

for (i in 1:n.graphs){

barplot (dbinom(0:total,n[i],.5), width=1, space=0, xlim=c(0,total+1), ylim=c(0,.3), xaxs=“i”, yaxs=“i”, yaxt=“n”)

ticks <- seq (0, 200, 10)

axis (1, ticks+.5, ticks)

axis (2, c(0,.1,.2))

text (total-5, .2, paste ("(n = ", n[i], ")", sep=""), cex=1.2)

}

Peter launched R. He typed in the suggested code, and lo! a plot appeared, a beautiful little array of histograms with the x axis labelled!

Glory!

He dashed off an e-mail to Gelman expressing his undying gratitude. Gift horse and all that, he could not quite see why the x axis ended at 70: an odd choice given that, when number of draws n=100, 0–37 and 63–100 are in fact symmetrical gaps — would one not want the symmetry to be visible? (But then he could not quite remember what he had said at the dark hour of 3 am.)

He scrutinized the code, yes, yes, yes, it was a simple matter of changing total to 100, and one might also, perhaps, want to have n in all multiples of 10 up to 100? Hey presto!

There’s an experience that’s common enough. One picks up a book, begins to read. When one looks up 5 hours have passed. One sits in a cold train on a siding; snow falls softly on a stubbled field.

5 hours later Peter found himself in a Korean diner flanked by 5 robots.

One of the robots was talking about Clovis, who ascended the throne at the age of 15.

It’s important to be rational.

Correlation is not causation, no. But what is to be done? What he has to go on is that, after a gap of over a year, all five robots had started talking again after this extraordinarily kind, helpful and above all elegant solution from Andrew Gelman, exactly the sort of assistance he might have hoped to find in a competent editor. But if, for the sake of argument, a book is worth a significant six figures or low seven, and if, for the sake of argument, a book depends in the first instance on being in communication with the robots, we can quantify the value of working with a Gelman-equivalent. But the first robot had spoken after he walked out of the

diner!!!!!!!!! Jim!!!!!!!!! There had been a man named Jim—

Ought he, perhaps, to go rushing back to, oh God, the other

But no, Jim (he was pretty sure it was Jim) would have gone back to his office. Ought he, in all decency, to drop off the correctly labelled chart at the office? Or call, perhaps he should

There was the matter of the briefcase, but it had only contained print-outs of PDFs which were on the laptop, so there was no particular need to retrieve, but

Wait. Wait wait wait wait wait wait wait.

As we were saying, before we were so rudely interrupted, the first robot made its presence known after one had walked out on Jim. We are unhappily not in the position of being able to run randomised blind trials. We can only proceed, with the utmost caution, on the evidence available.

A tentative conclusion is that there are compelling financial, as well as intellectual, reasons to abstain from communication with Jim. (Note that Jim had strictly confined his contribution to the financial element.) There would appear to be compelling financial reasons to communicate with Andrew Gelman (his friend), except that the man is not on the payroll of an agency or publisher. But.

But but but but but.

Surely.

If he understands the matter correctly, the plucky underdog, his first publisher, inveterate enemy that it is of e^iπ, can’t compel him to give them the second book.

Is this not what is meant by leverage?

Can he not, in fact, make any deal conditional on exclusive consultation with his friend (at a suitable fee) or some suitably numerate and computerate substitute?

He rather thinks he can.