## Communicating Numbers

Earlier this week I read this masterful blogpost on Andrew Gelman’s blog (though the post itself is not written by Andrew Gelman – it’s written by Phil Price) about communicating numbers.

Basically the way you communicate a number can give a lot more information “between the lines”. Take the example at the top of the article:

“At the New York Marathon, three of the five fastest runners were wearing our shoes.” I’m sure I’m not the first or last person to have realized that there’s more information there than it seems at first. For one thing, you can be sure that one of those three runners finished fifth: otherwise the ad would have said “three of the four fastest.” Also, it seems almost certain that the two fastest runners were not wearing the shoes, and indeed it probably wasn’t 1-3 or 2-3 either: “The two fastest” and “two of the three fastest” both seem better than “three of the top five.” The principle here is that if you’re trying to make the result sound as impressive as possible, an unintended consequence is that you’re revealing the upper limit.

Incredible. So 3 in 5 means one of them is likely to be 5th. And likely one is fourth as well. Similarly, if you see a company that calls itself a “Fortune 500 company”, it is likely closer to 500 than to 100.

The other, slightly unrelated, example quoted in the article is about Covid-19 spread in outdoor conditions. There is another article that says that “less than 10% of covid-19 transmission that happens indoors”. This is misleading because if you say “less than 10%”, people will assume it’s 9%! The number, apparently, is closer to 0.1%.

There are many more such examples that we encounter in real life. If you write on LinkedIn that you went to a “top 10 ranked B-school”, it means you DID NOT go to a “top 5 ranked B-school”.

Loosely related to this, I’ve got a bit irritated over the last year and a bit in terms of imprecise numerical reporting by the media (related to covid-19). I won’t provide links or quotes here, since what I can remember are mostly by one person and I don’t want to implicate her here (and it’s a systemic problem, not unique to her).

You see reports saying “20000 new cases in Karnataka. A majority of them are from Bangalore”. I’ve seen this kind of a report even when 90% of the cases have been from Bangalore, and that is misleading – when you say “majority”, you instinctively think of “50% + 1”. Another report said “as many as 10000 cases”. Now, the “as many as” phrasing makes it sound like a very large number, but put in context, this 10000 wasn’t really very high.

Communication of numbers is an art that is not very well spread. Nowadays we see lots of courses on “telling stories with data”, “data visualisation”, graphics, etc. but none in terms of communication of sheer numbers itself.

Maybe I should record an episode about this in my forthcoming podcast. If you know who might be a good guest for it, AND can make an introduction, let me know.

## Risk and data

A while back a group of <a large number of scientists> wrote an open letter to the Prime Minister demanding greater data sharing with them. I must say that the letter is written in academic language and the effort to understand it was too much, but in the interest of fairness I’ll put a screenshot that was posted on twitter here.

I don’t know about this clinical and academic data. However, the holding back of one kind of data, in my opinion, has massively (and negatively) impacted people’s mental health and risk calculations.

This is data on mortality and risk. The kind of questions that I expect government data to have answered was:

1. If I get covid-19 (now in the second wave), what is the likelihood that I will die?
2. If my oxygen level drops to 90 (>= 94 is “normal”), what is the likelihood that I will die?
3. If I go to hospital, what is the likelihood I will die?
4. If I go to ICU what is the likelihood I will die?
5. What is the likelihood of a teenager who contracts the virus (and is otherwise in good health) dying of the virus?

And so on. Simple risk-based questions whose answers can help people calibrate their lives and take calculated enough risks to get on with it without putting themselves and their loved ones at risk.

Instead, what we find from official sources are nothing but aggregates. Total numbers of people infected, dead, recovered and so on. And it is impossible to infer answers to the “risk questions” based no that.

And who fill in the gaps? Media of course.

I must have discussed “spectacularness bias” on this blog several times before. Basically the idea is that for something to be news, it needs to carry information. And an event carries information if it occurs despite having a low prior probability (or not occurring despite a high prior probability). As I put it in my lectures, “‘dog bites man’ is not news. ‘man bits dog’ is news”.

So when we rely on media reports to fill in our gaps in our risk systems, we end up taking all the wrong kinds of lessons. We learn that one seventeen year old boy died of covid despite being otherwise healthy. In the absence of other information, we assume that teenagers are under grave risk from the disease.

Similarly, cases of children looking for ICU beds get forwarded far more than cases of old people looking for ICU beds. In the absence of risk information, we assume that the situation must be grave among children.

Old people dying from covid goes unreported (unless the person was famous in some way or the other), since the information content in that is low. Young people dying gets amplified.

Based on all the reports that we see in the papers and other media (including social media), we get an entirely warped sense of what the risk profile of the disease is. And panic. When we panic, our health gets worse.

Oh, and I haven’t even spoken about bad risk reporting in the media. I saw a report in the Times of India this morning (unable to find a link to it) that said that “young are facing higher mortality in this wave”. Basically the story said that people under 60 account for a far higher proportion of deaths in the second wave than in the first.

Now there are two problems with that story.

1. A large proportion of over 60s in India are vaccinated, so mortality is likely to be lower in this cohort.
2. What we need is the likelihood of a person under 60 dying upon contracting covid. NOT the proportion of deaths accounted for by under 60s. This is the classic “averaging along the wrong axis” that they unleash upon you in the first test of any statistics course.

Anyway, so what kind of data would have helped?

1. Age profile of people testing positive, preferably state wise (any finer will be noise)
2. Age profile of people dying of covid-19, again state wise

I’m sure the government collects this data. Just that they’re not used to releasing this kind fo data, so we’re not getting it. And so we have to rely on the media and its spectacularness bias to get our information. And so we panic.

PS: By no means am I stating that covid-19 is not a risk. All I am stating is that the information we have been given doesn’t help us make good risk decisions

## Kneel down

When Colin Kaepernick knelt down during the national anthem, it was cool, and a strong sign of protest against racial violence in the United States. When other athletes, in the US and elsewhere decided to copy him (and did so on their own volition), it was cool as well.

What I find not so convincing is that after the Floyd murder earlier this year, sports organisations across the world decided to institutionalise the kneel down. When the English Premier League restarted after the covid-19 induced break, it was decided that all players and referees would kneel for a minute at kickoff.

Now it seems like it has been decided that the gesture will continue for the 2020-21 season as well – players and officials will take a knee for a minute at the beginning of each game. Of course, it has also been decided to make it “non-mandatory” – players who choose not to not join the protest will be free not to kneel.

The problem with the institutionalisation of the protest is that the protest loses its information content. Prior to the institutionalisation in June, if a player knelt, he/she was making a statement that he/she believed that “black lives matter”. Now that kneeling has become standard practice, there is no way for a player to convey this information.

Alternatively, it is possible now for a player to send out the opposite information (that he/she doesn’t believe in this protest) by refusing to join the protest. However, given the PR repercussions of such a move, it is unlikely that any player is going to take that stance (no pun intended).

Actually – by institutionalising the kneel, the protest level is getting changed, from individual players to leagues. I can see why the protest is going to be continued – it will be a continuing statement by the sporting leagues that they believe in the cause. However, individual players will not have the opportunity to show their protest (or dissent) any more.

I also wonder if and when this protocol is reversed, since it takes effort for some team or league to “bell the cat”. Even saying that “this is mere symbolism” is bound to attract wrath of protestors elsewhere, so teams are all caught in a Nash equilibrium where they continue to kneel down in protest.

And the longer this kneeling down protest continues, the more the meaning that it will lose. Rather than serving to make a statement, it will end up as yet another ritual.

## Jio, Amazon and Information Content

A long long time ago I had installed the Jio Cinema app on my Fire TV Stick. I had perhaps watched two movies on that, and then completely forgotten about it. This evening, I had to look for a movie to watch my the wife, and having exhausted most of the “compatible content” (stuff we can watch together on Netflix) and been exhausted by the user experience on Prime Video, I decided to give this app a try.

I ended up selecting a movie, which I later found out has a 4.5 IMDB rating and doesn’t even have a Wikepedia page. Needless to say, we abandoned the movie midway. That’s when the wife went in to put the daughter to bed and my fun began.

So Jio Cinema follows what I call the “Amazon paradigm for product management”. Since Amazon tries to sell every product (or service) as if it is a physical book, it has one single mantra for product management. “Improve selection and they will come”.

The user experience doesn’t matter. How easy the product is to use, and how pleasing it looks on the eye, and whether it has occasional bugs, is all secondary. All that matters is selection. Given that the company built its business on the back of selling “long tail” books, this is not so surprising, except that it doesn’t necessarily work in other categories.

I’ve written about Amazon’s ineptitude in product management before, in the context of that atrocity of an app called Sony Liv. The funny thing is that the Jio Cinema app (on Fire TV Stick) looks and feels pretty much exactly like Sony Liv. Maybe there is an open source shitty fire TV app that these guys have based themselves on?

In any case, I started browsing the Jio Cinema app, and I found something called “movies in 15 minutes“. Initially I thought it was a parody. The first few movies I noticed there were things I had never heard of. “This is perhaps for bad movies”, I reasoned. I kept scrolling, and more recognisable names popped up.

I decided to watch Deewana, which was released just before the start of my optimal age of movie appreciation, and which, for some reason, we didn’t get home a video cassette of.

It’s basically a collage of scenes from the movie. It’s like someone has put together a “highlights package”, taking all the important scenes and then putting them together.

And for a movie like Deewana it works. The 15 minute version had all the necessary plot elements to fully follow the movie. It is a great movie, for 15 minutes. Maybe at 30 minutes as well it might be a great movie. However, I can’t imagine having watched it in the full version.

That was two hours back. I’ve since gone crazy watching 15 minute versions of many other movies (mostly from the 70s and 80s, though they have movies as recent as Jab We Met). It’s been fantastic.

However, I have one crib. This has to do with information content. Essentially, the premise behind “movies in 15 minutes” is that the information content in these movies is so little that the whole thing can be compressed into 15 minutes.  The problem is that not every movie has the same amount of information.

15 minutes was perfect for Deewana. It was also appropriate for Kasam Paida Karne Waali Ki, which I watched only because it gets referenced in Gangs of Wasseypur. Between these two, I “watched” Namak Halaal, and I didn’t understand the head or tail of it. I had to go to Wikepedia to understand the plot.

Essentially the plot of Namak Halaal is complex enough, I imagine, that compressing it into 15 minutes is impossible without significant information loss. And the loss of information was so much that I couldn’t understand the summary at all. Maybe I’ll watch the movie in full some day.

I’m writing this blogpost after watching the 15 minute version of Don. I guess whoever made the summary realised that the movie is so complex that it can’t really be compressed into 15 minutes – and so they have added a voiceover to narrate the key elements.

In any case, I’m feeling super thrilled. I normally don’t watch movies because the bit rate in most movies is too low. Compression means that I can happily watch the movies without ever getting bored.

I wish they made these 15 minute versions of all movies. Jio, all (your Amazon-style product maangement) is forgiven.

Now on to Amar Akbar Anthony.

## Half-watching movies, and why I hate tweetstorms

It has to do with “bit rate”

I don’t like tweetstorm. Up to six tweets is fine, but beyond that I find it incredibly difficult to hold my attention for. I actually find it stressful. So of late, I’ve been making a conscious effort to stop reading tweetstorms when they start stressing me out. The stress isn’t worth any value that the tweetstorms may have.

I remember making the claim on twitter that I refuse to read any more tweetstorms of more than six tweets henceforth. I’m not able to find that tweet now.

Anyways…

Why do I hate tweetstorms? It is for the same reason that I like to “half-watch” movies, something that endlessly irritates my wife. I has to do with “bit rates“.

I use the phrase “bit rate” to refer to the rate of flow of information (remember that bit is a measure of information).

The thing with movies is that some of them have very low bit rate. More importantly, movies have vastly varying bit rates through their lengths. There are some parts in a movie where pretty much nothing happens, and a lot of it is rather predictable. There are other parts where lots happens.

This means that in the course of a movie you find yourself engrossed in some periods and bored in others, and that can be rather irritating. And boredom in the parts where nothing is happening sometimes leads me to want to turn off the movie.

So I deal with this by “half watching”, essentially multi tasking while watching. Usually this means reading, or being on twitter, while watching a movie. This usually works beautifully. When the bit rate from the movie is high, I focus. When it is low, I take my mind off and indulge in the other thing that I’m doing.

It is not just movies that I “half-watch” – a lot of sport also gets the same treatment. Like right now I’m “watching” Watford-Southampton as I’m writing this.

A few years back, my wife expressed disapproval of my half-watching. By also keeping a book or computer, I wasn’t “involved enough” in the movie, she started saying, and that half-watching meant we “weren’t really watching the movie together”. And she started demanding full attention from me when we watched movies together.

The main consequence of this is that I started watching fewer movies. Given that I can rather easily second-guess movie plots, I started finding watching highly predictable stuff rather boring. In any case, I’ve recently received permission to half-watch again, and have watched two movies in the last 24 hours (neither of which I would have been able to sit through had I paid full attention – they had low bit rates).

So what’s the problem with tweetstorms? The problem is that their bit rate is rather high. With “normal paragraph writing” we have come to expect a certain degree of redundancy. This allows us to skim through stuff while getting information from them at the same time. The redundancy means that as long as we get some key words or phrases, we can fill in the rest of the stuff, and reading is rather pleasant.

The thing with a tweetstorm is that each sentence (tweet, basically) has a lot of information packed into it. So skimming is not an option. And the information hitting your head at the rate that tweetstorms generally convey can result in a lot of stress.

The other thing with tweetstorms, of course, is that each tweet is disjoint from the one before and after it. So there is no flow to the reading, and the mind has to expend extra energy to process what’s happening. Combine this with a rather high bit rate, and you know why I can’t stand them.

## Super Deluxe

In my four years in Madras (2000-4), I learnt just about enough Tamil to watch a Tamil movie with subtitles. Without subtitles is still a bit of a stretch for me, but the fact that streaming sites offer all movies with subtitles means I can watch Tamil movies now.

At the end, I didn’t like Super Deluxe. I thought it was an incredibly weird movie. The last half an hour was beyond bizarre. Rather, the entire movie is weird (which is good in a way we’ll come to in a bit), but there is a point where there is a step-change in the weirdness.

The wife had watched the movie some 2-3 weeks back, and I was watching it on Friday night. Around the time when she finished the movie she was watching and was going to bed, she peered into my laptop and said “it’s going to get super weird now”. “As if it isn’t weird enough already”, I replied. In hindsight, she was right. She had peered into my laptop right at the moment when the weirdness goes to yet another level.

It’s not often that I watch movies, since most movies simply fail to hold my attention. The problem is that most plots are rather predictable, and it is rather easy to second-guess what happens in each scene. It is the information theoretic concept of “surprise”.

Surprise is maximised when the least probable thing happens at every point in time. And when the least probable thing doesn’t happen, there isn’t a story, so filmmakers overindex on surprises and making sure the less probable thing will happen. So if you indulge in a small bit of second order thinking, the surprises aren’t surprising any more, and the movie becomes boring.

Super Deluxe establishes pretty early on that the plot is going to be rather weird. And when you think the scene has been set with sufficient weirdness in each story (there are four intertwined stories in the movie, as per modern fashion), the next time the movie comes back to the story, the story is shown to get weirder. And so you begin to expect weirdness. And this, in a way, makes the movie less predictable.

The reason a weird movie is less predictable is that at each scene it is simply impossible for the view to even think of the possibilities. And in a movie that gets progressively weirder like this one, every time you think you have listed out the possibilities and predicted what happens, what follows is something from outside your “consideration set”. And that keeps you engaged, and wanting to see what happens.

The problem with a progressively weird movie is that at some point it needs to end. And it needs to end in a coherent way. Well, it is possible sometimes to leave the viewer hanging, but some filmmakers see the need to provide a coherent ending.

And so what usually happens is that at some point in time the plot gets so remarkably simplified that everything suddenly falls in place (though nowhere as beautifully as things fall in place at the end of a Wodehouse novel). Another thing that can happen is that weirdness it taken up a notch, so that things fall in place at a “meta level”, at which point the movie can end.

The thing with Super Deluxe is that both these things happen! On one side the weirdness is taken up several notches. And on the other the plots get so oversimplified that things just fall in place. And that makes you finish the movie with a rather bitter taste in the mouth, feeling thoroughly unsatisfied.

That the “ending” of the movie (where things get really weird AND really simplified) lasts half an hour doesn’t help matters.

## Why Brits talk so much about the weather

One stereotype about British people is that they are always talking about the weather. In the absence of any other topic to talk about, they get back down to talking about the weather.

Having lived here for a day after a half after moving here yesterday, I can offer one explanation about why Brits talk so much about the weather – the high information content. In the last day and half, the weather here has been so volatile that the information content in statements about the weather can be rather high.

Most places in the world have rather predictable weather. Delhi has hot summers and cold winters. It almost always rains in the tropical rain forests. It almost always rains in Mumbai during the monsoon. And so on. And when the weather is predictable, the information content in describing it is rather low.

For example, if there is a 90% chance that it will rain in Mumbai one monsoon day, a statement on the presence or absence of rain contains only 0.47 bits of information/entropy (-0.9 log 0.9 – 0.1 log 0.1). If the probability that a summer day in Delhi will be sunny is 99%, then the information content in talking about the weather is just 0.08 bits.

The thing with London weather, based on my day and half of observation, is that it is wildly volatile. This afternoon, for example, there was a hailstorm. And only a couple of minutes later there was bright sunshine. And then there was another hailstorm. I can see heavy rain from my window as I write this now.

Crow and fox getting married in London

A post shared by Karthik S (@skthewimp) on

So given how crazy and volatile the weather in London is, the information content in talking about the weather is rather high. As I write, there’s sunshine streaming through my window, and heavy rain outside. And I’m chatting with a friend who lives not very far from here, and whatever I tell her about the weather here is “information” to her, since it’s not the same there.

It’s this craziness and high volatility in weather in Britain that makes it worth talking about. The information content in a statement about the weather is always high. And this is not the case elsewhere in the world. And so people elsewhere get annoyed by Brits talking about the weather.

PS: What does it tell you that I’m blogging about the weather a day and half after landing in Britain?

## The one bit machine

My daughter is two weeks old today and she continues to be a “one bit machine”. The extent of her outward communication is restricted to a maximum of one bit of information. There are basically two states her outward communication can fall under – “cry” and “not cry”, and given that the two are not equally probable, the amount of information she gives out is strictly less than one bit.

I had planned to write this post two weeks back, the day she was born, and wanted to speculate how long it would take for her to expand her repertoire of communication and provide us with more information on what she wants. Two weeks in, I hereby report that the complexity of communication hasn’t improved.

Soon (I don’t know how soon) I expect her to start providing us more information – maybe there will be one kind of cry when she’s hungry, and another when she wants her diaper changed. Maybe she’ll start displaying other methods of outward communication – using her facial muscles, for example (right now, while she contorts her face in a zillion ways, there is absolutely no information conveyed), and we can figure out with greater certainty what she wants to convey.

I’m thinking about drawing a graph with age of the person on the X axis, and the complexity of outward information on the Y axis. It starts off with X = 0 and Y = 1 (I haven’t bothered measuring the frequency of cry/no-cry responses so let’s assume it’s equiprobable and she conveys one bit). It goes on to X = 14 days and Y = 1 (today’s state). And then increases with time (I’m hoping).

While I’m sure research exists some place on the information content per syllable in adult communication, I hope to draw this graph sometime based on personal observation of my specimen (though that would limit it to one data point).

Right now, though, I speculate what kind of shape this graph might take. Considering it has so far failed to take off at all, I hope that it’ll be either an exponential (short-term good but long-term I don’t know ) or a sigmoid (more likely I’d think).

Let’s wait and see.

## Curation, editing and predictability

One of my favourite lunchtime hobbies over the last one year has been watching chess videos. My favourite publishers in this regard are GM Daniel King and Mato Jelic. King is a far superior analyst and goes into more depth while analysing games, though Jelic has a far larger repertoire (King usually only analyses games the day they were played).

In some ways I might be biased towards Jelic because his analysis and focus are largely in line with my strengths back during my days as a competitive chess player. Deep opening analysis, attacking games, the occasional tactical flourish and so on. He has a particular fondness for the games of Mikhail Tal, showering praises on his (Tal’s) sometimes erratic and seemingly purposeless sacrifices.

Once you watch a few videos of Jelic, though, you realise that there is a formula to his commentary. At some point in the game, he announces that the game is in a “critical position” and asks the viewer to pause the video and guess the next move. And a few seconds of pause later, he proceeds to show the move and move on with the game.

While this is an interesting exercise the first few times around, after a few times I started seeing a pattern – Jelic has a penchant for attacking positions, and the moves following his “critical positions” are more often than not sacrifices. And once I figured this bit out, I started explicitly looking for sacrifices or tactical combination every time he asked me to pause, and that has made the exercise a lot less fun.

I’d mentioned on this blog a few weeks back about my problem with watching movies – in that I’m constantly trying to second-guess the rest of the movie based on the information provided thus far. And when a movie gets too predictable, it tends to lose my attention. And thinking about it, I think sometimes it’s about curation or editing that makes things too predictable.

To take an example, my wife and I have been watching Masterchef Australia this year (no spoilers, please!), and I remarked to her the other day that episodes have been too predictable – at the end of every contest, it seems rather easy to predict who might win or go down, and so there has been little element of surprise in the show.

My wife remarked that this was not due to the nature of the competition itself (which she said is as good as earlier editions), but due to the poor editing of the show – during each competition, there is a disproportional amount of time dedicated to showing the spectacularly good and spectacularly bad performances.

Consequently, just this information – on who the show’s editors have chosen to focus on for the particular episode – conveys a sufficient amount of information on each person’s performance, without even seeing what they’ve made! A more equitable distribution of footage across competitors, on the other hand, would do a better job of keeping the viewers guessing!

It is similar in the case of Jelic’s videos. There is a pattern to the game situation where he pauses, which biases the viewer in terms of guessing what the next move will be. In order to make the experience superior for his viewers, Jelic should mix it up a bit, occasionally showing slow Carlsen-like positions, and stopping games at positional “critical positions”, for example. That can make the pauses more interesting, and improve viewer experience!

What are other situations where bad editing effectively gives away the plot, and diminishes the experience?

## Doctors and correlation-causation

One of the common cribs about the medical profession is that most doctors don’t have enough grounding in mathematics and statistics (subjects they typically don’t study beyond high school). Given the role of mathematics and statistics in medicine, in terms of gathering evidence, medical testing, etc. the lack of mathematical or statistical knowledge can have serious consequences in terms of interpretation of techniques and symptoms and all that.

In the field of statistics we have this adage that goes that we should “treat the disease and not the symptom”. This is no less true in the medical profession – let’s say that you have a bacterial infection which causes a fever, a poor doctor would diagnose your fever by taking your temperature, assume that it is the fever thanks to which you are sick and give you medication to lower the fever without realising that there is a “third variable” that might be causing both – your fever and your sickness. Thus, your fever might come down and consequently your sickness but both would presently appear.

I’ve had chronic pain in my heels for a few months now. It’s especially severe whenever I put my feet on the ground from a raised position. Someone had told me that it occurs due to calcification near the Achilles Tendon, and I must take medication for that. Having pushed it for a few months now I finally went to see my uncle who is an orthopaedic yesterday (this is the same guy who told me about my Boxer’s Fist).

He promptly diagnosed me with Plantar Fasciitis, and wrote down some medication, and told me what I need to do in order to reduce the pain in my feet. After a short conversation on what else I need to do, and any precautions, and all such, I asked him about the calcification thingy – whether he had ruled out that calcification of the Achilles Tendon was causing this problem.

“I’m sure there will be some calcification”, he said, “and I’m not sending you for an X-ray because I have a very good idea of what it will show and it won’t add much value”. And then he proceeded to explain that calcification is a “result” of plantar fasciitis and not a cause of it. He didn’t use the terms “correlation” or “causation” but he explained that when you suffer from plantar fasciitis you end up with both calcification of your Achilles Tendon and also shooting pain in your heels, especially immediately after waking up. The two are thus related, he said, but neither causes the other, but there is a third factor (fasciitis) that causes both, and that is the one that he is treating me for!

I was doubly impressed with him – first for understanding “information theory” in terms of understanding that the X-ray wouldn’t add much information, and secondly for recognising that there was a third factor and that correlation should not be mistaken for causation. Or perhaps I had a particularly low prior for mathematical and statistical skills of doctors!

Postscript

He refused to charge me a fee, since I’m his nephew. While on my way out I was thinking about it and wondering on what circumstances I would waive my professional fees for my consulting. And I realised it would be hard to do so for anyone! It made me wonder what made my uncle waive his medical fees, while I’m extremely unlikely to do that.

I realised it has to do with the investment. He spent about five to ten minutes with me (perhaps a bit longer), but essentially his marginal cost of treating me was quite low. And this was a marginal cost that he was willing to sacrifice in return for the goodwill he gets for treating the extended family for free. Considering the size of my engagements, though, the marginal cost is usually high and is seldom justified by goodwill!