One of the great contributors to the profits of high-end restaurants is the fact that bottled water comes in two types, enabling waiters to ask ‘still or sparkling?’, making it rather difficult to say ‘just tap’. I had the idea of turning up at an apartment with five smoke detectors; the fire officer was to casually carry in all five, before saying, ‘I think we can make do with three here . . . How many would you like, three or four?’ We are highly social creatures and just as we find it very difficult to answer the question ‘still or sparkling?’ with ‘tap’, it is also difficult to answer the question about ‘three or four’ smoke detectors with with ‘one’. As Nassim Nicholas Taleb remarks, ‘the way a question is phrased is itself information’.
1.8: ‘A Change in Perspective Is Worth 80 IQ Points’
So said Alan Kay, one of the pioneers of computer graphics. It is, perhaps, the best defence of creativity in ten words or fewer. I suspect, too, that the opposite is also true: that an inability to change perspective is equivalent to a loss of intelligence.*
I was once walking down a suburban street in Wallingford, Pennsylvania. In American suburbia, there are no hedges to obscure the houses – a whitewashed fence about two feet high is all that marks property boundaries. So I was slightly alarmed when a large unleashed dog lurched towards me across one of the lawns, barking loudly. Clearly he was not going to have much difficulty clearing the little fence, after which he would be free to tear me to shreds. My companion, however, seemed unperturbed and sure enough, about two feet before the fence, the dog skidded to a halt on the lawn and continued its furious barking. As my friend knew, the dog was fitted with a collar that would detect the presence of a wire buried beneath the lawn boundary and administer an electric shock to the dog if it came too close. Although the fence was only two feet high, the dog was terrified to approach it.
A similar constraint also applies to decision-making in business and government. There is a narrow and tightly limited area within which economic theory allows people to act. Once they reach the edges of that area, they freeze, rather like the dog. In some influential parts of business and government, economic logic has become a limiting creed rather than a methodological tool. As Sir Christopher Llewellyn Smith, the former director of CERN, remarked after he had been tasked with changing the patterns of energy consumption in Britain, ‘When I ask an economist, the answer always boils down to just bribing people.’
Logic should be a tool, not a rule.
At its worst, neo-liberalism takes a dynamic system like free market capitalism, which is capable of spectacular creativity and ingenuity, and reduces it to a boring exercise in ‘how we can buy these widgets 10 per cent cheaper’. It has also propelled a narrow-minded technocratic caste into power, who achieve the appearance of expert certainty by ignoring large parts of what makes markets so interesting. The psychological complexity of human behaviour is reduced to a narrow set of assumptions about what people want, which means they design a world for logical rather than psycho-logical people. And so we have faster trains with uncomfortable seats departing from stark, modernist stations, whereas our unconscious may well prefer the opposite: slower trains with comfortable seats departing from ornate stations.
This is not the fault of the markets; it is the fault of the people who have hijacked the definition of what markets should do. Strangely, as we have gained access to more information, data, processing power and better communications, we may also be losing the ability to see things in more than one way; the more data we have, the less room there is for things that can’t easily be used in computation. Far from reducing our problems, technology may have equipped us with a rational straitjacket that limits our freedom to solve them.
Sometimes we grossly overrate ‘the pursuit of reason’, while at other times, we misdeploy it. Reasoning is a priceless tool for evaluating solutions, and essential if you wish to defend them, but it does not always do a very good job of finding those solutions in the first place. Maths, for instance, has the power to mislead as well as to illuminate. The inherent flaws of mathematical models are well understood by good mathematicians, physicists and statisticians, but very badly understood by those who are merely competent.*
Whenever I speak to a very good mathematician, one of the first things I notice is they are often sceptical about the tools which other mathematicians are most enthused about. A typical phrase might be: ‘Yeah you could do a regression analysis, but the result is usually bollocks.’ An attendant problem is that people who are not skilled at mathematics tend to view the output of second-rate mathematicians with an high level of credulity, and attach almost mystical significance to their findings. Bad maths is the palmistry of the twenty-first century.
Yet bad maths can lead to collective insanity, and it is far easier to be massively wrong mathematically than most people realise – a single dud data point or false assumption can lead to results that are wrong by many orders of magnitude.
In 1999 a British solicitor named Sally Clark was convicted for the murder of her two infant sons. Both had died of what was assumed to be cot death (or sudden infant death syndrome), a little over a year apart. After the death of the second child, suspicions were raised and she was prosecuted for murder. In a now-discredited piece of statistical evidence, the paediatrician Professor Sir Roy Meadow gave evidence at her trial that the chance of both deaths arising from natural causes was one in 73 million, or equivalent to ‘four different horses winning the Grand National in consecutive years at odds of 80 to 1’. There are 700,000 live births in the UK every year, and there’s a one in 8,453 chance of a child from a middle-class, non-smoking home dying from SIDS; Meadow had multiplied the two numbers together to get odds of one in 73 million, and claimed that you would only expect a double cot death in a family once in a hundred years.
One medical expert for the defence later described the figure as a gross statistical over-simplification, but it stuck – and there was a clear implication that there was only a statistically tiny chance that Clark was innocent. In a courtroom packed with scientists and lawyers, no one sought to rule this figure of one in 73 million inadmissible, but let’s look at just how wrong it is. Firstly, it assumed odds of one in 8,453, taken from a single, selective source – whereas a more accurate figure might be closer to one in 1,500 or so. It also made no allowance for the fact that both victims were male, which reduces the odds further. Worst of all, it made no allowance for the possibility that a common combination of genes or an environmental factor – such as an aspect of the house in which they both died that was common to both tragedies – might have played a part. Genetic factors are believed to be involved in SIDS – it may run in families, making a double incidence much more likely.
As the journalist Tom Utley pointed out in the Daily Telegraph, he himself, among a circle of perhaps 10,000 acquaintances, knew two people who had each innocently lost two infants for inexplicable reasons, so the likelihood that this was as rare an occurrence as Professor Meadow had suggested seemed implausible. Although Sally Clark may have been unlucky in that on both occasions she was alone at home with her child and had no witnesses to defend her, if you make the corrections above you might reasonably expect double deaths to occur by chance several times a year in the UK – and thus it seemed less likely that Clark was guilty.
This, however, is still wrong. If you wish to prove the murderous intent of Sally Clark, it is not enough to prove that the cot-death theory is improbable. To do this is to fall victim to what is known as ‘the prosecutor’s fallacy’, where the prosecution can imply that a similarity between the perpetrator and the accused carries more statistical weight than it deserves. (For instance, it may seem conclusive to suggest that a DNA marker shared by the perpetrator and a suspect is possessed by only one in 20,000 people, but if the suspect had been identified by trawling through a DNA database of 60,000 people, you would expect to find three people who exhibited this property, of whom at least two would be completely innocent.)
In Sally Clark’s case, it is not enough to prove that double cot death is unlikely: you also have to prove that it is more unlikely than double infanticide. With the accurate statistical comparison, where you compare the relative likelihood of double cot death or double infanticide, the implied odds of her innocence fall from one in 73 million to perhaps two in three. She might still be guilty, but there is now more than enough reasonable doubt on which to acquit. Indeed, the most likely explanation is that she is innocent.
Notice, though, how just a few wrong assumptions in statistics, when compounded, can lead to an intelligent man being wrong by a factor of about 100,000,000 – tarot cards are rarely this dangerous. This miscarriage of justice led Professor Peter Green, as President of the Royal Statistical Society, to write to the Lord Chancellor, Britain’s senior legal authority, pointing out the fallaciousness of Meadow’s reasoning and offering advice on the better use of statistics in legal cases. However, the problem will never go away, because the number of people who think they understand statistics dangerously dwarfs those who actually do, and maths can cause fundamental problems when badly used.
To put it crudely, when you multiply bullshit with bullshit, you don’t get a bit more bullshit – you get bullshit squared.
One thing this means is that everyone should know at least one seriously good mathematician; when you meet them, it is usually a revelation. I am proud to have met Ole Peters, a tremendous German physicist attached to the Santa Fe Institute and the London Mathematical Laboratory, in the last year. He recently co-authored a paper* pointing out that a huge number of theoretical findings in economics were based on a logical-sounding-but-entirely-erroneous assumption about statistical mechanics. The assumption was that, if you wished to know whether a bet was a good idea, you could simply imagine making it a thousand times simultaneously, add up the net winnings, and subtract the losses; if the overall outcome is positive, you should then make that bet as many times as you can.
So a bet costing £5 which has a 50 per cent chance of paying out £12 (including the return of your stake) is a good bet. You will win on average £1 every time you play, so you should play it a lot. Half the time you lose £5 and half the time you win £7. If a thousand people play the game just once, they will collectively end up with a net gain of £1,000. And if one person plays the game 1,000 times, he would expect to end up around £1,000 richer too – the parallel and series outcomes are the same. Unfortunately this principle applies only under certain conditions, and real life is not one of them. It assumes that each gamble is independent of your past performance, but in real life, your ability to bet is contingent on the success of bets you have made in the past.
Let’s try a different kind of bet – one where you put in a £100 stake, and if you throw heads, your wealth increases by 50 per cent, but if you throw tails it falls by 40 per cent. How often would you want to toss the coin? Quite a lot, I suspect. After all, it’s simple, right? All you have to do to calculate the expectation value over 1,000 throws is imagine 1,000 people taking this bet once simultaneously and average the outcome, like we did last time. If, on average, the group is better off, it therefore represents a positive expectation. But apparently not.
Let’s look at it in parallel. If a thousand people all took this bet once, starting with £100 each (meaning a total of £100,000), typically 500 people would end up with £150 and 500 people would end up with £60. That’s £75,000 + £30,000 or £105,000, a net 5 per cent return. If someone asked me how often would I like them to toss the coin under those conditions, and how much would I like to put in, I would say: ‘All the money I have, and throw the coin as fast as you can. I’m off to Mauritius with the winnings.’ However, in this case the parallel average tells you nothing about the series expectation.
Put in mathematical language, an ensemble perspective is not the same as a time-series perspective. If you take this bet repeatedly, by far the most likely outcome is that you will end up skint. A million people all taking the bet repeatedly will collectively end up richer, but only because the richest 0.1 per cent will be multi-billionaires: the great majority of the players will lose. If you don’t believe me, let’s imagine four people toss the coin just twice. There are four possible outcomes: HH, HT, TH or TT, all of equal likelihood. So let’s imagine that each of the four people starts with $100 and throws a different combination of heads and tails:
HH
HT
TH
TT
The returns on these four are £225, £90, £90 and £36. There are two ways of looking at this. One is to say, ‘What a fabulous return: our collective net wealth has grown over 10 per cent, from £400 to £441, so we must all be winning.’ The more pessimistic viewpoint is to say, ‘Sure, but most of you are now poorer than when you started, and one of you is seriously broke. In fact, the person with £36 needs to throw three heads in a row just to recover his original stake.’
This distinction had never occurred to me, but it also seems to have escaped the attention of most of the economics profession, too. And it’s a finding that has great implications for the behavioural sciences, because it suggests that many supposed biases which economists wish to correct may not be biases at all – they may simply arise from the fact that a decision which seems irrational when viewed through an ensemble perspective is rational when viewed through the correct time-series perspective, which is how real life is actually lived; what happens on average when a thousand people do something once is not a clue to what will happen when one person does something a thousand times. In this, it seems, evolved human instinct may be a much better at statistics than modern economists.* To explain this distinction using an extreme analogy, if you offered ten people £10m to play Russian roulette once, two or three people might be interested, but no one would accept £100m to play ten times in a row.
Talking to Ole Peters, I realised that the problem went wider than that – nearly all pricing models assume that ten people paying for something once is the same as one person paying for something ten times, but this is obviously not the case. Ten people who each order ten things every year from Amazon will probably not begrudge paying a few dollars for delivery each time, while one person who buys 100 things from Amazon every year is going to look at his annual expenditure on shipping and decide, ‘Hmm, time to rediscover Walmart.’*
One of our clients at Ogilvy is an airline. I constantly remind them that asking four businessmen to pay £26 each to check in one piece of luggage is not the same as asking a married father of two* to pay £104 to check in his family’s luggage. While £26 is a reasonable fee for a service, £104 is a rip-off. The way luggage pricing should work is something like this: £26 for one case, £35 for up to three. There is, after all, a reason why commuters are offered season tickets – commuting is not commutative, so 100 people will pay more to make a journey once than one person will pay to make it 100 times. In the same way, the time-saving model used to justify the UK’s current investment in the High Speed 2 rail network assumes that 40 people saving an hour ten times a year is the same as one commuter saving an hour 400 times a year. This is obviously nonsense; the first is a convenience, while the second is a life-changer.
1.9: Be Careful with Maths: Or Why the Need to Look Rational Can Make You Act Dumb
I would rather run a business with no mathematicians than with second-rate mathematicians. Remember that every time you average, add or multiply something, you are losing information. Remember also that a single rogue outlier can lead to an extraordinary distortion of reality – just as Bill Gates can walk into a football stadium and raise the average level of wealth of everyone in it by $1m.
The advertising agency I work for once sent out postal solicitations for a charity client, and we noticed that one creative treatment significantly outperformed the other in its net return. Since there was not much difference between the treatments, the scale of the difference in results surprised us. When we investigated, we found that one person had replied wit
h a cheque for £50,000.*
Let’s look at another example of how one rogue piece of data – a single outlier – can lead to insane conclusions when not understood in the proper context. I have a card which I use to pay for my car’s fuel, and every time I fill up, I record the car’s mileage on the payment terminal. After a year, the fuel-card company started putting my mileage per gallon on my monthly statements – a lovely idea, except it started to drive me insane, because every month my car became less economical. I was puzzling over this for ages, agonising about fuel leaks and even wondering if someone was pilfering from my tank.
But then I remembered: soon after my company had given me the fuel card, I had forgotten to use it once, and had instead paid with an ordinary credit card. This meant that, according to the data held by the fuel-card company, in one period I had driven the distance of two tankfuls with one tankful of fuel. Because this anomaly was still sitting in the database, every subsequent month I drove was making my fuel-economy stats look worse as I was regressing to the mean – one anomalous data point made all the rest misleading.
But let me come back to my previous point. In maths, 10 x 1 is always the same as 1 x 10, but in real life, it rarely is. You can trick ten people once, but it’s much harder to trick one person ten times.* But how many other things are predicated on such assumptions? Imagine for a moment a parallel universe in which shops had not been invented, and where all commerce took place online. This may seem like a fantastical notion, but it more or less describes rural America a hundred years ago. In 1919 the catalogues produced by Sears, Roebuck and Company and Montgomery Ward were, for the 52 per cent of Americans in rural areas, the principal means of buying anything remotely exotic. In that year, Americans spent over $500 million on mail-order purchases, half of which were through the two companies.
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