When we pose a question, it is usual that we want an answer. Sometimes the answer is clear, because it is defined to be what it is, or else is true as a matter of logic.

For example, what is 2 plus 2? If your number system defines the answer to be 4, then it is 4.

If I roll two standard dice, the highest total I can achieve is 12. So if I am asked what is the probability that I will roll a total of 14, the answer is zero.

For most questions, however, which are not true by definition or logic, there is no sure answer to any question, only various levels of probability.

Similarly, for any set of observations, the rule or set of rules that gave rise to these observations might not be clear. There may be a large number of different explanations which are consistent with the data.

For example, what rule gives rise to the number sequence 1,3,5,7? If we know this, it will help us to predict what the next number in the sequence is likely to be, if there is one.

Two hypotheses spring instantly to mind. It could be: 2n-1, where n is the step in the sequence. So the third step, for example, gives 2×3-1 = 5. If this is the correct rule generating the observations, the next step in the sequence will be 9 (5×2-1).

But it’s possible that the rule generating the number sequence is: 2n-1 + (n-1)(n-2)(n-3)(n-4). So the third step, for example, gives 2×3-1 + (3-1)(3-2)(3-3)(3-4) = 7. In this case, however, the next step in the sequence will be 33.

So if this is all the information we have, we have two different hypotheses about the rule generating the data. How do we decide which is more likely to be true? In general, when we have more than one hypothesis, each of which could be true, how can we decide which one actually is true?

For the answer, we need to turn to some basic principles of scientific enquiry. I list these as Epicurus’ Principle, Occam’s Razor, Bayes’ Theorem, Popperian ‘Falsifiability’; Solomonoff Induction and the Vaughan Williams ‘Possibility Theorem.’

To address these, and how they contribute to a grand unified theory of scientific enquiry, is beyond the scope of this post, but I can at least provide a basic explanation of the terms.

Epicurus’ Principle is the idea that if there are a number of different possible truths, we should keep open the possibility that any of them might be true until we are forced by the evidence to do otherwise. Otherwise stated, it is the maxim that if more than one theory is consistent with the known observations, keep them all.

Occam’s Razor is the idea that the theory which explains all (or the most) and assumes the least is most likely. This is totally consistent with Epicurus’ Principle, with the additional insight that a simpler theory consistent with known observations is more likely to be true.

Bayes’ Theorem is the idea that the likelihood of a hypothesis being true is a combination of the likelihood of it being true before some new evidence arises and the likelihood of the new evidence arising if the hypothesis is true and if the hypothesis is false. Its most critical insight is that the probability of a hypothesis being true given the evidence is a very different thing to the probability of the evidence arising given that the hypothesis is true.

Popperian ‘Falsifiability’ is the idea that a scientific hypothesis should be testable and falsifiable. Otherwise stated, it notes that a single observational event may prove hypotheses wrong, but no finite sequence of events can verify them correct.

Solomonoff induction is the idea that the information contained in the various explanations consistent with known observations can in principle be reduced to binary sequences, and that the shorter the binary sequence the more likely that explanation of the observations is to be true.

The Vaughan Williams ‘Possibility Theorem’ states that: “If something that might exist can’t be observed, it is more likely to exist than if it can be observed (with any positive probability) but isn’t observed.” This is critical when assessing how the probability of a hypothesis being true might be affected by information which potentially exists and is relevant but is missing because it is for whatever reason unobserved or unobservable.

Combining these principles into a unified framework can help identify the truth based on known and potentially missing observations.

That is the next step.

There are four doors, red, yellow, blue and green.

Three lead the way to dusty death. One leads the way to fame and fortune. They are assigned in order by your evil host who draws four balls out of a bag, coloured red, yellow, blue and green. The first three out of the bag are the colours of the doors that lead to dusty death. The fourth leads to fame and fortune. You must choose one of these doors, without knowing which is the lucky door.

Let us say you choose the red door. Since the destinies are randomly assigned to the doors, this means there is a 1 in 4 chance that you are destined for fame and fortune, a 3 in 4 chance that you are destined for a dusty death.

Your evil host, who knows the doors to death, now opens the yellow door, revealing a door to death. That is part of his job. He always opens a door, but never the door to fame and fortune.

Should you now walk through the red door, the blue door or the green door?

There once was a problem like this, involving three doors, a car, two goats and a game show host called Monty Hall. This has come to be known as the ‘Monty Hall Problem’, and is best known by probability tutors for the number of students who know the answer to the problem without knowing why, but pretending that they do. This is because it featured in a movie called ‘21’, featuring Kevin Spacey as a tutor and some blackjack genius as a student who solves it.

So let me explain.

Intuition dictates that the chance that the red door leads to fame and fortune is 1 in 4 to start with, but must be more than this after the yellow door is opened. After all, once the yellow door is opened, only three doors remain, the red door, the blue door and the green door. Surely there is an equal chance that fortune beckons behind each of these. If so, the probability in each case is 1 in 3.

The evil host now opens a second door, by the same process. Let’s say this time he opens the blue door, which again he reveals to be a death trap. That leaves just two doors. So surely they both now have a 1 in 2 chance.

Take your pick, stick or switch. Does it really matter?

Yes, it does, in fact.

The reason it matters is that the evil host knows where the doors lead. When you choose the red door, there is a 1 in 4 chance that you have won your way to fame and fortune if you stick with it. There is a 3 in 4 chance that the red door leads to death.

If you have chosen the red door, and it is the lucky door, the host is forced to open a deathly door. He knows that. You know that. So he is introducing useful new information into the game.

Before he opened the yellow door, there was a 3 in 4 chance that the lucky door was EITHER the yellow, the blue or the green door. Now he is telling you that there is a 3 in 4 chance that the lucky door is EITHER the yellow, the blue or the green door BUT it is not the yellow door. So there is a 3 in 4 chance that it is EITHER the blue or the green door.

It is equally likely to be either, so there is a 3 in 8 chance that the blue door is the lucky door and a 3 in 8 chance that the green door is the lucky door. But there is a 3 in 4 chance in total that the lucky door is EITHER the blue door or the green door.

Now he opens the blue door, and introduces even more useful information. Now he is telling you that there is a 3 in 4 chance that the lucky door is EITHER the blue or the green door BUT that it is not the blue door. So there must be a 3 in 4 chance that it is the green door.

So now you can stick with the red door, and have a 1 in 4 chance of avoiding a dusty death, or switch to the green door and have a 3 in 4 chance of avoiding that fate.

It is because the host knows what is behind the doors that his actions, which are constrained by the fact that he can’t open the door to fame and fortune, introduces valuable new information. Because he can’t, he is forced to point to a door which is not the door to fortune, increasing the probability that the other unobserved destinies include the lucky one.

If he didn’t know what lay behind the doors, he could inadvertently have opened the door to fortune, so when he does so this adds no new information save that he has randomly eliminated one of the doors. If three doors now remain, they each offer a 1 in 3 chance of avoiding a dusty death. If only two doors remain unopened, they each offer a 1 in 2 chance of death or glory. So you might as well just flip a coin – and hope!

There is a red and a yellow door in the puzzle house which you have just entered. You are told that there is either a boy or a girl behind each of the doors. The choice of whether to place a boy or a girl behind each door is determined by the toss of a coin. Well, there are four possibilities:

- 1. Boy behind red door and boy behind yellow door.
- 2. Boy behind red door and girl behind yellow door.
- 3. Girl behind red door and boy behind yellow door.
- 4. Girl behind both doors.

So the probability that there is a girl behind one of the doors = ¾. The new information I now impart is that at least one of the doors has a boy behind it. So we can delete option 4. This leaves three equally likely options, two of which include a girl. So the probability that there is a girl behind one of the doors given this new evidence is 2/3. The following week, you see a man walking along the street, accompanied by a young boy. You think you recognise him and ask him if you know him. “Didn’t we meet last year at a Sales Convention? If you recall, you told me you had two children, and you mentioned a son. Pleased to meet you, sonny!” “No, you’ve got the wrong guy,” he replies. “I do have two children, but that’s pure coincidence.” Armed now with this information, you instantly work out the chance that the man at the sales convention has a daughter and the chance that this random guy on the street has one as well. It is just like the Two Doors problem. It’s 2 in 3 in both cases. Or is it? Does it matter how you found out about the boy? I think it does. If you know that the man at the convention has two children and at least one son, the chance his other child is a girl is 2 in 3. If you find out that the man in the street has two children, but you don’t know that he has a son until you see him, the chance his other child is a girl is probably closer to 1 in 2, but not definitely. The reason is, I believe, straightforward. If you just happen to find out he has a son, without knowing in advance that he has at least one son, it could be compared to asking him to toss a coin and produce one of his children at random. Maybe draw a card from a deck. Black card means boy, red card means girl. In the case that he has a boy and girl, we might assume there is a 50-50 chance that he will produce a boy, and so that leaves a 50-50 chance that the remaining child is a girl. So, in any case where you find out that he has a boy, without any other information except that he has two children, a reasonable estimate of the chance that the remaining child is a girl is 50-50. But if you know that he has *at least *one boy, that is different information. Different information changes everything. You now know that there is a 1 in 3 chance that he has two boys (BB,BG,GB) and so the chance the other child is a girl is 2 in 3. If you see the boy, and you know there was a 50-50 chance that you would do so if he had a boy and girl, that is different information to simply being told that he has at least one son. The different information sets can be compared to tossing a coin twice. The possible outcomes are HH, HT, TH, TT. If you already know there is ‘at least’ one head, that leaves HH, HT, TH. The probability that the remaining coin is a Tail is 2 in 3. If, on the other hand, you don’t have any information other than that a coin has been tossed twice, the possibilities are HH, HT, TH, TT. Equal chance of a Head or a Tail if you uncover one coin randomly. So there is a 50-50 chance you will see a Head if you uncover one of the coins randomly, and there is a 50-50 chance that the coin you haven’t uncovered is a Tail. What goes for coins goes also for people. Actually, it’s not quite that simple. This assumes that there was a 50-50 chance that the man in the street would choose a boy or a girl (if he had one of each) to accompany him on his walk! The more likely it is that he would choose to take a boy with him than a girl (if he had a girl), the more likely it is that the other child is a girl. It is like tossing two coins, but only ever choosing to show a Head if there is a Head and a Tail. In that case, the likelihood that the other coin is a Tail is 2/3. It is the equivalent of announcing that you have at least one Head (or one boy) whenever you do have one, but never allowing the possibility of a Tail (or a girl) to be observed. It can be stated this way. “If something that might exist can’t be observed, it is more likely to exist than if it can be observed (with any positive probability) but isn’t observed.” I term this the Vaughan Williams ‘Possibility Theorem.’ Bottom line is that when we only display the boy when he exists, but never display the girl when she exists, it is more likely that the unobserved child is a girl. If we display both with equal probability, and they exist with equal probability, then when we observe one, the probability that the other exists is the same, unless some other information exists. It’s the difference between knowing that a boy is behind the red or yellow door, or perhaps both, on the one hand, and opening the red door and finding a boy behind it, on the other hand, when we know that boys and girls have been assigned to the doors by the toss of coin. In the first case, the probability of a girl behind a door in the puzzle house is 2 in 3; in the case where we open the red door and reveal the boy, the probability of a girl behind the other door is 1 in 2. We are assuming, of course, that there is actually a 50-50 chance of a boy and a girl, a Head and a Tail. If we change the assumptions or change the information set, we change the answer. Paradox? Or common sense? You decide.

In 1694, the Bank of England was founded by Act of Parliament, with the original purpose of acting as the Government’s banker and debt-manager. 1694 was also the year that the French Enlightenment writer and philosopher, Francois-Marie Arouet, better known as Voltaire, was born.

It is also the year from which we can be trace the so-called ‘Halloween Effect’ in the UK. This is the effect seminally confirmed in the leading academic journal, the American Economic Review, in 2002, by researchers Sven Bouman and Ben Jacobsen. In a paper titled, ‘The Halloween Indicator, Sell in May and Go Away: Another Puzzle’, they test the hypothesis that stock market returns tend to be significantly higher in the November-April period than in the May-October period and find it to be true in 36 of the 37 developed and emerging markets studied in their sample between 1973 and 1998, though it can be observed in their UK data back to 1694. They further find the effect is particularly pronounced in the countries of Europe and that it persists over time. The puzzle was especially noteworthy because the anomaly has been widely recognised for years, though not previously rigorously tested, yet it still persists.

Some analysts trace this back to the practice of the landed classes of selling off stocks in May of each year as they headed to their country estates for the summer months, and re-investing later in the year. Times have moved on, but summer vacations and attitudes may have not. That’s a theory, at least, but a strange one to explain modern-day investment strategies if true.

In a bigger follow-up study by Jacobsen now working with finance professor, Cherry Yi Zhang, published in 2012, they looked at 108 countries, using over 55,000 observations, including more than 300 years of UK data. And guess what! They found that the effect is confirmed for 81 of the 108 countries, with the post-Halloween returns out-performing the pre-Halloween period returns by 4.52 per cent on average, and by 6.25 per cent over the past 50 years looked at alone.

Strange but true! The Halloween Indicator seems to be working better than ever.

So it’s Halloween today. Time to get stuck into the stock market? You decide!

If Mr. Smith wants to SELL me his horse, do I really WANT to buy it? It’s a question as old as markets and horses have existed, but it was for many, many years, one of the unspoken questions of economics.

It is a question Groucho Marx once posed in a slightly different way, when he declared that he refused to join any club which was prepared to accept him as a member.

So how do we solve this paradox? The paradox of the horse that is, not the Groucho paradox.

For most of the history of economics, the answer was quite simple. Simply assume perfect markets and perfect information, so the horse buyer would know everything about the horse, and so would the seller, and in those cases where the horse is worth more to the buyer than the seller, both can strike a mutually beneficial deal. Gains from trade, it’s called.

In the real world, of course, life is not so straightforward, and the person selling the horse is likely to know rather more about it than the potential purchaser. This is called ‘asymmetric information’, and the buyer is facing what is called an ‘adverse selection’ problem, as he has adverse information relative to the seller.

This is a classic case of what economics professor George Akerlof sought to address in 1970, in a seminal paper called ‘The Market for Lemons’.

Akerlof had become intrigued by the limited tools that economists were using in the late 1960s. Unemployment, so went much of the general thinking, was caused by money wages adjusting too slowly to changes in the demand for labour. This was the so-called ‘neo-classical synthesis’ and it assumed classic markets, albeit they could be a bit slow to work.

At the same time, economists had come to doubt that changes in the availability of capital and labour could in themselves explain economic growth. The role of education was called upon as a sort of magic bullet to explain why an economy grew as fast as it did. But that posed a problem for Akerlof. How can we distinguish the impact on productivity of the education itself from the extent to which education simply helped grade people, he asked. The idea here is that more able people will tend on average to seek out more education. So how far does education in itself contribute to growth, and how far does it help simply as a signal and a screen for employers? In the real world, of course, these signals could be useful because employers are like the horse buyers – they know less about the potential employees than the employees know about themselves, the classic adverse selection problem.

Akerlof turned to the market for used cars for the answer, not least because at the time a major factor in the business cycle was the big fluctuation in sales of new cars. He quickly spotted the problem. Just like in the market for horses, the first thing a potential used car buyer is likely to ask is “Why should I WANT to buy that used car if he wants so much to SELL it to me”. The suspicion is that the car is what Americans call a ‘lemon’, a sub-standard pick of the crop. Owners of better quality used cars, called ‘plums’, are much less likely to want to sell.

Now let’s say you’re willing to spend £10,000 on a plum but only £5,000 on a lemon. In such a case, the best price you’d be willing to pay is about £7,500, and only then if you thought there was an equal chance of a lemon and a plum. At this price, though, sellers of the plums will tend to back out, but sellers of the troublesome lemons will be very happy to accept your offer. But as a buyer you know this, so will not be willing to pay £7,500 for what is very likely to be a lemon. The prices that will be offered in this scenario may well spiral down to £5,000 and only the worst used cars will be bought and sold. The bad lemons have effectively driven out the good plums, and buyers will start buying new cars instead of plums. Just as with horses, asymmetric and imperfect information in the used car market has the potential, therefore, to severely compromise its effective operation.

What can be done about this? For the answer we must go back to part of the reason why people seek education, which is to signal personal qualities which might otherwise be difficult to discern. This is part of the wider theory of signalling and screening, and it takes us to another place, on another day.

Reference: Akerlof, G. (1970), The Market for Lemons: Quality, Uncertainty and the Market Mechanism. Quarterly Journal of Economics. 84:488-500.

In a game first popularized in the academic literature in the early 1980s, two people are invited separately to play a game, with a monetary prize. The players are acting anonymously, playing the game via a computer terminal. The game, widely known these days as the ‘Ultimatum Game’, involves one of the players, who we will call Jack, being given £50, say, by the experimenter. He must now decide whether or how much he should offer the other player, who we will call Jill. Remember that Jack and Jill don’t know each other, and will remain anonymous to each other. The game is only played once, so there is no comeback from Jill whatever Jack does. There is, however, one consideration for Jack to think about. If Jill turns down the offer, they both walk away empty-handed. So how much should Jack offer Jill? And how much will he? Traditional economic theory about rational behaviour would suggest that Jack, as a profit-maximizing agent, should offer Jill a very small amount, and that Jill should accept this very small amount rather than get nothing. In fact, early experimenters who put real people into this scenario usually found that the amount of money offered lay somewhere between 50-50 and 65-35. Sometimes, nevertheless, the second player was indeed offered only a small amount, and in those cases where this was less than 30% of the prize, usually refused. In other words, when Jill was offered less than £15 of the £50, she usually walked away from the deal, leaving both with nothing. Is this reconcilable with rational economic behaviour? One explanation that is often proposed is that offers of less than 30% or so are considered as desultory, even insulting, and Jill is getting utility (as economists would call it) from punishing Jack. Yet the low offer made by Jack is not in fact a personal insult, and arises as part of the design of the game. Indeed, neither player will ever know who the other person is. It is certainly not profit-maximizing behaviour by Jill. Is there another explanation? One explanation that makes some sense, proposed in the mid-1980s by the distinguished mathematician and game theoretician, Robert Aumann, is that people tend to evolve rules of thumb according to which they behave in their day-to-day lives. One such rule he identifies as “Don’t be a sucker; don’t let people walk all over you.” This might indeed work well as a general rule for Jill to live by, insofar as it helps build up her reputation for people’s future reference. But in this particular situation, turning down £15 does nothing to build up her reputation, because she is anonymous. Aumann’s explanation is that Jill doesn’t think like that. She has built up this rule-of-thumb behavioural code over a lifetime, and will not so easily abandon it a particular context, when the situation is different. This is what we might call ‘bounded rationality’, in that people do not usually consciously maximize in each decision situation, but instead use rules of thumb that work well “on the whole”. So, that leaves a couple of questions. The first is whether Jack is being rational when he offers a small slice of the cake to Jill, and the second is whether he is he being altruistic or self-interested when he offers her a bigger slice? Reference: Robert Aumann, Rationality and Bounded Rationality: The 1986 Nancy L. Schwartz Memorial Lecture.

Derren Brown, the illusionist, is no stranger to the use of the idea of the wisdom of crowds as part of his entertainment package. A few years ago, for example, he selected a group of people and asked them to estimate how many sweets were in a jar.

All conventional ‘wisdom of crowds’ stuff, albeit wrapped as part of a magical mystery tour. His relatively more recent venture into this world of apparent wisdom went down a rather singular avenue, however, as he explained how a group of 24 people could predict the winning Lottery numbers with uncanny accuracy.

The idea in essence was that each of the 24 would make a guess about the number on each ball and the average of each of these guesses would converge on the next set of winning numbers. It appeared to work – but that is the thing about illusionists; they are good at producing illusions.

I will not go into how he did generate the effect of predicting the lottery draw, because there is no point if you already know, and because it would spoil the fun if you don’t. What is sure, however, is that the musings of the crowd had nothing to do with it.

But why not? After all, if the crowd can accurately guess the weight of an ox or the number of jelly beans in a jar, why not the numbers on the lottery balls? The simple answer, of course, is because the lottery balls are drawn randomly. And the thing about random events is that they are unpredictable. This is at the heart of what economists term ‘weak form market efficiency’, i.e. that future movements in market prices cannot be predicted from past movements. In this sense, the series has no memory.

So what is likely to happen if you do get a group of friends around and ask each to guess the number that will appear on each of the balls drawn next Saturday? If you take the average of the guesses about each in turn, my best estimate is that you are likely to end up with a prediction for each ball that is about 30 or less. Why so? Partly this is because people tend to pick birthdays but it’s also because the averaging of a large number of guesses is likely to produce a number somewhere nearer the mid-point of the set of numbers than the extremes.

But if you do use these numbers and just happen to win, you’re likely to be sharing your winnings with a lot of other people who’ve chosen the same numbers as you. The better strategy is to populate your ticket with bigger numbers, which are likely to be less popular.

This strategy won’t alter your chance of winning but it will increase how much you can expect to win if you do win. And that is no illusion!