The Blindfolded Archer
- The (Mis)behaviour of Markets: A Fractal View of Risk, Ruin and Reward by Benoit Mandelbrot and Richard Hudson
Profile, 328 pp, £9.99, September 2005, ISBN 1 86197 790 5
A lycée in Lyon, 1944. A young Polish refugee is hiding in the school. His identity papers are forged, and deportation to the death camps may await him if he is caught. His attention, however, is not on the dangers outside but on a mathematics lesson. When he first started taking the classes, ‘he sat uncomprehending before the meaningless words and numbers on the blackboard.’ Today, though, a picture has coalesced in his mind. It renders redundant the teacher’s protracted algebraic manipulations. He raises his hand: ‘Sir, you don’t need to make any calculations. The answer is obvious.’
In later decades, the young refugee, Benoit Mandelbrot, was most famously to deploy his talent for geometric intuition in what came to be known as ‘chaos theory’. To the search for deep pattern underlying apparently utterly irregular phenomena, Mandelbrot’s key contribution was ‘fractal’ geometry. The word is his coinage: it invokes the Latin frangere, ‘to break’ (past participle: fractus). His is not the geometry of Euclid’s and Plato’s absolutely straight lines and perfect circles. ‘How long is the coast of Britain?’ – a question asked by Mandelbrot in what became a famous article in Science in 1967 – raises issues of a different sort from ‘what is the circumference of a circle?’ One will get different answers if one measures a coastline from aerial photographs, by walking round it with a ruler, or by considering every grain of sand. Finding pattern in coastlines – and in other intrinsically rough, jagged, fragmented, irregular shapes – requires a geometry of a new kind. One can’t hope for a definitive measure of the length of a coastline but, Mandelbrot showed, one can characterise its degree of roughness by generalising the notion of ‘dimension’ so that fractional dimensions are possible.
Fractal geometry’s most widely-known structure is the ‘Mandelbrot set’. This is a geometric configuration of enormous complexity – a ‘devil’s polymer’, in Mandelbrot’s words – generated by an algorithm so simple that it can be implemented in a few lines of computer program. The set looks as if it belongs not in the pages of a mathematics journal but in a gallery of abstract art, and LRB readers will almost certainly have seen its beautiful filigree intricacies in illustrations and screen savers. (If you haven’t seen it, Googling ‘Mandelbrot set’ will lead you to thousands of sites.)
The (Mis)behaviour of Markets, jointly written by Mandelbrot and a financial journalist, Richard Hudson, covers a much less well-known aspect of his work, his contributions to economics. At their core is an apparently esoteric issue that nevertheless has fundamental practical ramifications, intertwined as it is with the fate of pensions and of savings. The issue is how to characterise mathematically the nature of price changes in markets, especially financial markets. Such changes can reasonably be viewed as ‘random’ or stochastic: in retrospect, if we had perfect knowledge, we might fully understand their causes; in advance, with imperfect knowledge, they are unpredictable. But randomness of what kind?
One form of randomness is epitomised by what statisticians call the ‘normal distribution’. Imagine an archer aiming arrows at a vertical line on a wall that stretches to infinity in both directions. Only a small number of her arrows will hit the target line. (Mandelbrot and Hudson point out that the statistician’s term, ‘stochastic’, derives ultimately from stokhos, which is what the ancient Greeks called the stake that archers used for target practice.) Some arrows will veer to the left, others to the right, and if the archer’s aim is true, we expect their numbers to be roughly equal. Most of the archer’s misses will be close. Large misses will be infrequent: the larger the miss, the less frequent.
The extent to which arrows ‘spread’ around the target line has a measure that can be calculated even if the archer goes on firing for ever: the ‘standard deviation’, or its square, the ‘variance’. If the normal distribution applies exactly, and if the archer shoots endlessly, nearly a third of her arrows will fall outside one standard deviation of the target, but fewer than 5 per cent beyond two standard deviations, and only around 0.25 per cent beyond three. Soon, one gets to frequencies so low that one could watch the archer shooting every second for as long as the universe has existed and not expect to see an impact that far out. Plot the frequency of arrow strikes against their positions along the wall, and the statistician’s famous ‘bell-shaped curve’ – the graphical representation of the normal distribution – appears. Its tails are ‘thin’: extreme events (large misses by the archer) happen very seldom.