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Authors: Terry Pratchett
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laws is something of a black art, and the results are not totally convincing, but that just tells us that economics is hard. Poor results notwithstanding, the economistâs way of thinking is a phase space point of view.
Hereâs a little tale that shows just how far removed economic theory is from reality. The basis of conventional economics is the idea of a rational agent with perfect information, who maximises utility. According to these assumptions, a taxi-driver, for example, will arrange his activities to generate the most money for the least effort.
Now, the income of a taxi-driver depends on circumstances. On good days, with lots of passengers around, he will do well; on bad days, he wonât. A rational taxi-driver will therefore work longer on good days and give up early on bad ones. However, a study of taxi-drivers in New York carried out by Colin Camerer and others shows the exact opposite. The taxi-drivers seem to set themselves a daily target, and stop working once they reach it. So they work shorter hours on good days, and longer hours on bad ones. They could increase their earnings by 8 per cent just by working the same number of hours every day, for the same total working time. If they worked longer on good days and shorter on bad ones, they could increase their earnings by 15 per cent. But they donât have a good enough intuition for economic phase space to appreciate this. They are adopting a common human trait of placing too much value on what they have today, and too little on what they may gain tomorrow.
Biology, too, has been invaded by phase spaces. The first of these to gain widespread currency was DNA-space. Associated with every living organism is its genome, a string of chemical molecules called DNA. The DNA molecule is a double helix, two spirals wrapped rounda common core. Each spiral is made up of a string of âbasesâ or ânucleotidesâ, which come in four varieties: cytosine, guanine, adenine, thymine, normally abbreviated to their initials C, G, A, T. The sequences on the two strings are âcomplementaryâ: wherever C appears on one string, you get G on the other, and similarly for A and T. So the DNA contains two copies of the sequence, one positive and one negative, so to speak. In the abstract, then, the genome can be thought of as a single sequence of these four letters, something like AATG-GCCTCAG ⦠going on for rather a long time. The human genome, for example, goes on for about three billion letters.
The phase space for genomes, DNA-space, consists of all possible sequences of a given length. If weâre thinking about human beings, the relevant DNA-space comprises all possible sequences of three billion code letters C, G, A, T. How big is that space? Itâs the same problem as the cars in the car park, mathematically speaking, so the answer is 4 à 4 à 4 à ⦠à 4 with three billion 4s. That is, 4 3,000,000,000 . This number is a lot bigger than the 70-digit number we got for the car-parking problem. Itâs a lot bigger than L-space for normal-sized books, too. In fact, it has about 1,800,000,000 digits. If you wrote it out with 3,000 digits per page, youâd need a 600,000-page book to hold it.
The image of DNA-space is very useful for geneticists who are considering possible changes to DNA sequences, such as âpoint mutationsâ where one code letter is changed, say as the result of a copying error. Or an incoming high-energy cosmic ray. Viruses, in particular, mutate so rapidly that it makes little sense to talk of a viral species as a fixed thing. Instead, biologists talk of quasi-species, and visualise these as clusters of related sequences in DNA-space. The clusters slosh around as time passes, but they stay together as one cluster, which allows the virus to retain its identity.
In the whole of human history, the total number of people has been no more than ten billion, a mere 11-digit number. This is
Hereâs a little tale that shows just how far removed economic theory is from reality. The basis of conventional economics is the idea of a rational agent with perfect information, who maximises utility. According to these assumptions, a taxi-driver, for example, will arrange his activities to generate the most money for the least effort.
Now, the income of a taxi-driver depends on circumstances. On good days, with lots of passengers around, he will do well; on bad days, he wonât. A rational taxi-driver will therefore work longer on good days and give up early on bad ones. However, a study of taxi-drivers in New York carried out by Colin Camerer and others shows the exact opposite. The taxi-drivers seem to set themselves a daily target, and stop working once they reach it. So they work shorter hours on good days, and longer hours on bad ones. They could increase their earnings by 8 per cent just by working the same number of hours every day, for the same total working time. If they worked longer on good days and shorter on bad ones, they could increase their earnings by 15 per cent. But they donât have a good enough intuition for economic phase space to appreciate this. They are adopting a common human trait of placing too much value on what they have today, and too little on what they may gain tomorrow.
Biology, too, has been invaded by phase spaces. The first of these to gain widespread currency was DNA-space. Associated with every living organism is its genome, a string of chemical molecules called DNA. The DNA molecule is a double helix, two spirals wrapped rounda common core. Each spiral is made up of a string of âbasesâ or ânucleotidesâ, which come in four varieties: cytosine, guanine, adenine, thymine, normally abbreviated to their initials C, G, A, T. The sequences on the two strings are âcomplementaryâ: wherever C appears on one string, you get G on the other, and similarly for A and T. So the DNA contains two copies of the sequence, one positive and one negative, so to speak. In the abstract, then, the genome can be thought of as a single sequence of these four letters, something like AATG-GCCTCAG ⦠going on for rather a long time. The human genome, for example, goes on for about three billion letters.
The phase space for genomes, DNA-space, consists of all possible sequences of a given length. If weâre thinking about human beings, the relevant DNA-space comprises all possible sequences of three billion code letters C, G, A, T. How big is that space? Itâs the same problem as the cars in the car park, mathematically speaking, so the answer is 4 à 4 à 4 à ⦠à 4 with three billion 4s. That is, 4 3,000,000,000 . This number is a lot bigger than the 70-digit number we got for the car-parking problem. Itâs a lot bigger than L-space for normal-sized books, too. In fact, it has about 1,800,000,000 digits. If you wrote it out with 3,000 digits per page, youâd need a 600,000-page book to hold it.
The image of DNA-space is very useful for geneticists who are considering possible changes to DNA sequences, such as âpoint mutationsâ where one code letter is changed, say as the result of a copying error. Or an incoming high-energy cosmic ray. Viruses, in particular, mutate so rapidly that it makes little sense to talk of a viral species as a fixed thing. Instead, biologists talk of quasi-species, and visualise these as clusters of related sequences in DNA-space. The clusters slosh around as time passes, but they stay together as one cluster, which allows the virus to retain its identity.
In the whole of human history, the total number of people has been no more than ten billion, a mere 11-digit number. This is
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