Monday, October 07, 2002

The law of small numbers

Today on Life Matters I used the excuse of the grand finals to speak to Rebecca Gorman about what psychologist Amos Tverski calls "The Law of Small Numbers.''

It is an ironic reference. The so-called law of big numbers states correctly that when an experiment such as a coin toss is repeated (say) thousands of times, heads will come up about half the time.

Tverski finds that people wrongly believe this bo be the case for very small numbers as well.

If a couple has two children, both of them girls, people attach significance to this. It must be in the man's genes. Yet this is highly likely (25 per cent) to happen just by chance.

When it comes to runs of heads, they are far more likely than we think. If you toss a coin 20 times, a run of 4 heads in a row (somewhere in the sequence) is extremely likely - the probability of that happening by chance alone is about 50 per cent!

But Tverski thinks that because we are victims of belief in "the law of small numbers" we feel the need to attach significance to a run of four wins in a row.

In sport in the US, this is called belief in "the hot hand". Tverski has shown that for basketball in the US the hot hand seems not to exist. Sone study reports that after a run of successes a certain player had a 75 per cent chance of having another success. After a run of failures the player also had a 75 per cent chance of success.

Neither players nor their supporters believe this.

We seem to have an almost programmed-in need to look for meaning in what might be meaningless noise....

Even the Australian Bureau of statistics publishes trend estimates, some of which are meaningless. I remember in 1993 that the direction of the trend for the current account deficit used to point up in some months, down in others. In reality the underlying movement in the current account deficit probably wasn't changing at all.

Which brings us to the Australian Securities and Investments Commission and its position paper on the advertising of investment returns. ASIC wants the rules governing the advertising of past performance tightened up.

It has commissioned a survey from the Financial Policy Research Centre which examines 100 surveys of the performance of funds managers over time. It says about half of the studies found no correlation at all between good past and future performance. "Good performance seems to, at best, a weak and unreliable predictor of good performance over the longer term."

There are all sorts of reasons why this should be the case. Without knowledge that the market doesn't have (which is illegal in equities) it should be impossible to consistently better predict where the market will end up than the market itself. An investment style which works in one set of market conditions may not work in the next. Successful funds managers will face a run on their staff, everyone will copy them, they will believe their hype. (Remember BT?)

I said on Life Matters that if a firm wins "Funds Manager of the Year" two years in a row, ditching it might be as good advice as keeping it.

Tverski points out that for sport there is no reason why a run of wins shouldn't signal something, it is just that the statistics show that it doesn't.

I must say that I personally found Tverski's findings about perception challenging. Until now I must have believed (subconsciously) that "god's hand" manipulates the outcome of a coin toss to ensure that a run of heads is always followed by a run of tails. I now see (late) that that isn't how it happens. A run of heads isn't reversed by the hand of god, its effect is diluted over time by other results, so that when the number of tosses gets very large the results are about 50-50. (If you toss a coin and get five heads in a row, and then keep tossing until you have tossed 1,000 times in total, probablity theory does not predict that the total number of heads will be 500, it predicts that the number will be about 502.)

Actually I still find the workings of probability hard to get my head around. Perhaps because I know from quantum physics that the hand of god does manipulate the results of wave/particle experiments to give us the result it wants us to see. So maybe our intuitive belief in a hand of god isn't so wrong after all.