Two victories for Randomness

I recently came across two smallish examples of where randomness was the solution to two perplexing problems. That is, rolling the dice seems to help you out of a situation where a planned method was not giving you what you wanted.

The first issue is the problem of how to board passengers on a plane. Finding the best way to board people is actually a well-studied problem, both theoretically and in practice, and you can see some of the work here. At the top of the same page there is a nice simulation program which shows you how different boarding strategies play out, and random boarding (just calling out people to board at random) is better than the usual front-to-back boarding that most of us are familiar with.

The reason is that random boarding gives a better utilization of the space in the plane whereas front-to-back boarding piles people into one part of the plane, eventually causing jams in the aisles. The full set of strategies examined are

• Back-to-front
• Rotating-zone
• Random
• Block
• Outisde-in
• Reverse-pyramid

On another topic, a Freakonomics blog post describes how researchers in South Africa are using a randomness trick to get truthful answers from farmers who are suspected of illegally killing leopards and hyenas. The method is called randomized response surveying, where when the farmers are asked potentially incriminating questions they first flip a coin, and based on the result give a yes or no answer to either the incriminating question if it was heads, or a harmless question (do you think the Springboks will win the RWC?) if it was tails. The farmers actually used a die, taking specific actions on which value from 1 to 6 was thrown, but the principle is the same as I have described it.

The trick here is that the person asking the question cannot tell which question the farmer is answering, but the farmer’s answer can be recorded. Statistical methods can then be used to determine the distribution of answers for the two questions, and actually make inferences about the proportion of positive answers to the incriminating question. This method was devised in the 60’s, and by the early 80’s it was being taught at my undergraduate university as part of a first year course.

The Fabled 25 Sigma Event

Last week I was reading a document published by my company called Dealing with the Unexpected, which gives some lessons learnt from the recent credit crisis. Early in the paper the authors speak about a 10 sigma event, or an event that we only expect to see once in 10,000 years. This piqued my interest because I remembered an infamous statement made by some financial leader during the onset of the financial crisis to the effect that we are now experiencing repeated 25 sigma events. Already a 10 sigma event is quite unlikely, but a 25 sigma event is just absurd –exponentially smaller, but just exactly how much? A one in a million year event? One in a billion years?

First, what is sigma? In statistics and probability, the lower case Greek letter sigma is used to denote the standard deviation of a distribution, which as the name implies, is the accepted unit to measure how much an outcome can vary from its mean or average. A Wikipedia article lists the following table for the likelihood of sigma deviations for the standard normal distribution

So at 6 sigma deviations we are already talking about events that occur once every 1.5 million years. Note that the scale is not linear, and the difference between 2 and 3 sigma events is less than the difference between 3 and 4 sigma events.

A 25 sigma event must be very unlikely indeed – so unlikely that the references I searched don’t even bother listing this value, including the news articles that carried the original quote. But after searching directly for sigma events I found a wonderful paper from researchers at the business school of the University College Dublin. The researchers are a little more pessimistic (by a factor of 2) in their calculations since they are only concerned with positive deviations away from the mean, as shown in the diagram below for the the 2 sigma case.

The paper reminds us that the 25 sigma quote came from David Viniar, CFO of Goldman Sachs, who actually said in the Financial Times in August 2007 that

We were seeing things that were 25-standard deviation moves, several days in a row.

Not just one 25 sigma event, but several! So the likelihood of those higher deviations was calculated to be

or 1-in-10^{135} years for a 25 sigma event. For example, this is much less likely than guessing an AES-256 key in one attempt. The researchers offer the following comparison as to how unlikely such an event is

To give a more down to earth comparison, on February 29 2008, the UK National Lottery is currently was offering a prize of £2.5m for a ticket costing £1. Assuming it to be a fair bet, the probability of winning the lottery on any given attempt is therefore 0.0000004. The probability of winning the lottery n times in a row is therefore 0.0000004^n , and the probability of a 25 sigma event is comparable to the probability of winning the lottery 21 or 22 times in a row.

Either Mr. Viniar was very confused or his models were very confused. Or potentially both. A final quote from the paper

However low the probabilities, and however frequently 25-sigma or similar events actually occur, it is always possible that Goldman’s and other institutions that experienced such losses were just unlucky – albeit to an extent that strains credibility.

But if these institutions are really that unlucky, then perhaps they shouldn’t be in the business of minding other people’s money. Of course, those who are more cynical than us might suggest an alternative explanation – namely, that Goldmans and their Ilk are simply not competent at their job. Heaven forbid!

Yes Heaven forbid!