The Flaw of Averages

A couple of years ago, I had the privilege of working with the remarkable
Sam Savage from Stanford University.   Professor Savage and I had developed a set of games for the Exploration arm of an oil company to demonstrate the creation and valuation of portfolios, and he opened my eyes to Jensen’s Inequality, an old statistical chestnut that he has repackaged as ‘the Flaw of Averages’. 

The implication of this is are that just about every business case you’ve ever seen (or written) is wrong.

Counting Your Chickens

Here is an example illustrating the danger of counting your chickens before they are hatched. Let’s suppose that you like an omelette in the mornings, and that you need three hens – no more, no less – to provide you with the raw materials. 

The hens die regularly (Roast Chicken! Yum!) so you need to replace them. You have the option of buying female chicks from your local farmer at £3 each, or of buying fertilised eggs at £1 each. You obviously want to minimise the cost of your omelette, and since each egg has a 50% probability of hatching into a girl, the obvious solution is to buy six eggs: this costs £6, and so is a much better option than paying £9 for three chicks.

If you did that a number of times you would find that getting three hens through buying six eggs costs, on average, £7.37, not £6.  Uh? Saying that each egg has a 50% probability of hatching into a girl is NOT the same as saying that 50% of the eggs will turn out to be girls unless you are talking about very large numbers. Any more than three female hatchlings (and any males) are useless to you for anything other than a moment’s sick golfing pleasure.  However, if less than three females hatch – which will happen one-third of the time - you have to make up the numbers with expensive, pre-hatched, chicks.

So if you can’t guarantee getting three females from six eggs, why not buy
more eggs to make sure of three girls?  Nope, buying 8 eggs costs (on average) £9.43 and 10 eggs costs £11.40, because there is still a chance of ending up with less than three female hatchlings.  In fact, the optimum solution is to buy four eggs: you’ll probably need to buy at least one chick, but overall this mix costs an average of £7.16. 

The cost curve is shown below. If you want to see how the model works, click
here and an Excel spreadsheet will open.

What this tells us is that constraints (such as a minimum or maximum number of chicks) decrease the average value.  So, do options increase the average value? 

Working in a Gold Mine

Suppose a kindly uncle gave you a gold mine for your birthday.  The good news is that it contains about 100 million shovelfuls of ore and that each shovelful, once refined, contains about a gram of pure gold.  You know that the price of gold averages about £8 per gram, but you discover that to refine the gold will also cost you £8.  So the mine is worthless, right?

Dead wrong!  Nobody said you had to keep digging.  The £8 gold price is only an average, and if the price is less than the refining cost you should exercise the option of going to the beach for the day.  Only when the price rises about £8 should you dig.  Even if we take a narrow range of prices (say, a standard deviation of £2) having the ‘stop digging’ option makes the gold mine worth about £17,000.  A spreadsheet illustrating this can be found here.

Now think about the theory of ‘peak oil’ in these terms - the amount of oil available to humanity depends not just on geology but also on whether or not it is economically feasible to extract it. When oil is $30 a barrel, oil companies tend to ‘stop digging’ unless the oil is very easy to extract. When oil reaches $300 a barrel (and it will) then we will find that previously uneconomic reserves are now very attractive. This means that we aren’t going to run out of oil anytime soon, even if we have stopped finding new supplies. Whether we
should burn all those hydrocarbons is another mater, of course.

Stop Using Average Values as Inputs

So here, then, is the Flaw of Averages – almost all numbers have some uncertainty associated with them, and using the
average value for the input number (instead of recognising the distribution) delivers false results for the outputs.  Constraints reduce value, options increase it.

The only exception to this rule is perfectly linear models – ones with no constraints, no price breaks, no options you can exercise.  But for anything more complex than calculating the cost of your grocery cart, you should be explicitly modelling the uncertainty to take the Flaw of Averages into account.

Professor Savage says: “Consider a drunk walking the centre line of a busy road.  He staggers back and forth about three feet from the white line.  The state of the drunk at his average position is alive, but the average state of the drunk is dead!”.

As dead as your business case, in fact.