The Monty Hall problem, behavioural economics and the painful truth

Occasionally, the obvious eludes everyone until some bright spark looks at it with fresh eyes, concludes everyone else was wrong all along, and tells it to the world. Everyone then slinks away looking a bit sheepish. This seems to have happened when Keith Chen, a behavioural economist, pointed out that classic demonstrations of cognitive dissonance were an experimental artefact, generated by researchers' misunderstanding of the Monty Hall problem. More here.

The Monty Hall problem is named after a gameshow hosted by someone called Monty Hall. It runs like this:

• There are three doors, one of which hides a car, the other two hide goats.
• Monty Hall lets you choose a door. If you choose the one which hides the car, you win the car.
• Before revealing your choice, Monty Hall opens one of the other two doors, revealing a goat.
• Do you stick with your original choice of door, or do you switch?
• The answer is that you switch to maximise your chances of winning the car.
• The Wikipedia link has the Bayesian analysis, which is quite clear, but in words, you only win the car 1/3 times but select the goat 2/3 times, therefore you should switch, because odds are that you selected the goat the first time.

I then began to wonder if I'd unknowingly been caught by the Monty Hall problem. Have I faced situations where I've not actually recognised it? A painful thought then struck me - I have a habit of "sticking" when investing. That is, I don't change my investments even if other investments in the universe have underperformed by more than my choice, showing that I made the right decision in not picking them.

This is not cognitive dissonance - I am aware that I have that bias and I try to correct for it. It's a conscious recognition that changing my decision has costs - trading costs, opportunity cost, and stress - so I try to minimise portfolio churn. However, Monty Hall logic suggests that expected payoffs on switching are twice as large as expected payoffs from not switching. So have I been fooled by the Monty Hall problem after all?

Here's an example. There are 3 mining stocks which I could buy - BHP Biliton, Anglo American and Xstrata. I choose BHP. Anglo American underperforms, but BHP and Xstrata stay in line with the index. I stick with BHP, as is my wont. The thing is, if Anglo American was a goat, and my research is no better than the average analyst (as is quite likely most of the time), I should switch if I want to go for the car.

If the car was definitely going to be there, I should of course switch. But is it? Probably not. Maybe I should invest in tobacco instead, or defence, if mining stocks are all going to be duds (or goats, according to the analogy). However, working through the combinations gives much the same solution, only this time the probability of picking the outperforming stock if there is a possibility all three stocks are duds is 1/3 when switching, 1/6 when not. Same goes when I increase the number of stocks - it's always better to switch.

Ouch. This is embarrassing - I have been caught out by the Monty Hall problem after all. At least I've realised it though. Another weapon for the investing arsenal.