Don’t box yourself in when making decisions

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For people in business and in financial services it might be a disturbing conclusion, but even in very simple cases, it is impossible to be certain that a particular mathematical representation of a real problem is a correct description.

Monty Hall, the quiz show host who bemused many FT readers a year ago by trying to persuade them to switch boxes in the hope of winning a car, has a new game. There are only two boxes and one contains twice as much money as the other. When you choose one, he shows you that it contains £100. Will you stick with your original choice, or switch to the other box?

This problem is real. Anyone who has changed jobs, bought a house or planned a merger has encountered a version of the two-box game; keep what you know, or go for an uncertain alternative. But familiar problems are not necessarily easy to model. Monty points out that you can lose only £50 but might gain £100. He says he does not know which box has the bigger prize. If untrue, is he trying to save his employer money, or trying to help you win? You have no way of judging whether the £50 loss is more or less likely than the £100 gain.

Decision theory tells you there is an expected gain of £25 from an equal chance of winning £100 or losing £50. But many people do not see the problem this way. They dislike the prospect of losing £50 more than they like the prospect of gaining £100. This, decision theorists claim, is irrational. If you accepted 100 gambles like this, you are virtually certain to end up with a substantial gain.

But, you may say, I am not playing this game 100 times. I am only playing it once and you cannot guarantee a gain in a single trial. That is true, but it illustrates “the fallacy of large numbers”. On the 100th trial, you are in the same position as someone who is offered the chance to do it once. So you should not do it the 100th time. But then you should not do it the 99th time, or the 98th – or the first.

Even if it is irrational to be more depressed by losses than elated by equivalent gains, this is how many people behave. That may be the best explanation of why the equity premium is so high – volatile assets need to show much higher returns to compensate for the pain of frequently seeing small losses. The argument also suggests a strategy for benefiting from this irrationality. Stop looking at share prices so often and in the long run you will get the benefit of the higher return without the pain of observing volatility.

So perhaps you can steel yourself to follow decision theory and go for the higher expected value. But then you encounter another, deeper, difficulty in Monty’s argument. It cannot make sense that, if you choose the first box, you should always switch to the second and if you choose the second, you should always switch to the first. True, you get the information that your chosen box contains £100, but Monty could have used his argument to persuade you to switch even before he opened the first box: your potential gain is twice your potential loss however much the first box holds.

So the probabilities of gain and loss cannot be equal. Then what are they? You can show that you cannot define a probability distribution for this problem, but this is not a satisfying answer; in several decades, no one has come up with a simple and compelling explanation of why Monty’s characterisation of the two-box problem is wrong.

The message of both the original Monty Hall problem and of this one is that, even in very simple cases, it is impossible to be certain that a particular mathematical representation of a real problem is a correct description. For people in business who rely on models and for people in financial services who must choose between boxes with uncertain contents every day, that is a disturbing conclusion. Next week, however, I will describe a successful strategy for the two-box problem.

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