The Monty Hall problem

I have received a large correspondence on Monty Hall, which has been extremely instructive (at least for me) in elucidating not just the Monty Hall question itself, which is of no consequence, but how we model business decisions more generally. The central reason the problem seems complex, and is so controversial, is the difficulty of establishing, or agreeing, on the extent to which a particular mathematical representation of a real problem accurately describes that problem. This issue is fundamental to all analytic approaches to decision making.

There are three broad approaches to solution. The first (which generates the standard ‘correct’ answer, which is to switch) seeks for the single most plausible representation of the problem. In this representation, Monty knows which box has the car, is obliged to open one of the boxes, and chooses at random between these boxes which have not been chosen and do not contain the car. However this is only one possible account of Monty’s motives and behaviour. There are other, perhaps less plausible, accounts.

That leads directly to a second approach to the problem, which is to attempt to construct an exhaustive list of every possible complete description of the problem. This involves speculation about the possible states of Monty’s knowledge, Monty’s motives, and in turn requires us to take a view of Monty’s expectations of the contestant’s behaviour. There is generally no way of establishing whether or not such a list of possible descriptions is truly exhaustive.

Once the list has been compiled, the next step is to see if there is any strategy for contestants which is robust to the assumptions made, i.e. which gives a ‘good’ result in all possible complete descriptions of the problem. I was initially inclined to think that there was no plausible model in which the contestant was better off sticking than switching – this was the basis of my assertion on Tuesday 23rd August that you must switch – but on reflection, after reading examples suggested in correspondence, I am not so sure. If my conjecture has been right switching would be a (weakly) dominant strategy, i.e. you can never lose by switching but you may gain.

Game theory is one way of dealing with the problem when there is no dominant strategy, i.e. there are some representations in which you gain, others in which you lose. We might look for solutions that (like a Nash equilibrium) that are based on mutual expectations of each player’s behaviour. To do this, of course, we would have to take a view of Monty’s motives, and of the motives of the production company, which presumably determines the rule of the game and issues some – not necessarily complete – instructions to Monty. Alternatively, and more simply, one can just attach probabilities to the likelihood that each problem description is correct. But how do you derive such probabilities?

The third approach follows from this. It would attempt to model the more general problem which gives rise to Monty’s offer and the contestant’s dilemma. ‘Let’s Make a Deal’ is a television show, it must make itself as interesting as possible to contestants, such a result best achieved if the outcome is varied and unpredictable, that evidence shows that a tendency to stick with established choices even in the face of contrary evidence is a powerful human trait, an enjoyable show would have some (but not too many) contestants who lost by switching, and so on. This model may suggest that there isn’t much to choose between strategies but perhaps an edge to switching.

None of these three approaches is the ‘right’ or ‘wrong’ way to look at an issue such as this. Each gives some insight into the problem. In a practical situation where the decision mattered one should consider each approach and be influenced by all of them.

John Kay