I’m in the middle of reading Andrew Crumey’s rather intruiging novel “Mr Mee”:http://www.amazon.com/exec/obidos/ASIN/0312268033/junius-20 at the moment. One minor point of interest is that this may be the first work of fiction to contain a description of the Monty Hall problem (see “Brian’s post below”:https://www.crookedtimber.org/archives/002210.html ) in the form of a letter, supposedly written in 1759 from a Jean-Bernard Rosier to the Encyclopedist d’Alembert:

bq. Sir, you may know that many years ago one of our countrymen was taken prisoner in a remote and barren region of Asia noted only for the savagery of its inhabitants. The man’s captors, uncertain what to do with him, chose to settle the issue by means of a ring hidden beneath one of three wooden cups. If the prisoner could correctly guess which cup hid the gold band, he would be thrown out to face the dubious tenderness of the wolves; otherwise he was to be killed on the spot. By placing bets on the outcome, his cruel hosts could enjoy some brief diversion from the harsh austerity of their nomadic and brutal existence.

bq. The leader of the tribe, having hidden his own ring, commanded that the unfortunate prisoner be brought forward to make his awful choice. After considerable hesitation, and perhaps a silent prayer, the wretch placed his trembling hand upon the middle cup. Bets were placed; then the leader, still wishing to prolong the painful moment of uncertainty which so delighted his audience, lifted the rightmost cup, beneath which no ring was found. The captive gave a gasp of hope, and amidst rising laughter from the crowd, the leader now reached for the left, saying that before turning it over he would allow his prisoner a final opportunity to change his choice. Imagine yourself to be in that poor man’s position, Monsieur D’Alembert, and tell me, what would you now do?

{ 11 comments }

Ian 07.22.04 at 4:21 pm

Anyone considering the Monty Hall problem might like to look at this photograph

http://www.fotolog.net/av_producer/?pid=8134635

Brian Weatherson 07.22.04 at 4:38 pm

So what is the reasonable answer for Monseiur D’Alembert? Let’s take it as given that using the mathematical and philosophical resources we now have at our disposal it is provable that swapping is the uniquely justified and justifiable play. The theory of probability was nowhere near as well developed back then as it was now, and from memory conditional probability was a particularly bad mess. Bayes’s paper which started sorting things out wasn’t published until 1764, so D’Alembert wouldn’t have had access to that. Now one could argue that the reasonable thing for D’Alember to have done would have been to figure out on the spot (and with the murderous tribe looking on!) the theory of conditional probability and then correctly apply it. But here’s a more interesting (if slightly vague) question.

Given just the resources available in 1759, is there a mathematical argument that swapping is the right play? I don’t know the answer, but I think such an argument would be quite tricky to come up with. (This being CT I bet someone will now produce such an argument in seconds…)

dermot casey 07.22.04 at 4:43 pm

Not sure when the book you mention was published. Mark Haddons “The Curious Incident of the Dog in the Night-time” describes the Monty Hall Problem in some detail. Its about a child with aspergers syndrome. Its also wonderful

dermot

Chris Bertram 07.22.04 at 5:38 pm

I think Crumey has priority, as his book came out in 2000. But yes, I have heard that Curious Incident is v. good and intend to get round to it.

Mike G 07.22.04 at 5:50 pm

Many times gambling is less about mathematics and probability than about correctly assessing the character of your opponent. The leader of the tribe presumably knows where he hid his own ring. Clearly, he wants to prolong the game. But why give you the option to change your mind? What would bring his men more pleasure â€“ the thrill of victory or the agony of your defeat? Humans seem to take more pleasure in seeing others fail than they do in seeing others succeed. Which would be the greater agony â€“ steadfastly making the wrong choice or knowing that you abandoned the correct choice because you panicked? For the brief moment before your death the tribe would revel in your agony. Thatâ€™s what the leader wants. Stand your ground. And since you really donâ€™t win either wayâ€¦..

will 07.22.04 at 6:07 pm

It seems like the Monty Hall problem can be explained without deploying any real mathematical framework. Perhaps I’m missing something? Here’s what I would argue:

In the case of the game show, one has a 1/3 chance of initially selecting the right door and a 2/3 chance of a wrong selection. Monty will never open the “right” door. Thus if you select a “wrong” door the other alternative he leaves is certain to have the car behind it — which is 2/3 of the time.

(If the tribal chief hadn’t initally known that the right was empty, but simply seized upon an opportunity for a twist on the game, nothing is changed. The situation is exactly as if he had known. The captive not knowing of the twist in advance, as the contestants presumably do, doesn’t change anything, either, as the initial choice is arbitrary and unaffected by such knowledge.)

will 07.22.04 at 6:10 pm

…and I apologize for being redundant in saying “the other alternative.”

will 07.22.04 at 6:19 pm

More difficult to explain, I think, is why this result seems counterintuitive. Initially each door has equal probability of hiding the car, and I suppose the assumption is that opening one door doesn’t change that the two other doors each have a 1/3 chance of hiding the car. Call it “probabilistic independence” or something.

But, in fact, the initial choice is a choice of which door to exclude from the information revelation round where Monty shows that one of two doors doesn’t have the car…

John Quiggin 07.22.04 at 9:45 pm

There is a further problem, which is that the story gives no clear indication that the chieftain was, like Monty Hall, sure to pick a cup that had no ring underneath it.

If you ignore this problem, you could do the whole thing without an explicit theory of conditional probabilities, just by counting, as follows.

Initially, there are three possible locations for the ring. Now consider the chief’s choices. If the ring was under the right or left cup, he has only one choice of cup to lift. If the ring is under the middle cup, the chief has two choices. So there are four cases.

By lifting the right cup, the chief eliminates one case, leaving only three, of which two are favorable to switching. Now a valid application of the principle of insufficient reason will do the job.

Of course, if you did this kind of exercise more than a few times, you’d be bound to discover conditional probability and Bayes theorem. Maybe we’ll discover new evidence that the good reverend was an active follower of 18th century game shows.

Chris Bertram 07.22.04 at 10:22 pm

I’m not sure that’s right John – I mean the “no clear indication” bit. We’re told (a) that the leader had hidden the ring and (b) that he wished to prolong the painful moment of uncertainty.

John Quiggin 07.22.04 at 10:56 pm

Chris, at this point we may need to call in a professional epistemologist. Crumey has the fictional Rosier tell the fictional d’Alembert that the doubly-fictional chief wants to prolong the uncertainty, but Rosier gives no clear indication that the doubly-fictional French captive knows this.

Meanwhile, we, your readers, have been asked to place ourselves in the position of a pre-Bayesian d’Alembert and respond to Rosier. How should we treat the aside about the chief’s intentions?

Hre’s my response:

A Bayesian captive would presumably have a prior which placed weight on both a Monty-Hall chief and a random choice. Since there’s zero effect from switching if the chief had made a random choice, a dominance argument suggests acting as if the chief was Monty Hall.

The fact that no ring was under the cup lifted by the chief increases the posterior weight on the Monty hypothesis, but this is just a bonus.

But I wouldn’t like to try and formulate this part of the argument in anything other than Bayesian terms.

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