Science Magazine has now published a letter to the editor in response to the review, published in early October, of the big Monty Hall book. The letter writer is Simon Levay, of West Hollywood, California. Here it is in full:
In his Book Review “Two doors and a goat” (9 October, p. 231), the answer D. O. Granberg offers to the Monty Hall problem is incorrect. He assumes that the contestant should try to win the car. In reality, a car pollutes the environment and adds nothing to the car the contestant already owns. In contrast, a goat replaces noisy lawnmowers and provides milk, cheese, and (if absolutely necessary) a tasty curry.
Anticipating this very point, I was carfeul, when stating the problem in the book, to say explicitly that your goal was to maximize your chances of winning the car. Not a worthy goal perhaps, but that's a moral question, not a mathematical one.
Interestingly, on the actual game show, contestants who wound up with a goat-concealing door really did win the goat. The contestant was given the option either of claiming the goat, or of accepting a one hundred dollar consolation prize in its place. In the history of the show, no one ever opted for the goat.
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I would have taken a goat. I would have actively TRIED to win the goat and not the car. I actually used to have goat once, a pigmy Cameroon billy-goat, as a stall-mate for my spirited Thoroughbred horse. The goats ar the coolest animals evah.
Goats are totally awesome. I would have gone for the goat even if I won the car!
The player's goal wasn't to win the car. For most, the goal was not to go home with nothing. There's a difference.
Monty's goal wasn't to prevent the player from winning a car. His goal was to make the show entertaining and fun for the home viewer, and winners are more entertaining than losers.
Remember the origial Jeopardy when all three players could keep their winnings. So a player's strategy in Final Jeopardy was not necessarily an attempt to win, in many cases they merely played to go home with some money rather than risking all for a chance to win.
That goat must have felt ... so rejected. Poor goat.
One thing that I have never seen made clear with 'the Monty Hall Problem' is whether the game was rigged. All of the explanations seem to assume, but never state, that the 'Monty Hall' player is attempting to influence what choices the contestant makes and that therefore the 'change your decision' offer somehow reveals something about where the prize is or is not. It would be nice to know if the motivation of the 'Monty Hall' player is an influence on the probabilities, but assuming that it is supposed to be an influence is the only way that the discussion of the problem has made any sense to me.
In the traditional formulation, there are two possibilities: either the contestant has picked the winning door, or one of the two losing doors.
If the player picks either losing door, Monty opens the other losing door. In this case to change is to win.
If the player picks the winning door, the host opens either one of the losing doors. In this case to change is to lose.
The host's motivation doesn't enter into it if these rules are followed. If you're already winning (1/3) then to change is to lose; if you're already losing (2/3) then to change is to win. The host doesn't have the option to just let the player go ahead with their original choice without interference.
Mind you, that's the maths puzzle version of the game. Apparently the game played on the show didn't work quite like that.
I would have taken a goat.Thanks.
A cheap goat is probably less thn $100 dollars. You could take the money and buy a goat and still have some left over.
Even if the contestant prefers a goat to a car, winning the car is the more reasonable goal - they could just sell the car and buy a goat (or several goats) from the proceeds, plus have a bunch of money left over.
But is the letter writer the Simon Levay?
How do you know you are going to get milk and cheese? Isn't it a 50% chance? Should we add that probability to the problem and figure out the strategy if the goal is to win milk and cheese? 8^)
Note the hidden assumption.
It's good to see Science can find space to print important criticism like this when they can never seem to find enough space to print my research articles.
On a side note, if you haven't seen QualiaSoup's "Flawed thinking by numbers" YouTube video, it has a pretty good graphical explanation of the Monty paradox. One of the better I've seen...
I just got my copy of The Monty Hall Problem and was very disappointed to find that the chapter on "Full Monty" was missing . . .
"In the traditional formulation, there are two possibilities: either the contestant has picked the winning door, or one of the two losing doors.
If the player picks either losing door, Monty opens the other losing door. In this case to change is to win.
If the player picks the winning door, the host opens either one of the losing doors. In this case to change is to lose.
The host's motivation doesn't enter into it if these rules are followed. If you're already winning (1/3) then to change is to lose; if you're already losing (2/3) then to change is to win. The host doesn't have the option to just let the player go ahead with their original choice without interference."
So the host acts differently (choosing the one 'losing door', or choosing one of the 'losing doors') depending on what the person has already chosen. I knew there had to be some sort of 'do this if...' change in the problem, but I've never seen it stated as clearly as you did. Thank you.
Goatherds generally slaughter most male kids, since nannies produce and billies are pretty much just a pain in the keester. (Too damn gruff.) A billygoat wouldn't count as a "prize."
Jason,
On what basis do you differentiate morals from mathematics? Oxford mathematician Richard Swinburne thinks moral facts are analogous to mathematical facts. Also, Russell, who you just quoted in a previous entry, famously tried to derive all moral truths from mathematical truths, albeit ultimately in failure.
Yours,
NS
Unless you have have a problem with trolls :)
"...no one ever opted for the goat."
I guess there just aren't any goat too people around any more! Just kidding....
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