Tuesday, April 08, 2008

Religion and the Monty Hall Problem

Hirhurim links to a simulator of the counterintuitive Monty Hall Problem (MHP.) Because he is Hirhurim, and not Kottke, I thought about how the MHP might apply to religion.

The MHP can be summed up as follows (via Wikipedia:)
Suppose you're on a game show and you're given the choice of three doors. Behind one door is a car; behind the others, goats. The car and the goats were placed randomly behind the doors before the show. The rules of the game show are as follows: After you have chosen a door, the door remains closed for the time being. The game show host, Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it. If both remaining doors have goats behind them, he chooses one randomly. After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door. Imagine that you chose Door 1 and the host opens Door 3, which has a goat. He then asks you "Do you want to switch to Door Number 2?" Is it to your advantage to change your choice?


Most people's intuition tells them that it doesn't matter what they do -- both doors have a 50% chance of winning. If you do the math, though, (or run the simulator) it becomes clear that switching gives you a 66.7% of winning the car.

Here is the explanation: when you first pick a door, you have a 33.3% chance of identifying the correct door. But things change when the host has to open a door with a goat. If you have the correct door, which happens 33.3% of the time, he may choose either of the other doors. If you have the wrong one, 66.7% of the time, he has no choice and must pick the only remaining goat-door. In other words, 66.7% of the time, the door he does not pick contains the car.

So how does this apply to religion?

Well, when you are born, you are assigned a random category (a specific religion or atheism or agnosticism or nothing) based on your parents' religion*. This is akin to choosing one of the three doors, because your category was "chosen" out of all existing categories**. However, once you learn a little bit about the world and gain the ability to reason, it's relatively easy to have Monty Hall open some of the false doors -- i.e. you may safely eliminate categories which are clearly incoherent (to you) or simply extremely improbable, like Scientology, Mormonism, and various denominations that hold of Biblical literalism.

You now have a choice akin to the two doors. By the same logic as the MHP, then, and with the added assumption that your religion isn't necessarily more likely to be true than the others***, then your best bet is to switch once you reach the age of knowledge and reason.

Things get more complex if you attempt to rate the categories you can't eliminate according to probability (e.g. you might think Catholicism is twice as likely to be true as Islam, or that Presbyterianism is slightly more likely than Anglicanism) but the overall logic doesn't change. Unless you think you can safely eliminate all other categories or you believe you can fairly rate your own as exceedingly probable, the right move is to switch.


* Obviously, this is an oversimplification, for the sake of clarity.

** The assigned categories are weighted by numbers of adherents and birthrate, of course, but that doesn't necessarily change the underlying logic, since we don't know whether popular religions are more likely to be correct or not.

*** Understanding that people are biased towards (or sometimes against) their given religion should play into this. Also, if you can eliminate your own category, you clearly have to switch.



(See also Bad Religious Arguments: Pascal's Wager, in which I argue that if your primary concern is optimizing your afterlife, you should choose a relatively probable religion that has the worst hell for nonbelievers and the surest way into heaven for adherents.)

EDIT: It occurred to me to google "Monty Hall religion" right after I posted, and it turns out at least one other person had a similar idea.

15 comments:

Mark said...

JA,
Doesn't work.

Iteration for one. Your parents had the same choices, and those before them. They've also used "reason" or other methods to eliminate.

Citing reason as the Monty Hall element is also an error. Take the example from science. We don't choose our explanatory theories based on what is reasonable. Quantum mechanics is completely reasonable, and for that matter remains incoherent. We don't choose them because they are complete (a hypothetical catalog listing the results of every possible experiment would be unsatisfying as a TOE).

You need to run your Monty Hall statistics with a non-omniscient MC and include iteration. I think the results of that will be less weighted toward "switch." (says the guy who has switched no less).

Jewish Atheist said...

Mark,

Iteration for one. Your parents had the same choices, and those before them. They've also used "reason" or other methods to eliminate.

They may have used reason or other methods, but I don't think they used the Monty Hall logic. Also, all the parents from every other religion may have used reason or other methods.

Citing reason as the Monty Hall element is also an error. Take the example from science. We don't choose our explanatory theories based on what is reasonable.

Science requires data. I'm assuming we don't have any good data on the religion question.

You need to run your Monty Hall statistics with a non-omniscient MC

It's true that eliminating, for example, Scientology offers some risk to non-omniscient mortals, I think my strategy of only eliminating religions which are clearly (to the "player") very improbable would lead to decent results. I guess that does rest on an assumption that there isn't a trickster God. If God exists and he's trying to trick us, all bets are off.

littlefoxling said...

an absolutely brilliant analysis.

Random said...

Did you read the comments at "God Snot"(!) as well? Because "327th male" (who's an atheist too) pretty much nailed it as to why the Monty Hall problem doesn't work for this sort of thing.

Jewish Atheist said...

Random:

He nailed God Snot's version, which did rest on a faulty assumption. I don't think mine does. I'm treating atheism as just another category.

Brandon said...

Very, very interesting. As a matter of curiosity, how are you thinking one would determine which door to switch to in cases where there are more doors than three? In the three door Monty Hall problem, the switch is determined by there being only one door left. So if you're randomly assigned to category 'atheist', then 67% of the time the theistic door Monty doesn't open would be right. But if there are say, ten doors, then if Monty opens one, you have eight to switch to, and no information about any of them.

One could do it randomly, perhaps; in such a case, unless I'm mistaken (this is off the top of my head, so I might be), as the number of doors increase the benefit of switching over staying, i.e., the increase in your chances, diminishes. If you're willing to take any increase in chances, however minute, a random switch would still be an increase, no matter how many doors. If we put a cut-off point, so that we consider an improvement of chances not worth the trouble unless it is above a certain threshold, then the number of doors becomes crucial to deciding whether to switch or stay, since there is a certain number of doors-unopened-by-Monty beyond which one would prefer to stay rather than switch because switching would improve your chances so little. And if that were so, the atheist would have no reason (sufficient to act on) to switch from the atheistic door to a theistic door. (Also, as the unopened doors increase an ever smaller superiority of the probability of the original door over the unopened doors is needed to swamp out the advantage of switching.)

Suppose you have atheist, Hindu, Christian, Sikh, Muslim, and Scientologist doors. First choice: atheist, by upbringing. Monty opens the Scientologist door. Now that's four doors left. Instead of having a certain chance of getting the right door (as with three doors), you have a 25% chance. But your original chance of being right with the atheist door was about 17%, so if atheism were (say) 1.5 times as probable as the most probable of the alternatives, you have incentive to stay rather than switch. If, on the other hand, there are 100 doors, your original chance is 1%, but if Monty only opens one door, your chance of being right by randomly switching to another door is only slightly larger than that, and atheism only has to be a tiny bit more probable than the most probable alternative to merit a stay rather than a switch.

Random said...

"He nailed God Snot's version, which did rest on a faulty assumption. I don't think mine does."

It does, to be precise the one 327 raises in his second post. Basically, the problem is that the Monty Hall problem relies on Monty (or you, as the author of the post) knowing what the correct answer is, while the contestant does not. Now you may believe atheism is the correct answer, and you may feel it is the simplest and most logical explanation for the world as you see it, you may be secure in your own conscience that it is, but you cannot know as a matter of objective fact that it is until you die.

Basically, you are making an assumption, asserting it as fact for the purpose of the problem and then claiming that the outcome of the problem proves the initial assumption. I trust I don't have to point out the flaw in this logic?

Somewhat ironically, the only entity with the knowledge required to function as Monty in your version of the problem would be God (or Allah, or Brahma, or Amaterasu if you must) himself! And if He's playing the game you've already lost...

Jewish Atheist said...

Random:

But I'm not saying the correct move is to switch to atheism. I'm saying it's to switch to anything other than what you started with. And there may be no Monty Hall, but I think most of us are confident enough about eliminating Scientology, Biblical literalists, etc., that the math would still work.

Random said...

"But I'm not saying the correct move is to switch to atheism. I'm saying it's to switch to anything other than what you started with."

Ah, got you - you're making a somewhat more subtle point than the one I thought you were making. I still think my objection is valid though, as the Monty Hall problem only works if Monty knows what the correct answer is and only opens doors that reveal incorrect answers. If he doesn't know and instead opens doors at random, then by the time there are only two doors left there's no benefit to switching. So the validity of the problem basically boils down to how much you trust Monty to know what the correct answer is.

Theresa said...

Rethinking switching religions is something that everyone should do when they come to the age they can reason. I think most everyone could agree on this. If they have made a good decision, then it cements them in their belief, if not, it helps them really seek out the truth.

Brandon's comment is good and accurate (I'm talking about how towards the end he incorporates the idea that you have some knowledge of the correctness of your door and what is behind it) and I was going to write something similar until I read it. With more doors, the benefit of changing becomes less in the Monty Hall problem although there still is one.

More importantly, in religion, the doors aren't thick wood you can see through. They are like opaque surfaces that become more and more translucent as you search further. If you could even dimly see the shape of a car behind your door and the shape of a goat behind the other, it wouldn't be a good idea to change.

Theresa said...

I guess overall, my criticism is that your reason which is necessary in order to open the door to see the goat religion also is able to work to let you know what is behind your door. So the slight probability benefit of changing is way overshadowed by the ability to use reason and to stay with your own religion assuming it looks like a car.

Stephen said...

Evangelical Christians have a saying: "Bloom where you're planted."

(It isn't to warn people against changing religions; it's to warn people against changing churches. The idea is, the grass isn't actually greener on the other side of the fence.)

Obviously the saying is self-serving. Nonetheless, I agree with Theresa. I think it's important, when one reaches the age of reason, to begin to think critically about one's own religion, and to learn to appreciate the strengths of other religions — and yes, of atheism.

But I'm not sure that switching on the basis of the Monte Hall problem is really such a wise strategy. I might better build on the strengths of my faith, while guarding against its weaknesses. Even if my religion turns out to be false (a mathematical probability), its strengths are likely to be an asset to me.

Tigerboy said...

stephen:
"I think it's important, when one reaches the age of reason, to begin to think critically about one's own religion"

That's a wonderful idea, Stephen.

However, to use that age-won reason, and to actually think critically about one's religion, will only lead one to its abandonment.

There is not a shred of concrete evidence for any of the world's religions.

Reason, and a critical eye, are the opposite of "faith."

"Faith" is required in order to believe unreasonable things.

Anonymous said...

The monty hall problem is a bad analogy.

The game show host must know which door the prize is behind in order for it to be a 67% chance to switch. If the game show host doesn't know which door its behind its a 50/50 chance.

In the case with religion we don't know which door is definitely false to eliminate, it is a completely random guess. Sticking or switching religions in this case is a 50/50 guess.

Meyer E.

Tatarize said...

Yeah, I (Ssnot) came to think it doesn't quite fit. There are some real pressing points of disanalogy and it really does sort of assume the conclusion by making the atheist the equivalent of Monty Hall who properly knows the truth of the matter.

Although, there are some real oomph along these lines. It really is wrong to assume your religion must be more likely to be right, just because you deem other religions to be wrong.

http://godsnotwheregodsnot.blogspot.com/2009/12/modicum-of-thought-parable-of-cards.html