Friday, July 28, 2017

Monty Hall Revisited

Follow-up To: Monty Hall Problem and Intuitive Solutions

Full details: wikipedia

The Monty Hall problem was posed and solved by Steve Selvin in 1975. It is loosely based on the TV show Let's Make a Deal. It was correctly answered in Marilyn vos Savant's column in 1990. Approximately 10,000 readers wrote to the magazine, most of them claiming vos Savant was wrong.

I'm interested in the properties of this problem that cause people to resist the solution.

Here is the problem as published in Marilyn vos Savant's column:

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. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

First off, we might forgive her readers since they weren't provided with the standard assumptions. But since no party in several iterations of the controversy ever used the word assumption, I'm inclined to damn them all.

Standard assumptions:

1. The player has no special knowledge, so the original choice is random.
2. The host must always open a door that was not picked by the player.
3. The host must always open a door to reveal a goat.
4. The host must always offer the chance to switch between the originally chosen door and the remaining closed door.

Under the above conditions that game can be restated as follows:

Let's play a game. There are three white boxes. Two are empty. One contains something good. First, a monkey randomly throws red paint at a box. Next, you have a choice:
1. Receive the contents of the red box.
2. Receive the contents of both white boxes.

In either case, since at least one white box is empty, and to enhance excitement for the viewers, you must suffer through the prize-hider opening an empty white box before opening either the maybe-non-empty red box or the remaining maybe-non-empty white box, based on your choice.

I hope that when phrased in this way, the best choice is obvious to everyone.

Why am I allowed to say "both white boxes"? When the red box is empty, the white opening order is completely determined by the random red assignment. When the red box contains something good, either white box can be opened first, but since they are both empty the distinction is meaningless. The action of the prize-hider's first box opening is effectively completely pre-determined by the random red assignment. This random assignment is independent of the player's choice of potential prizes.

Why is this phrasing easier to understand? I think it's because the illusion of the prize-hider's box opening choice is revealed by moving the player's prize assignment choice to before the first white box reveal.

I can't decide if this tells us that human logic is easily confused by insignificant temporal rearrangement or that humans prefer to believe that agents act by mystical self volition rather than admitting the external causes of an agent's action, although I favour the latter.

In either case, I'll end by saying that the matrix has you.

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