Donny Hall Puzzle

The Game Room
1

I came across this puzzle posted by plasmid at Brainden.com:

Originally posted by **plasmid **:

The Donny Hall Puzzle
Monty Hall was out sick today, so his brother Donny Hall filled in as the host of Let’s Make a Deal. But people quickly discovered why Monty was chosen as the regular host in the first place. It was a bad sign when Donny asked a contestant “Would you like to keep that $100, or trade it for the trip to Hawaii inside the mystery box?” Worse was when he offered prizes for drawing a heart or spade and zonks for drawing a diamond or club, but presented the deck of cards face-up for contestants to draw from. Although the producers are sobbing with their heads in their hands off in the production booth, they haven’t yet had a commercial break to give young Donny a crash course in hostmanship. And now he comes to you with an offer to win a car, a donkey, or a lifetime supply of air behind doors #1, #2, or #3. You choose door #2. Donny continues with “All right, the first door you didn’t pick was door #1 so let’s see what’s behind that door.” It’s opened to reveal a lifetime supply of air.

Now you’re given the chance to stay with door #2 or switch to door #3. (Donny also offered the chance to switch to door #1, but let’s ignore that option for the purpose of this puzzle.) Which door should you pick?

So I am always interested in the latest twist in our favorite riddle here at the SDMB. What’s the answer this time folks?

2

I don’t see how Donny has revealed (or even has) any information about the initial distribution of prizes, so switching shouldn’t matter.

3

A life time of air is essentially guaranteed, not something you win on a game show, so you’re not going to switch to that. Obviously that Donny gives away beneficial choices is key. That he would only makes a suggestion to benefit the player, and gives him a choice to change, suggests the player should change.

4

I agree, the car still has a 50-50 chance of being behind your door.

5

Here’s a link directly to the puzzle, not just the website:

http://brainden.com/forum/topic/17868-the-donny-hall-puzzle/

6

The key reasons the Monty Hall problem works are that we’re guaranteed that:

  1. We know that Monty knows the correct answer
  2. We know that he will always reveal a door without a prize.

(Arguably, 1 and 2 are just the same point restated in different ways, but whatever)

We don’t have any guarantees with Donny. It really depends on your interpretations of his demeanor. Basically, it depends on exactly what nature you assume his incompetence takes. If you assume he’s incompetent in that he only gives beneficial hints, you change (in fact, in this case it’s basically the exact same problem). if you assume he is essentially chaos manifest, just randomly fucking up, then we have no new information and staying and switching are equivalent.

7

Which means switching is the better choice. At worst it changes your odds not at all. And, at best, it’ll help you out.

8

I don’t see that this is true. He could be (momentarily) malicious, offering the choice to switch only because he knows your first choice was the best door.

9

Then it changes to a lateral thinking puzzle, disregarding the spirit of the puzzle’s introduction. You’re offering testimony not in evidence.

10

I read it that Donny is completely clueless, and will always show the lowest numbered unchosen door first… even if it has the car! And he would, if the car showed up behind door #1, let you trade for it.

11

But we don’t what his motivation is (if any), but we do know that all of his suggestions favor the player.

12

Unless you know what rules Donny is following, you can’t make any clear declaration about odds.

If you know that the rules of the game are that you choose a door, then Donny opens the lowest-numbered door that you didn’t choose (regardless of what’s behind it) then it makes no difference whether you switch or not (assuming Donny doesn’t let you switch to the door that was just opened, which it sounds like he might do. In which case switch to that one if it shows the car. Or if it shows the donkey and you value a donkey at least half as much as a car).

13

Yes. If Donny’s motivation is to help the player completely, then he would open the door with the car first and allow the player to switch. Because he does not do this, that would imply the car is behind the player’s door.

But I think the wording indicates that Donny is just incompetent. He says he is opening the lowest numbered unchosen door first because it is the lowest numbered unchosen door – and would do so in every game.

To me, this indicates the player unknowingly had a 66% chance of winning, but that the chance has been reduced to 50% with the reveal.

14

Helping the player doesn’t suggest dumbness, so, even though he overtly helps the player, he’s smart enough to (attempt to) conceal his actions. IMHO

15

And that’s my point, thank you. You have to judge on personality because you aren’t given sufficient numbers to mathematically figure it out.

16

Maybe he knows what prize is behind each door but is not allowed to choose which to open, so, given the first open door, he nudges the player to switch instead of stay.

17

I think Donny knows what is behind the doors. After all, he knew Hawaii was in the mystery box. But I believe we should take him at his word for the reason he opened door #1.

18

I feel like we should have heard a quote from Donny offering the chance to switch. Hearing his actual phrasing would likely tell us whether he is clueless or trying to help us.

As it stand, my interpretation is that he is doing his best to help the contestant, and has full knowledge. To that end, in the case where we picked the donkey or the air, he’d have have opened the door with the car and let us choose it. In the event where we pick the car, he does not have a way to give us an easy win, other than opening both doors, but maybe that is expressly forbidden to him. So he does the best he can, which is opening one door and downplaying the option to switch.

The second path most closely matches what happened, so that’s my guess. Stay.

19

The comments there are similar to here, except someone came up with the perfect solution to the puzzle:

“Thanks, Donny,” you say. “I pick the door with the car behind it.”

20

Okay, I have to admit that at this point,

  1. I was laughing out loud, and
  2. In my head, Donny Hall is played by Steve Buscemi.

Anyway, the answer the way I interpret the situation is that it doesn’t matter if you switch. The Monty Hall Puzzle operates on the extremely important assumption that Monty Hall knows where the car is, and always opens a door he knows to be a booby prize. It’s clear his brother Donny is just doing shit at random, and he has not been given any training per the OP, so the way I’m seeing this he opened Door #1 at random and it might well have had the car behind it, but it didn’t. So the remaining doors are fifty-fifty.

If in fact Donny is following some sort of script, I could be wrong, but the OP is giving me enough information to tell me all of this is really quite random.