Jigsaw Trolley Problem

Sorry fam, I don’t negotiate with game theorists.

So the only winning move is not to play? Caught you, game theorist.

I think the ball is at C3.

Albert knows that Bernard doesn't know where it is exactly, that means the ball can't be in the first two rows, because if it was that would mean there's a chance for the ball to be in column 5 or 6, in which case Bernard would known right away where it is exactly (as those columns have only one ball). This means the ball must be in row C or D and column 1, 2, 3 or 4. We know Bernard was able to deduce the exact position of the ball from this information, which means it can't be column 1 (as there are 2 balls still remaining there). That means it must be in column 2, 3 or 4. We know Albert was able to deduce the exact position if the ball from this, and since row D has 2 balls still in play (collumn 2 and 4), meaning Albert wouldn't be able to guess which one it was, that means it's in row C, as row C has only one ball left in play.

Now idk what the correct choice would be statistically. If I remember correctly, the Monty Hall problem states that there's 2/3 of a chance you'll get it correctly if you switch doors, but since a gold ball was pulled out from behind the first door (the ball at C3 is gold), I think that means there's a 2/3 chance of the first door (the one from which the ball was pulled out of) being the one with all gold balls (since it contains 2/3 of all gold balls). In that case I think the probabilities cancel out? Which means it doesn't increase your chances whether you switch or not.

Beautifully written!
I think the probabilities cancel out?
My conclusion as well. Except that since Jigsaw has taken one gold ball out, door 2 must have slightly better chance of gold ball remaining on average (75% vs 72.2%?).
What I don't understand is how Albert got the information from Bernard.
Written as it is, we can deduce the C3.
But how did Albert understand from Bernards sentence that it's in the 3rd column?
- When Albert says he knows Bernard doesn't know, he is saying that in Albert's row, every single ball has a ball below/above it. This eliminates rows A and B because if Bernard had heard 5 or six, Bernard would immediately know the exact location of the ball (but Albert is 100% sure that Bernard is confused).
  This elimination of A and B is new knowledge for Bernard. He confirms that he was confused, but now he isn't. This means it's not 1 because he recently discovered that it has to be C or D, but there are balls in both C1 and D1 which means he would still be confused if that were the case.
  With all this knowledge, all that's left is a row with one ball, and a row with two balls. Albert says he is not confused anymore, so it has to be the row with one ball.

I'm so confused by the added complexity here. If there's a gold ball behind every door, the choice of door doesn't matter.

We were asked to pull a ball from behind one of three doors, not figure out what two dudes know about balls in a grid.

I choose the following option:

Kick the madman in the nuts, open all the doors and take all of the balls, switch the trolly track myself, save the people, sell the balls, then go back and tie the madman to the tracks and ask him to pick the door with the black and blue balls. If he doesn't choose his own fly as the door, TROLLEY.

I'm think it's implied that you can only take pull out one ball at random from behind a door.
- We don't even get that far. Everytime we try, the dude does something before we get the chance and then changes the rules lol

Commenting before reading other comments

I had the grid solution wrong, but your explanation is sound! Thanks!

Hoping to select the door with a goat, that will headbutt the switch for me.

C3. I used to do these logic puzzles at work.

Both Albert and Bernard know the ball is at a certain row and at a certain column, respectively. Albert first admits two things:

He doesn't know where the ball is. Aka, the row containing the ball has more than one ball in it. This isn't important now, but will be re-used when he asserts it again.
Bernard doesn't know either. If Bernard knew where the ball was simply from the column, it'd be because that column only had one ball in it. Since he knows that Bernard doesn't know given just the row, each ball in that row is in a column that contains more than one ball.

This eliminates rows A and B, since B5 and A6 are the only balls in their columns. For this to work, Bernard now has to understand the above.

Then Bernard admits that now he knows where the ball is. Since we can eliminate or ignore each ball in A and B, it can either be balls D2, C3, and D4 and not any ball in column 1 using logic from #2.

At this point, Albert knows whether the ball is in C or D. If the madman told him the ball was in row C, then he would instantly know the ball is C3 given the elimination of C1 and D1. If he was told it was row D, then he still wouldn't know.

However, Albert admits the former, which tells us it's C3.

Yes and because C3 is a golden ball, you should confidently switch to the second door. Because now it's just the Monty Hall problem with balls instead of goats. When the madman chose a door to opened, he deliberately chose a bad (mixed) door, otherwise he would have given away the correct location. The fact, that he opened the third instead of the second gives you new information, that you can take advantage of by switching, increasing your chances. Had the ball been silver, it might have been revealed to come from a bad door.
Edit: damn I just realized he picked the ball from the first door, not the second. Okay in that case we might actually have to calculate probabilities, but I'm too lazy for that.
- I have no idea what the ball thing is about. I just assumed that since he was a madman, he was just doing madman things.
Thanks for the explanation, but I cannot follow on this line
Since he knows that Bernard doesn't know given just the row, each ball in that row is in a column that contains more than one ball.
Why is that? Why couldn't it be A2 or A3? In this case neither Albert nor Bertrand could tell what row/column this was either, because it would be in a row/column with another ball. How can you exclude any row with overlap with any single-ball columns?
- It couldn't be row A or B because there would then be a possibility the ball was in A6 or B5 and that Bernard actually did know where the ball was. Albert says he knows Bernard doesn't know where the ball is, which means he is 100% confident, and the only rows that allow for that confidence are rows C and D, because each ball in each row is not alone in its column.
  Scenario 1: The madman tells you the ball is in row A. Then the madman whispers the column to Bernard. Can you really be sure that Bernard knows or doesn't know where the ball is?
  Scenario 2: The madman tells you the ball is in row C. From this, you can be completely sure that Bernard doesn't know where the ball is, because any column he would be given doesn't narrow it to a single row.

Can I try again until I get a silver ball? Please.

Yeah, problem doesn't specify whether you want your family to live. Common mistake.
- That's like the old joke of the CIA loyalty test.
  A CIA recruit reached the final stages, at the last step he found his mother tied to a chair and gagged. His prospective boss hands him a gun and says "To prove you are ready for this job and will follow any orders, You'll need to kill your mother, I'll be outside.
  10 seconds after leaving the recruiter hears the bang of a gun... followed by other bangs, clangs of metal.
  Prospective recruit walks out the door.. "The god damn gun you gave me had blanks in it, so I had to beat her to death with the chair"

I don't have that many family members, so I don't know who all those people are. Anyway I'm just gonna pick at random because I am too old and too tired for this shit.

Knowing that the ball was gold gives you Bayesian knowledge about the boxes behind the door, since the prior probability of the host pulling a gold ball from a 6-gold door is different than from the 3/3 door. So you have to multiply Monty Hall probabilities and Bayesian probabilities together.
That assumes the host pulled a ball at random, of course, and not a deliberately gold ball.
- Hmm, you're quite right. My intuition is that the Bayesian portion would exactly offset the Monty hall portion. I think, at a glance, Bayes would give door 1 a 2/3 probability of having 6 gold, but Monty Hall would give door 2 the same probability, so we can effectively cancel these out and just consider a raw probability
  You either have 5 gold or 2 gold 3 silver behind door 1, and 6 gold or 3 and 3 behind door 2, which gives door 2 a very slight edge. Does that check out?

Sounds to me like you're allowed to see the color of the ball before you take it out. This means it doesn't matter which door you open, a gold ball will be available to pull and you win. Checkmate.

Doesn't say it's your family tied to the tracks either.
- It does in the first line?

I don't see no question.

Uh, deal? I walk up to the third door and pull a gold ball from it. I shrug at the madman.

Also, it doesn't say I can only open one box, and all three doors should still have at least one gold ball remaining.

Doesn't say the trolley is a runaway. Don't even need to make deals, you can safely walk away.
- It also doesn't say that the track isn't already headed away from the people.

Smullyan would be proud.

For the rest, well, there isn't even actually a question, I suspect you'd open a door and pick a box and hope that you've got a gold ball to pick, and it's not clear that he's following Monty Hall rules and always opening a bad door, but I think knowing which ball got thrown would make the rest of the odds fall into place.

it's not clear that he's following Monty Hall rules
Whenever I see a badly-specified Monty Hall Problem, I always imagine the host saying something like "Oh, you want to pick door number 1? Well, guess what - the car was behind door number 3 all along! You get nothing but a goat!" And only when you initially pick a door with the car does the host ask "Are you sure you wouldn't rather switch to door number 2? Look, there is even a goat behind door number 3!" Switching doors 100% gurantees you get a goat... or 50% silver balls in this case.