The Monty Haul Problem(s): Debunking Distortion

From Wikipedia, the Monty Haul problem is pretty much this:

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?

A lot of people will now say they’ve seen the light on it – they’ll say it IS to your advantage to switch doors. That they were adamant it was 50/50 before, but it’s not!

Here is a revised consideration for that debunking, developed onward from the previous post found under the tilde.

Let’s establish some commitments first – is punching Monty in the face and opening every door until you find the car then driving off in it GTA style a legitimate solution to the Monty Hall problem?

I’m guessing most would say no, since it essentially breaking the rules of the game.

How about this – what if he presented you with two doors, showed you what’s behind them but then covered your eyes and closed and shuffled the doors. BUT you noticed a scratch on the door frame of the door with the car. Monty asks which of the two doors you will choose. He shows you again and dang, does that scratch just stand out to you!  Is that really a solution to the game?

I think people in general might vary on this one. But I think most mathematicians would probably say no, it’s not a solution. It’s a get around. Perhaps even a cheat. It’s breaking the rules of the game.

What about knowledge bleed over? Say it’s proposed to you to play a single player game of memory (trying to finish as quickly as possible), you complete the game then someone proposes a new game of single player memory ‘memory’, but all they do is flip all the cards on the table face down again. For a start, this isn’t how to play memory, it requires shuffling. And secondly you’ve seen all the cards – you are going to have knowledge bleed over from the previous game. You are neither playing memory plus you have bleed over from the previous game.

Is bleed over a legitimate solution? Again people might vary. But I think many mathematicians would probably say no, it’s not a solution. It’s a get around. Perhaps even a cheat.

Further how much are you going to be tempted to use that knowledge if it potentially means winning a new car? How biased towards using it are you going to be?

Indeed, how biased will you be towards saying the second game was a perfectly fine game of memory? Because you could use bleed over to your advantage in winning a car?

So here’s the issue – In the original game Monty asks you to pick a door. Then he kind of ignores what you’d pick as he doesn’t open that and he instead removes another door. He then asks which of the remaining doors you’d pick.

The question is, did Monty just offer you a new game?

Taking it he’s run a new game, it is exactly the same as the single player memory game issue – he’s run one game with you and now you have bleed over from the first game into the second game.

But the original debunking of the 50/50 relies on the knowledge from the first game to be valid for use in the second.

Does the original debunking actually prove there is only one game?

~~~

One of the main arguments I heard for it uses an exaggeration of the problem to highlight the ‘solution’. I think it does the opposite, so I’ll use it myself: Instead of three doors, imagine there are ten! Now there are two choices – the player’s original choice and the second choice Monty provides. If there are ten doors, Monty is going to take away 8 doors, leaving us with two doors.

The thinking goes that the right door is out there, amongst those other 9 doors that the first choice doesn’t cover. And for the second choice Monty has just taken away 8 bad doors. The second option has a 9 in 10 chance of having the winning door! It has to make sense to swap to the second choice!

While some will note that sure, you’re original choice had a 1 in 10 chance of being right. But that’s small odds, right? We can forget about that, right?

Indeed, forget about it so much we do bad math?

Let’s slow down the door removal instead of doing it all at once. Now The first choice has a 1 in 10 chance of being right.

Does that mean the second choice has 9 doors sort of assigned to it?

Well no, the second choice has a 1 in 10 chance by default as well. Monty doesn’t control if you guessed the right door to begin with.

BUT surely that first choice is only a 1 in 10 chance – let’s forget about it like we forget about our chance at winning the state lottery. Dismissible odds (bias).

Here’s part of the problem with the original debunking – there’s a cognitive predilection to assign the second choice a 9 out of 10 chance of having the right answer. An urge to apply a symmetry there. After all the second choice covers 9 of the other doors, right?

If you slow down the removal of the other doors you see that it doesn’t work that way.

If we remove one dud door from the second choice, we now have 9 doors. So does that increase the odds for the second choice?

Remove another and we have 8. So does that increase the odds for the second choice?

The problem here is a fixation on the second choice.

When you remove a door, the odds of the first choice being right go up, from 1 in 10 to 1 in 9.

When you remove another door, the odds of the first choice being right go from 1 in 9 to 1 in 8.

Lets go straight down to 3 doors.

The first choice has a 1 in 3 chance.

The second choice?

We remove a door. Now the first choice has a 1 in 2 chance.

And the second choice a 1 in 2 chance.

50/50

It’s because you’d done bad math and not increase the odds of the first choice as you redid the odds of the second choice. The first choice appeared to be dismissible odds – so they fell out of your work tray when it came to doing the math. You had such low odds of getting the right answer and Monty was removing all the doors but the winner (IF you hadn’t chosen the right choice already) that you automatically ignored looking at your chance of winning a second time. You didn’t increase it as you should because you’d dismissed it from the equation.

Wait, it’s worse than that! You applied a symmetry bias and a permanence bias on top of that – if the first choice has a 1 in 10 chance, you forced the idea of the second choice having a 9 out of 10 chance of succeeding and as if it would just stay that way. Then an icing of confirmation bias; of not going through each door removal one at a time and working out the odds. Or you did, but dismissible odds bias meant you ignored working out the improving odds of the first choice. One or more biases kicked you this way or that.

  • With 10 doors: The first choice has a 1 in 10 chance.
  • With 9 doors: The first choice has a 1 in 9 chance.
  • With 8 doors: The first choice has a 1 in 8 chance.
  • With 7 doors: The first choice has a 1 in 7 chance.
  • With 6 doors: The first choice has a 1 in 6 chance.
  • With 5 doors: The first choice has a 1 in 5 chance.
  • With 4 doors: The first choice has a 1 in 4 chance.
  • With 3 doors: The first choice has a 1 in 3 chance.
  • With 2 doors: The first choice has a 1 in 2 chance. 50% chance.

So how do we express the odds of the second choice without falling to symmetry bias or the other biases?

The trick is there is no second choice, you’ve been fooled (as have I). There is no need to express the odds of ‘the other choice’ because there is no other choice present. The above list shows the odds of winning or losing – that’s it.

Consider if I flipped a coin and asked you heads or tails. You call heads. Then I say ‘Do you maybe want to dance a jig first?’.

You’d think it an utterly redundant question.

The switch question is utterly redundant. It’s the final bias – a ‘red herring bias’, where information given is treated as if it has to matter somehow rather than potentially just be utterly redundant.

Do you want to switch? Do you want to dance a jig?

Neither makes any difference. Only by the (semi white lie) deception that is heavy emphasis does this red herring seem to matter. Are you sure you don’t want to dance a jig? Can you really be certain about your choice of heads on the coin if you don’t dance a jig? Why not work out the odds of how a jig would change the result?

Do you want to switch your understanding of the Monty Haul problem? Ironically if you disagree with this debunk of a debunk, you perhaps wont put much effort into thinking about whether you should switch. It’s a positive outcome bias – thinking about switching because switching might lead to a car seems good because positive outcome. Thinking about switching because it might lead to you being wrong just sucks. Which are you more inclined to put effort into thinking about?

Hopefully the slower, bullet pointed single door by door removal will help, as you would agree the first choice has a 1 in 10 chance when there are 10 doors. And the progression of the odds for that choice being correct will make sense as well. So before you go to argue your logical proofs, please say what is wrong with that bullet lists progression?

The hardest thing is likely you thought you were right about 50/50 to begin with – then you thought you’d eaten humble pie on it, hard earned your new knowledge on the matter, accepted you were wrong and now you proudly know the truth. Sunk cost fallacy.

There’s more pie. That’s a general rule of things in the world. There’s always more pie. Epistemic humility and peace out, ya’ll.

 

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Against the Giants: A snippet of actual play

This battle took about an hour. I posted this on facebook and it turned out so long I think it needed a blog entry as well:

I think a highlight last night was half the party down at the base of the fire giants throne room. The king, gravely hurt, decides to go lick his wounds and sends his guards to kill the interlopers.The wizard Annalena winks away, teleporting two hundred feet away down the smokey corridor leading to the throne room. Dell the rogue zips out to apply a healing potion to Jerek the sorcerer, who, revived, then pulls a cloak of elvenkind over herself to appear as a rock – next to the rock the king threw that felled Jerek! Then when no one is looking, casts invisibility on herself – as a rock just disappearing would look odd! Dell manages to administer a potion to Korall the palladin, but when he flees to the collumns to hide, the giants follow!

This is as Dell finds that while his hiding is great, enemies need only go behind the column he hid behind to find him – giant long swords hack at him! Wounding him fairly gravely!

However, their attention is distracted – and Korall gets to his feet behind them and walks to the last known place of the king, the throne. He stealths away (nat 20!)

To where invisible Jerek is looking for the secret exit the king took, after investigating the kings disappearance (nat 20!). Korall is disturbed by the voice from nowhere! And they find no secret door!

Meanwhile Dell has fled, evading the giants and running headlong down the smokey hallway. At the other end, invisible Annalena calls to him, trying to speak loudly enough to be heard and softly enough to not be heard by the giants, for Dell to come to her. And perhaps a GM was too nice about that working out, but indeed it does – Dell runs at full speed into the smoke of the corridor, clasp forearms with Annalena and teleport once again! To behind the throne! Korall sees Dell appear and gets another invisible voice talking to him as well! While the giants, who saw Dell running off in the other direction, move away from the throne room!

Annalena learns form Jerek the king has escaped by some kind of secret door and casts pass wall. Suddenly the secret exit is revealed to them and they press through, going from near total party death to instead hunting down the giant raider king!

Schrodinger’s Railroad

There is a recurring sentiment in gaming culture I find fascinating, since it seems an utter paradox. Here’s a recent and clear assertion of it:

Railroading is forcing players down a narrative route . If the players think it’s their decision, they’re not being forced. Agency is in the mind of the player; they can feel they have it when they don’t and they’ll be happy. But if they don’t feel they have any, even when they do, they’ll become disenfranchised.

A DM’s job is to make it so the players always feel like they have agency, to make them feel their characters are in danger. D&D is a game of illusions, and the DM is the man behind the curtain.

Original Comment

It is so odd. For example, how could players ever get disenfranchised? Consider the chronology of disenfranchisement

  1. The DM makes the players follow his decision
  2. The players feel it is their decision
  3. The players somehow begin to realise it is not their decision
  4. The players are disenfranchised

Okay, so if it is not railroading when the players feel it is their decision, how could players ever get to step 3 when by the logic of the quote in step 3 there is no railroading to detect?? Like Schrodinger’s cat being both alive and dead, somehow it is both railroading and not railroading at the same time?

Ultimately it’s probably pretty simple – the whole notion likely comes from the idea that the agency described is the best agency you can get. The idea being the best agency you can get is one where the GM is making the decisions – the only thing to consider is if the players have their nose rubbed in it that the GM makes the decisions or they are relatively witless that he makes the decisions.

The idea of an agency where the players actually make the decisions – it’d probably sound ludicrous to anyone who has advocated the quote for a long time.

Nearly completed ‘The Blasting Diamond’

It’s been interesting to convert GM advice into an actual adventure – instead of arguing against, for example, ‘you have to do the adventure’ thinking by GMs, actually writing an adventure area that supports PCs that don’t get ‘hooked’ but actually do what their character wants to do.

And calling it an ‘Adventure area’ rather than just ‘an adventure’ is even a result of turning GM advice into material. You don’t ‘have adventures’, you engage with an area of an imaginary world. What is an adventure is something that is seen in retrospect, like a story generated at the table. You don’t decide to have an adventure, you just engage in things that in retrospect was an adventure. And areas of imaginary world can be seeded to more likely trigger an actual adventure to occur.

Quantum Ogre: Redemption

Man, I thought I could find the original of this with a quick google, but it seems only refutations abound!

Okay, in short, the original problem was that you’re at a fork in the road. If you go left, the GM makes an ogre pop out. If you go right, the GM makes an ogre pop out.

The problem – well, it’s curious – I think the problem should be obvious. And I’m tempted to link a controversial forge post as a suggestion as to why some/many(?) don’t see it as a problem (or so google makes it seem).

Anyway, at that point you’re being played for a fool by the person who is GM, who probably has a Gordian knot of control issues about story, his own artistic agency and his perceived responsibilities and with that responsibility, sense of his own scope of control (yeah, I know I mentioned control already – I said it was a Gordian knot!). Of course peeps refute that it’s a problem because when you think when option A is having quantum ogres and option B is having a game that doesn’t work at all, of course you refute that A is a problem. Of course it seems like B is the only alternative, because the GMs in question perceive that anything else gets in the way of them telling a good story – ie, gets in the way of their creative agency. Yeah, again, I know, I mentioned GM agency already! I said it was a Gordian knot – it’s like the idea of ‘elephants all the way down’, but here it’s like there’s a responsibility to the players, which rests on top of the GM getting to tell a good story, which rests on top of B: not having a game at all, and you know why we can’t have that game…well, it’s because of the responsibility to the players, so B sits on top of responsibility, and responsibility sits on top of the GM telling a good story, which sits on top of ‘B’…and haven’t we been here before? How can everything be resting on everything else?? Elephants. All the way down!

What makes it more curious is that I think it doesn’t take much to redeem it – for example, if you went right and there’s an ogre there and if you went left there’s an ogre there, but he has no armour (or has more armour – I can’t remember if they wear any now, TBH!). Now ideally there would be a hint as to what you’ll get either way – if the left has a sign pointing to the ogre baths, maybe that’s a hint an ogre is there and has taken his Armour off to have a mud bath.

But even if there is no hint and it’s a blind choice, at least the choice is a randomiser and the results of either direction will be different from each other. Player input actually decides (part of) the story generated at the table on game night! Who’d a thunk it?

I guess though that such a difference could very well get in the way of ‘GM creative agency for telling a good story’. And then it’s sucked back into the knot from above.

Stories in Novels Vs Difficulty curves in games (and idiot RPG authors who say you can do the former)

I wrote this recently as a comment on reddit and it warrants it’s own place rather than being buried amidst a pile of comments.

The situation was the GM had a group of ‘bandits’ (actually a political faction causing trouble under the guise of bandits) . The players run right into the group and get defeated, one captured the others escaping. The players weren’t happy. And here is my reply to this:

The problem is in Venn diagram terms, what the players find fun is one circle and what you presented is another circle – they really didn’t overlap.

Imagine you’d done this instead – they ran into outlier camps of the bandits, who are in small groups that are more balanced to the PCs and wont be calling the main group. The players would win the battle – which they were looking for at least once otherwise they feel their new PCs are chumps.

Further imagine you make larger and larger groups, with an increasing chance of calling the next group along.

The players would encounter tougher and tougher resistance until they question whether they can take the next group. Maybe they should see if anyone in town can help – sellswords, for example? Exactly as it turned out, but with players being happy about it.

This is a smooth difficulty curve, rising from low to challenging. What you had was a difficulty spike – nobody can really handle that and enjoy it, precisely because it’s too realistic – if realism was fun, why are we playing fantasy rather than being out in the real world?

That said, the author of the books give the impression you could run the game exactly as you did (or so I guess – most RPGs do). And the authors are idiots for it. You were told what you did would work but you were told something that does not work because it’s not actually compatible with human psychology.

So many new gamer’s try to use the aspiring novel writer method of designing games – but it doesn’t work, because as a novel writer you can screw your characters over royally and no one bitches about it. What you did would work as a novel. As a game it doesn’t work. But you were given bad advice, so it’s not your fault.

 

Have a plot but you hate having passive players?

I’m posted this recently: I’d like to explain that having plots makes players passive lumps. Why is that? Because if a player ever goes to do something that would screw the plot up, the GM does one or both of the following A: Subtley or obviously chastises the player or B: Makes events stop the player from doing the action.

The thing is, the player is never sure what action will cause the plot to break. So what’s the best solution for the player if they don’t want the plot to break and to suffer A or B? Well….do nothing. Be a passive lump. That way you can’t break the GMs precious plot.

GMs who use plot but loathe passive players are their own worst enemy.