Hat problem

[drmwc]

4 bridge players are at a table. The tournament director offers them the following proposition:

They are all randomly and simulateously assigned black or white hats. Once they are given their hat, they may not communicate with the other 3 players in any way. If they violate this rule in any way, they will all be shot.

The chances of an individual player being given a hat of a particular colour is 50%; and the hat colours are chosen independently for the 4 players.

They cannot see their own hat colour, but they can see the other 3 players' hats. Any attempt to find their own hat colour by nefarious methods (such as mirrors) will result in them all being shot.

After a deliberation period, all 4 players have the option of guessing their hat colour. The guesses will be simultaneous (or they will be shot). Each player has the option of remaining silent - this will not result in anyone being shot.

If at least one player guesses, and all the guesses made are correct, they win a bottle of champagne. Wrong guesses have no ill effect - no-one is shot.

They may discuss strategy beforehand without being shot.

The problem is to find a strategy which maximises the probability of them getting champagne, with 0% chance of them being shot.

To get things started: They could nominate 1 person to guess and the other three will abstain. This has 50% chance of champagne. (The winning strattegy is better than 50%.)

[LocoRon]

The problem is to find a strategy which maximises the porbability of them getting champagne, with 0 chance of them being shot.

Get a job and pay for the champagne like a normal person.

[Magicwand]

4 bridge players ...... than 50%.)

i have a feeing that i am missing some important info.

[amnal]

I'm not sure if I'm supposed to be clever with the wording or not...I can't see a way of doing it taking it all completely at face value (though with this kind of problem, that's more likely to be my failing than anything else ), but...

...depending on the nature of 'communication' and the post-deliberation-period, the players can at least exchange a little temporal information. For instance, after the deliberation period player A will immediately shout BLACK (a 50% chance) unless all three other participants are wearing white hats...in which case he hesitates, and player B knows to shout WHITE to win the game. Assuming these guys are good at keeping track of time, the first three all get a chance to hesitate and pass on a little information. That gets you slightly better than 50%, and I expect there's something a little better by being clever with binary logic or whatever...

I'm honestly not sure if this is reasonable or not, within the bounds of the problem. Even if it is, I'm not sure it helps much . I'd have thought it counts as communication, since information is transferred, but if absolutely no information can be transferred then surely >50% is not possible. Of course, I use 'surely' in the vaguest of senses.

[tundra]

.

[Fedya]

If you see three hats, you guess the opposite color; otherwise you shut up. This should work 80% of the time; if you assume that the game is played until there's either a winning situation or wrong guesses. If the hats are given out 2/2, then nobody will ever guess, and there won't be any champagne given out.

[kaimat]

Some thoughts:

1. "Once they are given their hat, they may not communicate with the other 3 players in any way. If they violate this rule in any way, they will all be shot." and "They may discuss strategy beforehand without being shot."

... so it would seem they could talk (or communicate in any way) only before getting their hats. I don't know their being bridge players at a table is important (it should be noted that I don't play bridge, so maybe there is a bridge connection I'm missing) and they could just as easily be in separate rooms communicating with phones or something.

2. "The guesses will be simultaneous (or they will be shot)." and "Once they are given their hat, they may not communicate with the other 3 players in any way. If they violate this rule in any way, they will all be shot."

... so they would have to decide beforehand to have any guesses done as soon as the hats are placed on their heads in order to be simultaneous since once the hats are on they couldn't even do some kind of gesture to indicate when to guess.

3. "Each player has the option of remaining silent - this will not result in anyone being shot." and "If at least one player guesses, and all the guesses made are correct, they win a bottle of champagne. Wrong guesses have no ill effect - no-one is shot."

... so no one guessing means 0% of getting killed and 0% chance of getting champagne. If one person guesses you have a 50% chance of getting champagne and a 0% chance of getting shot, since wrong guesses don't matter. If more than one person guesses you still won't get shot and the chance of winning champagne is cut in half.

This leads me to conclude that it's best to have one person randomly guess. Please let me know if I'm missing something, but it seems like they only get shot if they try to communicate after getting their hats or trying to figure out their hat color. I don't understand why someone would risk possibly getting shot just to get some champagne.

What if not guessing also gets them shot?

And why are we talking so much about shooting people. This is supposed to be a peaceful forum

[rhubarb]

I'm not sure if I'm supposed to be clever with the wording or not...I can't see a way of doing it taking it all completely at face value (though with this kind of problem, that's more likely to be my failing than anything else ), but...

...depending on the nature of 'communication' and the post-deliberation-period, the players can at least exchange a little temporal information. For instance, after the deliberation period player A will immediately shout BLACK (a 50% chance) unless all three other participants are wearing white hats...in which case he hesitates, and player B knows to shout WHITE to win the game. Assuming these guys are good at keeping track of time, the first three all get a chance to hesitate and pass on a little information. That gets you slightly better than 50%, and I expect there's something a little better by being clever with binary logic or whatever...

I'm honestly not sure if this is reasonable or not, within the bounds of the problem. Even if it is, I'm not sure it helps much . I'd have thought it counts as communication, since information is transferred, but if absolutely no information can be transferred then surely >50% is not possible. Of course, I use 'surely' in the vaguest of senses.

Following up on this, a request for clarification:

When you say that the guesses will be simultaneous, which of the following do you mean?

1. There is a fixed time at which guesses are permitted. No one may guess at any other time.

-or-

2. At any time after the deliberation period ends, guesses are permitted. No guess may be made at a later time than the first guess, but multiple guesses may be made at once.

...because in case 2 the players can learn something via the time it takes to guess, which might be construed as not communicating. My guess is you've got option 1 in mind (otherwise you could probably get 100%, though I'm not positive; sounds like Muddy Children).

[bgrieco]

This problem is also known as the "muddy children". Instead of hats. Kids have mud on their faces, which they cannot see.
It's an interesting example on use of Kripke's structures for Modal Logic.
Not spilling the beans beyond this point though

[Joaz Banbeck]

This is not the "Muddy Children" problem. It that, the children's answers are sequential. In the bridge problem, the answers are simultaneous.

As the problem is stated, with simultaneous answers, and dreadful penalty for ilicit communication, there is no information transferred between players. Thus each player is effectively in his own informaton universe. The best one can do is 50%. If a second guesses, the chances of winning the bottle are 25%.

My conclusion is that Drmwc accidentally mis-stated the problem, or he just likes to shoot people.


My best guess about the mis-statement is that there are multiple rounds.

[Boidhre]

This is not the "Muddy Children" problem. It that, the children's answers are sequential. In the bridge problem, the answers are simultaneous.

As the problem is stated, with simultaneous answers, and dreadful penalty for ilicit communication, there is no information transferred between players. Thus each player is effectively in his own informaton universe. The best one can do is 50%. If a second guesses, the chances of winning the bottle are 25%.

My conclusion is that Drmwc accidentally mis-stated the problem, or he just likes to shoot people.


My best guess about the mis-statement is that there are multiple rounds.

Nope. A mistake he did make but you're incorrect.

[skydyr]

Is there a known pool of hats from which the players hats are selected?

[rhubarb]

Muddy Children does, I think, need the children to know at the outset that at least one of them is muddy. (Analogue: know that at least one of the participants has a white hat, or that at least one has a black hat.)

I'll probably feel dumb when the answer's posted, but I don't immediately see it.

@skydyr: I think it's to be understood as each of the participants' being selected by a separate (independent) coin toss.

[jlaire]

The best one can do is 50%. If a second guesses, the chances of winning the bottle are 25%.

Nope. Here's a simple counterexample.

The bridge players agree on the following strategy:

- If you see 3 hats of the same color, guess the other color.
- If you see 2 hats of the same color, guess that color.

This leads to everybody giving the correct guess when the colors are distributed 1:3. That distribution happens exactly 50% of the time, so there's a 50% chance that they will win the bottle, even though everybody makes a guess.

[Boidhre]

Nope. A mistake he did make but you're incorrect.

I was being dense, ignore me.