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

...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.

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.

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

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).

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.

I have found a strategy that gives 10/16 = 62.5%. Edit: make that 11/16 = 68.75%.

I think it's crazy how much effort people have spent on looking for loopholes in the wording. I think the problem statement was very clear on first reading.

I think the stament was clear except for how the guessing is implemented: whether a time element can be utilized or not. I believe not, but if that is the case then it would be clearer to say e.g. that each player writes down their guess so that no one can see when and what others write, and the guesses are revealed simultaneously; not writing is permitted and equals not guessing.

If a time element could be utilized, a simple strategy would be e.g. that the players decide who will go first. That player will look at the player on the right, and if he sees a white hat, he will immediately guess something. If he sees a black hat he will stay silent, and after a while the next player will guess black. This strategy has 75% chance of success (whenever the second player has a black hat and 50% of the time when he has a white hat)

If a player sees three hats of the same color, he immediately guesses the other color.

If all hats are the same color, this means all players immediately guess wrong.

If the colors are divided 3-1, only one player guesses immediately, and his guess is correct.

If, after 10 seconds, nobody has made a guess, then the hats must be divided 2-2. One player is preassigned to guess after 10 seconds the color that makes it 2-2 (i.e. if he sees WWB, he guesses B, if he sees WBB, he guesses W).

This strategy fails if all hats are the same color (1/8 chance), succeeds in all other cases (7/8 chance).