Heads, Tails or pass

Steven G

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Dec 30, 2014
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3 people have each been assigned H or T on their foreheads, based on the results of tossing a fair coin. Each member can see each others' letters but not their own. Their common goal is to win a game with the following rules:


  1. They must all make a statement at the same time, and no communication is allowed beforehand.
  2. Each member's statement can either be a guess of their own letter ("H" or "T") or "pass".
  3. They win if at least one person guesses correctly and no one guesses incorrectly.
With everyone applying the optimal strategy, what is their probability of winning?
 
I'm sure there's something clever I'm missing here, but my initial thoughts are:

- the coin tosses are supposed to be independent, which means that looking at other people's foreheads gives me zero information about what's on mine

- given the above fact, I have no idea what the optimal strategy would be

What I do know is that there's three things each person can do

- guess correctly
- pass
- guess incorrectly

Does this mean no one should ever pass? Because everyone's guess is 50/50, so why not just guess and have the probability of everyone being correct be (1/2)^3 instead of (1/3)^3?

On the other hand, if everyone guesses, there's only one way to win. Everyone has to be correct. If some people pass, there is more than one outcome that leads to winning. I can't think of how to enumerate those possibilities right now.
 
3 people have each been assigned H or T on their foreheads, based on the results of tossing a fair coin. Each member can see each others' letters but not their own. Their common goal is to win a game with the following rules:


  1. They must all make a statement at the same time, and no communication is allowed beforehand.
  2. Each member's statement can either be a guess of their own letter ("H" or "T") or "pass".
  3. They win if at least one person guesses correctly and no one guesses incorrectly.

With everyone applying the optimal strategy, what is their probability of winning?

I know the answer (that is, I knew where to look to find it, and now I know it, so I can't participate!). I don't want to give it away. For larger numbers, there is a rather advanced solution; for three, it should be possible to work out a strategy by simple logic; and maybe it's even possible to prove this to be optimal.

The idea here is not to guess randomly, but to make a rule, which may be the same for everyone, but may be different. For example, for one person, it could be something like "If you see TT, guess H; if you see HH, pass; if you see TH (or HT), guess T". (I made that up randomly.) For the second person, it might be different. The goal is to maximize the probability that everyone either passes or guesses correctly. It is possible for this probability to exceed 50%. (And for more people, the generalized strategy can give an even higher probability.)
 
As strategy, can they have one guy pass every time
he sees 2 same (and at no other time), which means
the other 2 guys would then know what's on their
foreheads...or is that prevented by the rule:

1.They must all make a statement at the same time,
and no communication is allowed beforehand.

They can certainly do that; but since the statements are simultaneous, they can't make use of the logic. What they've said, they've already said before they can conclude anything from what another said.

So the goal is not to be able to deduce anything, but only to say things, based only on what they themselves observe, that will lead to winning more than half the time.
 
I can get to 75% probability of success with the strategy that anyone who sees like guesses un-like; anyone who sees unlike passes. But I have only looked at deterministic strategies. I have not proved that the best deterministic strategy dominates all probabilistic ones.

The key point here is that three people or one person can see like. Three people can see like only if we have TTT or HHH, which has a probability of 25%. Only person 1 will see like if we have THH or HTT. Only person 2 will see like if we have THT or HTH. Only person 3 will see like if we have HHT or TTH. Only one person will see like with probability 75%.

There are only three choices open to someone who sees like, namely pass, guess like, or guess unlike.

There are only three choices open to someone who sees unlike, pass, guess H, or guess T. Nine non-probabilistic strategies.

If everyone passes, the probability of winning is zero.

If everyone guesses, the probability of winning is

0.5 * 0.5 * 0.5 = 12.5%.

If everyone who sees like passes and everyone else guesses, the probability of winning is

0.25(0) + 0.75(0.5 * 0.5) = 37.5%

If everyone who sees like guesses like and everyone else guesses, the probability of winning is

0.25(1) + .75(0) = 25%.

If everyone who sees like guesses unlike and everyone else guesses, the probability of winning is

0.25(0) + 0.75(0.5)(0.5) = 18.75%.

If everyone who sees like guesses like and everyone else passes, the probability of winning is

0.25(1) + 0.75(0) = 25%.

If everyone who sees like guesses unlike and everyone else passes, the probability of winning is

0.25(0) + 0.75(1) = 75%.

EDIT: I have tried without success to hide this answer. There used to be an eye icon, didn't there?
 
Last edited:
I think this should also work.

The 3 players all follow the same strategy:
if they see HH they guess T or if they see TT they guess H, else they pass.

There are 8 cases (all equally likely):
HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

For HHH and TTT, they lose since all 3 guess wrongly.

For all the other 6 cases they win.
As example, HHT: both H's will see a H and a T so will pass.
But the T will see HH, so will correctly guess T.

The other 5 cases will all work similarly.

So they win 6 out of 8, or 75%

Am I going to the corner? :rolleyes:
Only for copying my work.
 
I can get to 75% probability of success with the strategy that anyone who sees like guesses un-like; anyone who sees unlike passes. But I have only looked at deterministic strategies. I have not proved that the best deterministic strategy dominates all probabilistic ones.

The key point here is that three people or one person can see like. Three people can see like only if we have TTT or HHH, which has a probability of 25%. Only person 1 will see like if we have THH or HTT. Only person 2 will see like if we have THT or HTH. Only person 3 will see like if we have HHT or TTH. Only one person will see like with probability 75%.

There are only three choices open to someone who sees like, namely pass, guess like, or guess unlike.

There are only three choices open to someone who sees unlike, pass, guess H, or guess T. Nine non-probabilistic strategies.

If everyone passes, the probability of winning is zero.

If everyone guesses, the probability of winning is

0.5 * 0.5 * 0.5 = 12.5%.

If everyone who sees like passes and everyone else guesses, the probability of winning is

0.25(0) + 0.75(0.5 * 0.5) = 37.5%

If everyone who sees like guesses like and everyone else guesses, the probability of winning is

0.25(1) + .75(0) = 25%.

If everyone who sees like guesses unlike and everyone else guesses, the probability of winning is

0.25(0) + 0.75(0.5)(0.5) = 18.75%.

If everyone who sees like guesses like and everyone else passes, the probability of winning is

0.25(1) + 0.75(0) = 25%.

If everyone who sees like guesses unlike and everyone else passes, the probability of winning is

0.25(0) + 0.75(1) = 75%.


EDIT: I have tried without success to hide this answer. There used to be an eye icon, didn't there?
Change the font ink to "white" and it will hide in the white background (Like I did to your solution above - so that Denis cannot copy it)
 
Change the font ink to "white" and it will hide in the white background (Like I did to your solution above - so that Denis cannot copy it)
Thank you Subhotosh.

Of course then no one can ever know whether my answer is correct, but it is more important to keep Denis honest.

i have left a secret message for Denis below.

10 minutes in the corner for copying. It would be longer, but my leaving things unhidden was a great temptation for you.
 
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