A Mind-Expanding Probability Problem
Three mathematicians play a three-stage game. In the first stage, they talk to one another about the strategies they are going to use to play the next two stages.
Then, in the second stage, they each put on a hat. Each hat will either be red or blue, with equal probability, and the colors of the three hats are independent. Each mathematician does not see his own hat, but he can see the hats of his two friends. The mathematicians may not communicate to each other any information about the hats, or discuss any strategies to use in the game at this point.
Finally, in the third stage, each mathematician writes down on a slip of paper one word. Each may write ‘Red’, ‘Blue’, or ‘Pass’. He may not show this slip to his friends or communicate any information to them in any way.
Now the results come in. All strips of paper are shown. If at least one mathematician wrote down the color of his own hat, and no mathematician wrote down a color other than the color of his hat, then all mathematicians are awarded an equal sum of money. Otherwise, meaning if no mathematician correctly identified the color of his hat, or if even one identified incorrectly, then no mathematicians get any money at all.
What strategy should the mathematicians agree upon in the first stage so that they maximize their average earnings?