The hats of Ebert

1. They don't need all three to talk. In fact, in good strategy there is almost always only one person speaking.
2. The important thing is not to take advantage of all the information possible, but to reserve the floor for the only player who, in certain cases, can be sure to get it right.
3. If a player sees two hats of the same color, he is in a very different position than someone who sees two different colors.

Answer: the highest probability of success is $3/4$. The strategy is this: - if a player sees two hats of the same color, he says that his is the opposite color;

• If he sees two hats of different colors, he shuts up. There are $2^3=8$ possible distributions of hats, all equally probable. If all three hats are the same—all red or all blue—each player sees two identical hats and applies the rule. The three speak and the three fail. Those are the two lost cases. In any other layout, there are two hats of one color and one of the other. Then the player wearing the minority color sees two identical hats, speaks and gets it right. The other two see different colors and remain silent. Therefore, the group wins in 6 of the 8 cases: $$

\frac{6}{8}=\frac{3}{4}.
$$ It remains to be seen why he can't do better. The 8 cases can be grouped into 4 complementary pairs, where all the colors are inverted. A safe strategy cannot win in both cases of each pair if in both cases someone had to speak with the same inverted visual information. That forces at least 2 lost cases. Since the previous strategy loses exactly 2, it is optimal.