N players play a game: each player wears a hat that's either red or blue. Each player can see all the other N-1 players' hats but can not see his own hat. The game rule forbids any form of signaling between players during the game. But before the game (before they are put on hats), players can discuss and thus adopt some form of strategy. When the game begins, all players are required to report the color of his own hat simultaneously.
This is just the old hat color game, but now I ask a different question:
Suppose each player has 1/2 independent probability to be put on a red hat. What is the strategy that maximizes the expected number of people who report correctly?