A hat trick

Consider the following game. There are n players, each wearing a hat colored red or blue. Each player does not see the color of her own hat but does see the colors of all other hats. Simultaneously, each player has to guess the color of her own hat, without communicating with the other players. The players are allowed to meet beforehand, hats-off, in order to coordinate a strategy. We give an explicit polynomial time deterministic strategy which guarantees that the number of correct guesses is at least max{ n r , n b }- O ( n 1/2 ), where n r is the number of players with a red hat and n b = n - n r . This answers a question of Feige.