On hats and other covers

The following game puzzle has recently received attention from mathematicians, computer scientists, coding theorists, and even from the mass press: A team of n players is fitted with hats, which are either red or green. A player can see the other players' hat colors, but not his own. Each player is then asked to declare his hat color, or pass. All the players must declare simultaneously, with no inter-player communication allowed during the game. They are permitted, however, to hold a strategy coordination meeting before the game starts. The team wins if at least one player declares the correct color, and no player declares incorrectly. The goal of the team is to devise a strategy that maximizes the winning probability, under an assumption of uniform probability distribution on the hat color combinations.