Periodic reminder that & #39;the game is afoot& #39; does not mean & #39;the game has begun& #39;. It means & #39;the prey animal is running& #39;. Game is used here in the sense of gamekeeper, game pie, game bird.
The full Shakespeare quote, which Holmes only uses the last bit of, refers to coursing with greyhounds. So the & #39;game& #39; was probably a hare.