Information Theory, Inference, and Learning ... - Inference Group

Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links.13.14: Solutions 227of the locations of the winning and losing states on the three-dimensionalhypercube, then thinking laterally.If the number of players, N, is 2 r − 1, the optimal strategy can be definedusing a Hamming code of length N, and the probability of winning the prizeis N/(N + 1). Each player is identified with a number n ∈ 1 . . . N. The twocolours are mapped onto 0 and 1. Any state of their hats can be viewed as areceived vector out of a binary channel. A random binary vector of length Nis either a codeword of the Hamming code, with probability 1/(N + 1), or itdiffers in exactly one bit from a codeword. Each player looks at all the otherbits and considers whether his bit can be set to a colour such that the state isa codeword (which can be deduced using the decoder of the Hamming code).If it can, then the player guesses that his hat is the other colour. If the state isactually a codeword, all players will guess and will guess wrong. If the state isa non-codeword, only one player will guess, and his guess will be correct. It’squite easy to train seven players to follow the optimal strategy if the cyclicrepresentation of the (7, 4) Hamming code is used (p.19).