The Penney’s game [1,2] is a game that can be simply played with coins, betting on sequences of heads and tails (so it is a binary game). The players agree on the length of the sequence, at list of three bits, and then choose orderly which sequence they bet on. Then a coin is tossed several times, generating the binary sequence. The player whose sequence appears first wins. As odd as it may seem at first, The second player has always the possibility of winning over the first, statistically. Indeed, the Penney’s game is a well-known example of non-transitive games [3], so that for any given sequence of length three or more, one can always find another sequence that has a higher probability of occurring first. This game can be mapped onto a more general problem: given a Markov chain with a trap (the first sequence), where is more convenient to place a second trap in order to "shade" at most the first one? In this language, we are interested in interference among traps (intransitivity), and we developed some techniques to compute the degree of intransitivity [4]. For the Penney's game, we show that there is a sort of phase transition: if the coins are sufficiently biased, the game the becomes more fair, in the sense that the first player choice may not always be defeated by the second one.
[1] Walter Penney, Journal of Recreational Mathematics, October 1969, p. 241.
[2] https://en.wikipedia.org/wiki/Penney’s_game
[3] M. Gardner, On the paradoxical situations that arise from nontransitive relations., Sci. Am. 231 (1974) 120.
[4] Giulia Cencetti, Franco Bagnoli, Francesca Di Patti, Duccio Fanelli, The second will be first: competition on directed networks, Scientific Reports 6, Article number: 27116 (2016) doi:10.1038/srep27116