Optimal Penney Ante Strategy via Correlation Polynomial Identities

In the game of Penney Ante two players take turns publicly selecting two distinct words of length n using letters from an alphabet Ω of size q. They roll a fair q sided die having sides labelled with the elements of Ω until the last n tosses agree with one player’s word, and that player is declared the winner. For n ≥ 3 the second player has a strategy which guarantees strictly better than even odds. Guibas and Odlyzko have shown that the last n − 1 letters of the second player’s optimal word agree with the initial n − 1 letters of the first player’s word. We offer a new proof of this result when q ≥ 3 using correlation polynomial identities, and we complete the description of the second player’s best strategy by characterizing the optimal leading letter. We also give a new proof of their conjecture that for q = 2 this optimal strategy is unique, and we provide a generalization of this result to higher q.