Share
COMBINATORICS
2006

# Optimal Penney Ante Strategy via Correlation Polynomial Identities

9 years 9 months ago
In the game of Penney Ante two players take turns publicly selecting two distinct words of length n using letters from an alphabet of size q. They roll a fair q sided die having sides labelled with the elements of 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.
Daniel Felix
 Added 11 Dec 2010 Updated 11 Dec 2010 Type Journal Year 2006 Where COMBINATORICS Authors Daniel Felix