Penneys Game Odds From No-Arbitrage

Penneys game is a two player zero-sum game in which each player chooses a three-flip pattern of heads and tails and the winner is the player whose pattern occurs first in repeated tosses of a fair coin. Because the players choose sequentially, the second mover has the advantage. In fact, for any three-flip pattern, there is another three-flip pattern that is strictly more likely to occur first. This paper provides a novel no-arbitrage argument that generates the winning odds corresponding to any pair of distinct patterns. The resulting odds formula is equivalent to that generated by Conways leading number algorithm. The accompanying betting odds intuition adds insight into why Conways algorithm works. The proof is simple and easy to generalize to games involving more than two outcomes, unequal probabilities, and competing patterns of various length. Additional results on the expected duration of Penneys game are presented. Code implementing and cross-validating the algorithms is included.