Penney's Game Odds From No-Arbitrage

Joshua Miller

OSF Preprints | Published : 2019


Penney's 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 Conway's ``leading number'' algorithm. The accompanying betting odds intuition adds insight into why Conway's..

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University of Melbourne Researchers