# Penney's Game Odds From No-Arbitrage

## TL;DR

This paper introduces a simple no-arbitrage method to calculate winning odds in Penney's game, providing insights into the game's probabilities and extending to more complex scenarios.

## Contribution

It presents a novel no-arbitrage approach to derive Penney's game odds, aligning with Conway's algorithm and generalizing to complex pattern and outcome scenarios.

## Key findings

- The no-arbitrage odds formula matches Conway's algorithm.
- The method generalizes to multiple outcomes and unequal probabilities.
- Additional results include expected game duration analysis.

## Abstract

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 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 Penney's game are presented. Code implementing and cross-validating the algorithms is included.