A Markov Chain Analysis of a Pattern Matching Coin Game

TL;DR

This paper analyzes a non-transitive coin game using Markov chains, calculating expected rounds, probabilistic advantage, and proving the optimality of Penney's strategy.

Contribution

It introduces a Markov chain approach to analyze Penney's coin game, including new methods for advantage calculation and proof of strategy optimality.

Findings

Expected number of rounds for each game variation

Probabilistic advantage of Penney's strategy

Proof of strategy optimality

Abstract

In late May of 2014 I received an email from a colleague introducing to me a non-transitive game developed by Walter Penney. This paper explores this probability game from the perspective of a coin tossing game, and further discusses some similarly interesting properties arising out of a Markov Chain analysis. In particular, we calculate the number of "rounds" that are expected to be played in each variation of the game by leveraging the fundamental matrix. Additionally, I derive a novel method for calculating the probabilistic advantage obtained by the player following Penney's strategy. I also produce an exhaustive proof that Penney's strategy is optimal for his namesake game.