The Penney's Game with Group Action
Tanya Khovanova, Sean Li

TL;DR
This paper extends Penney's game to patterns defined by group actions on words, analyzing non-transitivity, expected wait times, and bounds under various group symmetries.
Contribution
It introduces a framework for Penney's game with group action-based patterns, providing new insights into non-transitivity and pattern-based probabilities.
Findings
Non-transitivity persists as pattern length increases.
Derived bounds on Conway leading number and wait times.
Analyzed game behavior under cyclic and symmetric group actions.
Abstract
Consider equipping an alphabet with a group action that partitions the set of words into equivalence classes which we call patterns. We answer standard questions for the Penney's game on patterns and show non-transitivity for the game on patterns as the length of the pattern tends to infinity. We also analyze bounds on the pattern-based Conway leading number and expected wait time, and further explore the game under the cyclic and symmetric group actions.
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Taxonomy
Topicssemigroups and automata theory · Algorithms and Data Compression · Computability, Logic, AI Algorithms
