The No-Flippancy Game

TL;DR

This paper investigates a deterministic coin-toss game where players select strings and aim to have their string appear first, analyzing conditions for game termination, strategies, and specific case outcomes.

Contribution

It introduces a detailed analysis of the no-flippancy game, including conditions for infinite play, strategic considerations, and case-specific results.

Findings

If the game exceeds 4n-4 turns without ending, it is infinite.

Strategies can be devised to force a desired outcome.

The paper characterizes outcomes for particular string choices.

Abstract

We analyze a coin-based game with two players where, before starting the game, each player selects a string of length $n$ comprised of coin tosses. They alternate turns, choosing the outcome of a coin toss according to specific rules. As a result, the game is deterministic. The player whose string appears first wins. If neither player's string occurs, then the game must be infinite. We study several aspects of this game. We show that if, after $4n-4$ turns, the game fails to cease, it must be infinite. Furthermore, we examine how a player may select their string to force a desired outcome. Finally, we describe the result of the game for particular cases.