On a dart game of Niedermaier

TL;DR

This paper analyzes Niedermaier's dart game, deriving the distribution of total throws and players' winning probabilities using generating functions and recursive formulas.

Contribution

It introduces a novel probabilistic analysis of Niedermaier's dart game, providing explicit formulas for game duration and winning chances.

Findings

Distribution of total throws is derived.

Recursive formulas for player win probabilities are established.

The analysis offers insights into game dynamics and outcomes.

Abstract

We analyze a game introduced by Andy Niedermaier, where $p$ players take turns throwing a dart at a dartboard. A player is eliminated unless his dart lands closer to the center than all previously thrown darts, in which case he goes to the back of the line, until only one player remains. Using generating functions, we determine the distribution of the number of throws in the game, and we obtain a recursive formula to compute the probability that each player wins.