The asymptotics of group Russian roulette

TL;DR

This paper analyzes the asymptotic behavior of the group Russian roulette problem, revealing that the probability of no survivors oscillates periodically with respect to the logarithm of the initial number of people, rather than converging.

Contribution

It proves that the probability of no survivors does not converge but exhibits asymptotic periodicity and continuity on the log scale, a novel insight into the problem's long-term behavior.

Findings

Probability $p_n$ does not converge as $n o

$ $p_n$ is asymptotically periodic on the $\log n$ scale

The behavior is continuous with period 1 in the log scale

Abstract

We study the group Russian roulette problem, also known as the shooting problem, defined as follows. We have $n$ armed people in a room. At each chime of a clock, everyone shoots a random other person. The persons shot fall dead and the survivors shoot again at the next chime. Eventually, either everyone is dead or there is a single survivor. We prove that the probability $p_n$ of having no survivors does not converge as $n\to\infty$, and becomes asymptotically periodic and continuous on the $\log n$ scale, with period 1.