The Asymptotics of Group Russian Roulette

#### W. Kager, R. Meester, T. van de Brug

2017, v.23, Issue 1, 35-66

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.

Keywords: group Russian roulette, shooting problem, non-convergence, coupling, asymptotic periodicity and continuity