Say we have a closed environment with a population $n$ of creatures. These creatures actively hunt each other. At each discrete step (round), each of them picks some other member to be attacked, with potential victims being selected with equal probability. It is allowed that multiple creatures have targeted the same individual. There is no reproduction, so the population can never increase.
The creatures have some probability $p$ of successfully dispatching their victim, so the probability an individual survives a round is $(1-p)^k$ where $k$ is the number of attackers of that individual in the round.
I am interested in the expected number of rounds until hunting ceases (that is, until zero or one creatures remain).
I am able to accomplish this via a Markov chain, but the generation of states becomes computationally onerous pretty quickly (~7 or 8 starting population).
Is there some clever way to arrive at the desired results that could be used for larger $n$?
For reference, the values I've found for $p={1,1/2,1/4}$ for $n=2...6$ are:
p=1:$\left\{1,1,\frac{35}{27},\frac{99}{64},\frac{48989}{28125}\right\}$
p=1/2:$\left\{\frac{4}{3},\frac{13}{7},\frac{20276}{8505},\frac{148963}{53568},\frac{1721759004187}{553675500000}\right\}$
p=1/4:$\left\{\frac{16}{7},\frac{916}{259},\frac{5551664}{1223775},\frac{1087130833}{203897232},\frac{2088276241150181}{348276166875000}\right\}$
I've also fiddled with various slicing & dicing of the results in attempt to find a pattern to no avail.
