$N$ players are in a gunfight. Starting from player 1, each player takes turns to act in the order of $1,2,...,N,1,2,...$. In their turn, a player randomly chooses one of the other remaining players as the target, and fires one shot at them. If hit, the target is eliminated. The game continues until there's only one survivor. All shots hit targets with probability $p$.
Is it true that when $p$ is small enough, player 1 has the highest surviving probability for any $N$?
---------- Edit
There seems to be some misunderstanding in the comments so let me clarify. I'm asking if there exists some $p^{*}\gt 0$, such that if everybody has hit probability $p^{*}$, player 1's position is the best position for any choice of $N$.
For example, if $p=1$, then player 1's position is not the best for $N=3$. If $p=0.5$, then player 1's position is not the best for $N=5$. But if $p=0.45$, player 1's position is the best for $N\leq 10000$. (I haven't checked for more because the algorithm is $\mathcal{O}(N^3)$ and $N\leq 10000$ took me more than 20 minutes)