Solving the 10 players duel game
Robert Israel's answer on Math Overflow provides the key insight:


If $S$ is the set of players remaining alive and it is player $j$'s turn, player $j$ should shoot at the player $k$ such that, if $k$ dies, the remaining set gives $j$ the best survival chances.


This enables us to build a dynamic programming algorithm, as Robert Israel states.  I'll explain in a bit more detail how that works.

Represent a configuration of the game by $(S,j)$, where $S$ is the set of players remaining alive and $j$ is the player whose turn it is to shoot next.
Define $f_i(S,j)$ to be $i$'s probability of survival, under the equilibrium strategy for all (under optimal play), in configuration $(S,j)$.  Then the insight above means that in configuration $(S,j)$, player $j$ should shoot at the player $k$ such that $f_j(S\setminus\{k\},j+1)$ is as large as possible.

(As a conventional notation, when I write $(S,j+1)$, you should interpret $j+1$ as the next person to shoot after $j$, given that players $S$ are alive.)

Based on this, we can write recursive equations for $f$:

$$f_i(S,j) = p_j \cdot f_i(S \setminus \{g(S,j)\}, j+1) + (1-p_j) \cdot f_i(S,j+1)$$

where

$$g(S,j) = \arg \max_k p_j(S \setminus \{k\}, j+1).$$

We'll start by calculating the values of $f_i(S,j)$ and $g(S,j)$ for all $S$ where $|S|=1$.  Then, we'll calculate the values of $f_i(S,j)$ and $g(S,j)$ for all $S$ where $|S|=2$.  Then, we'll calculate the values of $f_i(S,j)$ and $g(S,j)$ for all $S$ where $|S|=3$.

At the $k$th stage, to calculate the $f$-values, we solve a system of linear equations in $k^2 \times {n \choose k}$ unknowns.  (The unknowns are the terms $f_i(S,j)$ where $|S|=k$ and $i,j \in S$.  Note that at this point $f_i(S \setminus \{\text{anything}\},j$ is known, since the first argument to $f$ will be a set of size $k-1$, which we've already calculated in the previous stage.)  In your problem, $n=10$, so the $k$th stage involves solving a system of linear equations in $k^2 \times {10 \choose k}$ unknowns.  This has a maximum at $k=6$, where we have 7560 unknowns.  The system of linear equations can be solved using Gaussian elimination, which takes cubic time.  In particular, Gaussian elimination on a $7560 \times 7560$ matrix is entirely feasible on a computer (probably it will take at most seconds).

Therefore, this should provide an efficient algorithm to compute the survival probabilities and strategy you are looking for.