The rules of the game are the following:
There are 2 players, each of them have a gun and plays at the same time in every turn, in a turn they must perform one of three actions, shoot, charge or dodge.
However, you can only fire if you have at least one charge, and, as soon as you shoot, your charge count goes to 0, the two players start with 0 charges.
You win if you manage to shoot the other player while he is not shooting you back or dodging. If you have 5 charges you fire a super-bullet that the other player cannot dodge, and can neutralize only with another super-bullet.
What is the nash equilibrium strategy? Here's what I thought.
The strategy has to depend on the number of charges of each player, Moreover there are some states for which the best strategy is easy to find, namely:
-0 charges vs 0 charges, both players have to charge -4 vs 0, the first player has to charge and he wins
A more interesting state is 4 vs 4, where each player has to play each action with probability $1/3$, as any action can make you win, lose, or draw (in the sense that you go into a state that you can win with probability $1/2$), much like in paper-scissors-rock.
Another easy observation is that clearly, if both players are using an optimal strategy, the probability of winning in states of the kind $n$ vs $n$ will be $1/2$.
Unfortunately it isn't very clear what strategy to employ in other states, one way to find it out would be the classical way: let $s_1,s_2$ be the strategies of the two players, calculate the probability of winning for the first player in the state 0 vs 0 and call it $w(s_1,s_2)$ with Markov chains, and then find $s_1$ for which $w(s_1,s_2)\geq 1/2$ for all the strategies of the second player.
However this method is very time-consuming and I'm not able to implement it on, say, Octave.
So the real question is: is there a fast way to find or approximate the optimal strategy?
Note: I am not very knowledgeable about game theory and I may have abused terms and have been quite imprecise, forgive me for that, tell me if any clarification is needed, and be sure to point out any mistake I may have committed in the formulation of my question.