Army A has a single plane which can strike one of three possible targets, A, B and C. Army B has one anti-aircraft gun that can be assigned to one of the three targets to guard it. The value of each target, $v_k$ is $v_A>v_B>v_C>1$. Army A can destroy the target only when it is unguarded and A attacks. Army A wishes to maximize the damage whilst Army B wishes to minimize the damage. Find all Nash Equilibria.
I tried formulating the game into a $3*3$ payoff matrix game, by giving Army A $0$ if the target it aimed at was guarded and the value of the target if it was unguarded. Similarly, for payoffs of Army 2, I assigned $-v_k$ added to the valuations of other two target which were not destroyed, if Army A succeeded in destroying the target and $v_A+v_B+v_C$ if Army A failed to destroy any target, i.e., by aiming at a guarded post. Given this setup, I found that there was no Pure Strategy Nash Equilibria in the game. I do not know how to proceed with the Mixed Strategy Nash Equilibria, if any.
I'm very doubtful of the approach that I tried to employ. I would really appreciate a little help!