<p>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, <span class="math-container">$v_k$</span> is <span class="math-container">$v_A&gt;v_B&gt;v_C&gt;1$</span>. 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.</p>

<p>I tried formulating the game into a <span class="math-container">$3*3$</span> payoff matrix game, by giving Army A <span class="math-container">$0$</span> 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 <span class="math-container">$-v_k$</span> added to the valuations of other two target which were not destroyed, if Army A succeeded in destroying the target and <span class="math-container">$v_A+v_B+v_C$</span> 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. </p>

<p>I'm very doubtful of the approach that I tried to employ. I would really appreciate a little help! </p>

<game-theory><self-study><nash-equilibrium>
Nash Equilibria in Target Destroying-Guarding Game
S.Rana
# Nash Equilibria in Target Destroying-Guarding Game

(Asked by: username_0 on 2019-01-13)

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!

**Tags:** game-theory, self-study, nash-equilibrium

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## 0 answers

