An attack group commander has a single plane which can strike only one of three

Question

An attack group commander has a single plane which can strike only one of three possible targets. The defender's commander has one antiaircraft missile battery that can be assigned to only one of the targets. The plane will fail in its assignment if it attacks a defended target. If the attacked target is undefended, it is destroyed. The number of enemy troops at the three targets is 5, 4 and 1, respectively. The attacker wishes to maximize the defender's expected losses, so his payoff is the number of enemy casualties, while the defender wishes to minimize the expected troop losses, so his payoff is minus the number of casualties. Write the normal form of the game, and find all equilibria (pure and mixed).

Explanation / Answer

Below table displays all the option for attacker when he attacks on any of the target. Also it covers all the options when the defendant has secured any one of the station with the anti aircraft missile.

Station 1 (5 Troops)

Station 2 (4 troops)

Station 3 (1 Troop)

Attacks when station 1 is defended

(Fails, 0)

(Success, 4)

(Success, 1)

Attacks when station 2 is defended

(Success, 5)

(Fails, 0)

(Success, 1)

Attacks when station 3 is defended

(Success, 5)

(Success, 4)

(Fails, 0)

Result here is displayed from perspective of attacker. So if he attacks station 1, while it is defended by missile than he fails in the assignment and 0 troops are killed. Whereas if he attacks station 2 while station 1 is defended than he succeeds in assignment and kills 4 troops stationed at station 2. All the results can be read in same line.

Nash Equilibrium:

If the attacker doesn't know about defendant carrying anti-aircraft missile, than he will try to hit station 1 and maximize killing the troops. However defendant will also place missile in station 1 and try to defend it.

In other case if attacker is aware of defendant strategy of carrying the anti-aircraft missile than he will assume it to be at station 1 since defender will try to minimize the losses and hence attacker can choose to attack on station 2. Now defender can also change its strategy by analyzing attackers strategy and he can put the missile at station 2 to defend it.

Station 1 (5 Troops)

Station 2 (4 troops)

Station 3 (1 Troop)

Attacks when station 1 is defended

(Fails, 0)

(Success, 4)

(Success, 1)

Attacks when station 2 is defended

(Success, 5)

(Fails, 0)

(Success, 1)

Attacks when station 3 is defended

(Success, 5)

(Success, 4)

(Fails, 0)

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