Question

1. Army A and army B are engaged in a battle. There are three possible targets that army A can strike which army B will like to defend. Army A can decide to hit only one target and army B can decide to defend only one target. If both parties go for the same target, their forces cancel each other out and they both earn 0. If army A goes for target I while army B decides to defend another target, army A’s payoff becomes v_i and army B’s payoff becomes –v_i. Let v₁ = 1, v₂ = 2 and v₃ = 5. Calculate all mixed strategy Nash Equilibria of this game.

Answer 1

A's strategy $\downarrow$	B 's strategy $\rightarrow$
	Target 1 ( p )	Target 2 (1 - p)
Target 1 ( q )	(0,0)	(vi , -vi )
Target 2 (1 -q)	(vi , -vi )	(0,0)

where -

This game has two pure strategy Nash equilibrium i.e. when both either play Target 1 or Target 2.Here the payoffs are (0 , 0).

To compute mixed strategy equilibrium let A 's probability of to hit target 1 is q and B 's probability to defend target 1 is p .

The table now is -

A's strategy $\downarrow$	B 's strategy $\rightarrow$
	Target 1 ( p )	Target 2 (1 - p)	A's Payoff
Target 1 ( q )	(0,0)	(1, -1 ) (2 , -2 ) (5 , -5)	(1-p) , (2-2 p) , (5-5 p)
Target 2 (1 -q)	(1, -1 ) (2 , -2 ) (5 , -5)	(0,0)	p, 2 p, 5 p
B's payoff	(q-1) (2 q -2) (5 q - 5)	-q -2 q -5 q

In order to find Nash equilibrium in this game the strategy chosen by one should be such that the other cannot improve their payoffs.

So each will randomise their chances of hitting or defending the target.

For A being indifferent about hitting target 1 or 2, it is given by the probability, p  = 1/2.

(Note ; To calculate this, take 1-p = p ; 2 -2 p = 2 p ;  5 - 5 p = 5 , we get p = 1/2 )

Similarly for B being indifferent about defending target 1 or 2, it is given by the probability, q = 1/2.

(Note ; To calculate this, take q -1 = - q  ; 2 q -2 = - 2 q ; 5 q - 5 = -5 q , we get q = 1/2 )

As all these probabilities are independent probabilities, therefore to get the required mixed strategy probabilities for all the outcomes we have to multiply the respective probabilities for the outcomes , i.e. for outcome (i) p x q = 1/2 x 1/2 = 1 /4 or for outcome (ii) p x (1- q) = 1/2 x 1/2 = 1/4 and so on. The table summarises the results.

A's strategy $\downarrow$	B 's strategy $\rightarrow$		A's Payoffs are
	Target 1 ( p )	Target 2 (1 - p)
Target 1 ( q )	1/4	1/4	0.75, 1.5, 3.75
Target 2 (1 -q)	1/4	1/4	0.25, 0.5, 1.25
B's Payoffs are	-0.75 , -1.5 , -3.75	-0.25, -0.5 , -1.25

As Payoffs for A is greater when hitting Target 1 therefore it will hit Target 1, come what B does.

Similarly, For B defending Target 2 entails lesser negative payoffs therefore it will defend Target 2.