In: Economics
A's strategy | B 's strategy | |
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 | B 's strategy | ||
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 | B 's strategy | 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.