Question

4. Compute the mixed-strategy equilibria of the following games. 1 А в \ L M R 2,4 0,0 3. 6,3 | 3,5 5,5 B 1,6 | 3,7 4,8 5,2 | 3,7 4,9

Answer 1

1. Suppose player 1 choose A with probability p, and B with probability 1-p, if player 2 best responds with a mixed strategy then player 1 must make him indifferent between his strategies so that his expected payoff from his strategies is equal. That is E2(A) = E2(B).
So, 4p + 6(1-p) = 0p + 7(1-p)
So, 4p + 6 - 6p = 7 - 7p
So, 4p - 6p + 7p = 7-6
So, 5p = 1
So, p = 1/5 = 0.2
1-p = 1-0.2 = 0.8

Suppose player 2 choose A with probability q, and B with probability 1-q, if player 1 best responds with a mixed strategy then player 2 must make him indifferent between his strategies so that his expected payoff from his strategies is equal. That is E1(A) = E1(B).
So, 2q + 0(1-q) = 1q + 3(1-q)
So, 2q = q + 3 - 3q
So, 2q - q + 3q = 3
So, 4q = 3
So, q = 3/4 = 0.75
1-q = 1-0.75 = 0.25

So, the mixed strategy NE is (p, q) = (0.2, 0.75)

2. We first find the dominated strategy and eliminate them.
Given that player 1 choose U, player 2's best response is M(5).
Given that player 1 choose C, player 2's best response is R(8).
Given that player 1 choose D, player 2's best response is R(9).
So, player 2's dominated strategy is L as it is never chosen by player 2.

Given that player 2 choose L, player 1's best response is U(8).
Given that player 2 choose M, player 1's best response is C(5).
Given that player 2 choose R, player 1's best response is U(6).
So, player 1's dominated strategy is D as it is never chosen by player 1.

Now, we eliminate the two dominated strategies.

	2: M	2: R
1: U	3, 5	6, 3
1: C	5, 5	4, 8

Suppose player 1 choose U with probability p, and C with probability 1-p, if player 2 best responds with a mixed strategy then player 1 must make him indifferent between his strategies so that his expected payoff from his strategies is equal. That is E2(M) = E2(R).
So, 5p + 5(1-p) = 3p + 8(1-p)
So, 5p + 5 - 5p = 3p + 8 - 8p
So, 5p = 8-5
So, 5p = 3
So, p = 3/5 = 0.6
1-p = 1-0.6 = 0.4

Suppose player 2 choose M with probability q, and R with probability 1-q, if player 1 best responds with a mixed strategy then player 2 must make him indifferent between his strategies so that his expected payoff from his strategies is equal. That is E1(U) = E1(C).
So, 3q + 6(1-q) = 5q + 4(1-q)
So, 3q + 6 - 6q = 5q + 4 - 4q
So, q + 3q = 6 - 4
So, 4q = 2
So, q = 2/4 = 0.5
1-q = 1-0.5 = 0.5

So, the mixed strategy NE is (p, q) = (0.5, 0.5)