Discuss the following scenarios in the discussion forum:

   1.  Consider the following game in matrix form with two players. Payoffs for the row player Shelia are indicated first in each cell, and payoffs for the column player Thomas are second.

              C            D

A            10, 16   14, 24

B            15, 20    6, 12

       1.  Does either player have a dominant strategy? Explain your answer.

       2.  What are the pure-strategy Nash equilibria (if any) of this game? Justify your answer. If there is more than one pure equilibrium, which would Thomas prefer?

       3.  This game has a fully mixed strategy Nash equilibrium in which both Shelia and Thomas play each of their actions with positive probability. What are the mixed strategies for each player in this equilibrium? Show how you would compute such a mixed equilibrium and verify that your mixed strategies are indeed in equilibrium.

Solution:

1.  Dominant strategy. 

Assume the domination concept from the question is the strict variety.  

Test the pure strategies for domination.

For Sheila (player 1):  strategy A does not dominate strategy B because while u1(A,D)  > u1(B,D),

u1(A,C)   <  u1(B,C) .   The reverse logic is the reason that strategy B does not dominate strategy A.

for Thomas (player 2):  strategy C does not dominate strategy D because while u2(B,C) > u2(B,D),

u2(A,C) < u2(A,D).  Again, the reverse logic is the reason that strategy D does not dominate strategy C.

Therefore, neither player has a dominant strategy. 

2.  What are pure strategy Nash equilibria (PSNE), if any? 

Test all pure strategies:

A, C - B is a better response to C than A, so A, C is not PSNE

A, D - A is a best response to D and D is a best response to A.  Therefore, A, D is a PSNE

B, C - B is a best response to C and C is a best response to B.  Therefore, B, C is a PSNE

B, D - C is a better response to B than D, so B, D is not PSNE

Of the two pure strategy Nash equilibria, Thomas prefers A, D because in that equilibrium his utility is greater.

3.  Find positive probability mixed strategy Nash Equilibrium

Suppose both players randomize choice of actions.

Suppose player 1 chooses A at random with probability x.  Therefore, probability of B is (1 – x).

Suppose player 2 chooses C at random with probability y.  Therefore, probability of D is (1 – y).

Let m1 and m2 be the mixed strategy value for choices.  Mixed strategy equilibrium is when

m1(A) = m1(B)   and

m2(C) = m2(D)

m1(A) = m1(B)

u1(A,C) P(A,C) + u1(A,D) P(A,D) = u1(B,C) P(B,C) + u1(B,D) P(B,D)

10(xy) + 14(x(1-y)) = 15((1-x)y) + 6((1-x)(1-y))

m2(C) = m2(D)

u2(A,C) P(A,C) + u2(B,C) P(B,C) = u2(A,D) P(A,D) + u2(B,D) P(B,D)

16(xy) + 20((1-x)y) = 24(x(1-y)) + 12((1-x)(1-y))

Using Wolfram equation solver

https://www.wolframalpha.com/input?i=systems+of+equations+calculator&assumption=%7B%22F%22%2C+%22SolveSystemOf2EquationsCalculator%22%2C+%22equation1%22%7D+-%3E%22%22&assumption=%22FSelect%22+-%3E+%7B%7B%22SolveSystemOf2EquationsCalculator%22%7D%7D&assumption=%7B%22F%22%2C+%22SolveSystemOf2EquationsCalculator%22%2C+%22equation2%22%7D+-%3E%22%22

Choose the positive closed form solution:

x = 1/55 (sqrt(8749) - 68)  approx. =  0.464

y = 1/58 (sqrt(8749) - 65)  approx. = 0.492