Math 486  Game Theory. Miderm 3.  April 21, 2011. 
Name______________________________________

Find the equilibria in pure strategies, the pure Pareto optimal payoffs,
the characteristic function,  the Shapley values, and the Nash bargaining solution.
 

1 (30 pts).   Players: the row player Ann; the column player Bob.

0, 1

1, 0

0, 0

1, 3

1, 1

2, 3

2, 2

2, 3

0, 3

0, 0

1, 3

3, 3

1, 0

0, 0

0, 0

1, 4


Solution.   Two  equilibriua in pure joint strategies, (row 2, column 2; row 3, column 4):

0, 1

1, 0

0, 0

1, 3*

1*, 1

2*, 3*

2*, 2

2, 3*

0, 3*

0, 0

1, 3*

3*, 3*

1*, 0

0, 0

0, 0

1, 4*

 

Two Pareto optimal pure payoffs: (3, 3) and (1,4).

v(Ann)  is the value  the value of the  following matrix game  

0

1*

0

1

1*'

2*

2*

2

0

0

1

3*

1

0

0

1

There is a saddle point, row 2 column 1, so v(Ann) = 1.

v(Bob) = 3 = the value of the matrix game (where Bob is the row player)

1*

1

3

0'

0’

3

0'

0'

0’

2*

3

0’

3*

3

3*'

4*

 

v(empty)=0, v(Ann,Bob) = 6.

Nash bargaining:   (u-1)(v-3)  -> max , u >= 1, v >= 3,    (u, v) is a mixture of  (3, 3) and (1,4).,
u + 2v =9,    (u – 1) + 2(v – 3) =2,  (u-1)= 2( v -3)=1;  u=2, v=3.5  optimal solution= Nash solution=   the arbitration pair.

This arbitration pair is a mixture of given payoffs: (2, 3.5) =(1/2)(1, 4) + (1/2)(3.3).

                            Ann          Bob 

Ann Bob                   1          5

Bob Ann                   3          3

------------------ 

Shapley values         2         4


2 (45 pts).   Players: A, B, C
strategies             payoffs
1  1  1                      2  3  2
1  1  2                      1  0  1
1  2  1                      2  3  3
1  2  2                      1  2  3
2  1  1                      0  0  1
2  1  2                      1  1  1
2  2  1                      2  3  3
2  2  2                      2  3  2
3  1  1                      0  0  0
3  1  2                      0  0  0
3  2  1                      1  1  1
3  2  2                      2  3  0

Solution.
Three equilibria and one Pareto optimal triple (2,3,3):
strategies                     payoffs
1  1  1                      2* 3*  2*  equilibrium  strategy
1  1  2                      1  0  1
1  2  1                      2*  3*  3*  equilibrium &  Pareto optimal
1  2  2                      1  2  3
2  1  1                      0  0  1
2  1  2                      1  1  1
2  2  1                      2*  3*  3*equilibrium &  Pareto optimal
2  2  2                      2  3  2
3  1  1                      0  0  0
3  1  2                      0  0  0
3  2  1                      1  1  1
3  2  2                      2   3 0    

v(empty)=0, v(A,B,C)) = 8.


v(A) = 1 = the value of the matrix game

2

1*'

2

1

0

1

2

2

0

0

1

2

v(B) = 1 = the value of the matrix game

3

0

0

1

0

0

3

2

3

3

1*'

3

v(C) = 0 = the value of the matrix game

2

3

1

3

0*'

1

1

3

1

2

0

0

v(A,B) = 5= the value of the matrix game

5

1

5

3

0

2

5

5*'

0

0

2

5

v(A,C) = 4= the value of the matrix game

4*'

5*

2

4

1

5

2

4

0

2

0

2

v(B,C) = 3 = the value of the matrix game

B&C vs A

c1

c2

c3

r11

5

1

0

r12

1

2

0

r21

6

6

2

r22

5

5

3*'

order         contribution
                 A   B   C
ABC          1   4   3
ACB          1   4   3
BAC          4   1   3
BCA          5   1   2
CAB          4   4   0
CBA          5   3   0
---------------
            10/3 17/6  11/6    Shapley values.


The Nash solution is (2, 3, 3)., the only Pareto optimal payoff.