Math 486  Game Theory Miderm 3  April 22, 2003.  2 problems, 30 pts each.
Name

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

1.   Players: the row player Ann; the column player Bob.
 0, 1 1, 1 0, 0 1, 1 1, 1 1, 3 2, 3 2, 2 0, 0 0, 0 1, 3 3, 2 1, 0 0, 0 0, 0 0, 0
(payoffs for  Ann and Bob)

Four equilibria: (row 1, column 2) with payoff (1,1), (row 2, column 2) with payoff (1,3),
(row 2, column 3) with payoff (2,3), and (row 4, column 1) with payoff (1,0).
 0, 1* 1*, 1* 0, 0 1, 1* 1*, 1 1*, 3* 2*, 3* 2, 2 0, 0 0, 0 1, 3* 3*, 2 1*, 0* 0, 0* 0, 0* 0, 0*

Two Pareto optimal solutions: (2, 3) and (3,2).

v(Ann) =  the value of  the matrix game

Bob
 0 1 0 1 1*' 1*' 2 2 0 0 1 3 1 0 0 0
= 1.

v(Bob) =  the value of  the matrix game
Ann
 1 1 0 0* (also min in row) 1 3 0 0 0 3 3 0 1 2 2 0
= 0.

v(Ann, Bob) = 5  (maximal total payoff).

Nash bargaining, starting from (1, 0) gives  (3, 2) (arbitration pair).
Bob's payoff
^
|                           (2,3)
|                                  slope -1
|                                         (3,2)
|                             slope 1
|                          (1,0)
--------------------------> Ann's payoff

The Pareto optimal mixed solutions are the mixtures of (2, 3) and (3, 2).

Ann            Bob
Ann Bob                    1                4
Bob Ann                    5                0
------------------------
Shapley values            3               2

2.   Players: A, B, C
strategies                                                payoffs
1  1  1                     0  1  2
1  1  2                      3  0  1
1  2  1                      2  3  0
1  2  2                      1  2  3
2  1  1                      0  0  0
2  1  2                      1  1  1
2  2  1                      2  3  3
2  2  2                      3  3  2
3  1  1                      0  0  0
3  1  2                      0  0  0
3  2  1                      1  1  1
3  2  2                      0  0  0

One equilibrium:  (2, 2, 1) with payoff (2,3,3).
1  1  1                     0*  1  2*
1  1  2                      3*  0  1
1  2  1                      2*  3*  0
1  2  2                      1  2*  3*
2  1  1                      0*  0  0
2  1  2                      1  1  1*
2  2  1                      2*  3*  3*
2  2  2                      3*  3*  2
3  1  1                      0*  0  0*
3  1  2                      0  0*  0*
3  2  1                      1  1*  1*
3  2  2                      0  0*  0

Two Pareto optimal solutions: (2, 3, 3) and  (3,3, 2).

The characteristic function:
v(A,B,C) = 8 (the maximal total payoff);
v(A) = the value of the game
B & C
1,1  1,2  2, 1   2, 2
1             0      3     2      1
A   2             0       1    2      3
3             0       0    1      0
= 0  ( (row 1, column (1,1)) is a saddle point);

v(B) = the value of   game
A & C
1,1   1,2     2, 1   2, 2   3, 1  3,2
1             1      0         0      1       0      0
2             3       2        3      3       1     0

= 0  ( (row 2, column (3,2) ) is a saddle point);

v(C) = the value of   game
A & B
1,1   1,2     2, 1   2, 2   3, 1  3,2
1             2      0         0      3       0      1
2             1       3        1      2       0      0

= 0  ( (row 2, column (3,1) ) is a saddle point);

v(A,B) =  the value of   game
C
1        2
1,1                  1        3
1,2                  5        3
2,1                  0        2
2,2                  5        6
3,1                  0        0
3,2                  2        0
= 5  ( (row (2,2), column 1 ) is a saddle point);

v(A,C) =  the value of   game
B
1        2
1,1                  2        2
1,2                  4        4
2,1                  0        5
2,2                  2        5
3,1                  0        2
3,2                  0        0
= 4  ( (row (1,2), column 1 ) is a saddle point);

v(B,C) =  the value of   game
A
1        2      3
1,1                  3        0      0
1,2                  1        2      0
2,1                  3        6      2
2, 2                 5        5      0
= 2  ( (row (2,1), column 3 ) is a saddle point);

The Pareto optimal mixed payoffs are the mixtures of  (2  3  3)   and   (3  3  2).
The Nash bargaining solution (the arbitration triplet) is
((2  3  3)   +   (3  3  2))/2 = (2.5, 3, 2.5).

A    B    C
---------------------
A B C                    0      5      3
A C B                    0      4      4
B A C                    5      0      3
B C A                    6      0      2
C A B                    4      4      0
C B A                    6      2      0
--------------------
Shapley values     3.5   2.5    2