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, 0	0, 0	1, 1
1, 1	1, 3	2, 0	2, 2
0, 3	0, 0	1, 3	3, 2
1, 0	0, 0	0, 0	1, 0

(payoffs for  Ann and Bob)

Solution.   Two equilibria:

0, 1*	1*, 0	0, 0	1, 1*
1*, 1	1*, 3*	2*, 0	2, 2
0, 3*	0, 0	1, 3*	3*, 2
1*, 0*	0, 0*	0, 0*	1, 0*

 --- (row 2, column 2) and (row 4, column 1).

Two Pareto optimal payoffs: (3, 2) and (1,3).

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

0	1	0	1
1	1*'	2	2
0	0	1	3
1	0	0	1

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

1	1	3	0*'
0	3	0	0*'
0	0	3*	0*'
1	2	2	0*'

v(empty)=0, v(Ann,Bob) = 5.

Nash bargaining:   (x-1)y -> max , x >= 1, y >= 0,    (x,y) is a mixture of  (3, 2) and (1,3).,
x+2y=7,   (x-1)+2y=6,  x-1=2y=3;  x=4, y=1.5  optimal solution on line,; x=3, y=2 the arbitration pair.

                         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                      1  0  1
1  2  1                      2  3  4
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  4  3

Solution.
Three equilibria, two Pareto optimal:
strategies                                                payoffs
1  1  1                      0  1  2
1  1  2                      1  0  1
1  2  1                      2*  3*  4*  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
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*  4*  3*  equilibrium   Pareto optimal

v(empty)=0, v(A,B,C)) = 9.
v(A) = 0 = the value of the matrix game

0*'	1	2	1
0	1	2	2
0	0	1	2

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

1	0	0	1	0	0
3	2	3	3	1*'	4

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

2	4	1	3	0*'	1
1	3	1	2	0	3

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

1	1
5	3
0	2
5*'	5
0	0
2	6

v(A,C) = 2= the value of the matrix game

2*'	6
2	4
1	5
2	4
0	2
0	5

v(B,C) = 16/3 = the value of the matrix game

B$C vs A	c1	c2	c3	(5c2+c3)/6
r11	3	1	0
r12	1	2	0
r21	7	6	2	16/3
r22	5	5	7	16/3
(r21+2r22)/3		16/3	16/3	16/3*'

order         contribution
                 A B C
ABC          0 5 4
ACB          0 7 2
BAC          4 1 4
BCA    11/3 1 13/3
CAB         2 7 0
CBA  11/3  16/3 0
---------------
     20/9  79/18  43/18     Shapley values.

Nash bargaining:
x(y-1)z -> max, x >=0 , y>= 1,  z > = 0,  (x,y,z) is a mixture of  (2,3,4) and (2,4,3), so x = 2.
y+z=7, (y-1)+z=6,  y-1=z=3, (x,y,z)= (2,4, 3) the arbitration triple.