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 |
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 |
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 |
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