Math 486 Game Theory. Miderm 3. April 22, 2010.
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, 1 |
|
1, 1 |
1, 3 |
2, 4 |
2, 2 |
|
0, 3 |
0, 0 |
1, 3 |
3, 2 |
|
1, 0 |
0, 0 |
0, 0 |
1, 4 |
Solution. One equilibrium in pure joint strategies, (row 2, column 3):
|
0, 1* |
1*, 0 |
0, 0 |
1, 1 |
|
1*, 1 |
1*, 3 |
2*, 4* |
2, 2 |
|
0, 3* |
0, 0 |
1, 3* |
3*, 2 |
|
1*, 0 |
0, 0 |
0, 0 |
1, 4* |
Two Pareto optimal pure payoffs: (3, 2) and (2,4).
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) = 1 = the value of the matrix game (where Bob is the row player)
|
1* |
1 |
3 |
0' |
|
0’ |
3 |
0' |
0' |
|
0’ |
4* |
3 |
0’ |
|
1*’ |
2 |
2 |
4* |
v(empty)=0, v(Ann,Bob) = 6.
Nash bargaining: (u-1)(v-1) -> max , u >= 1, v >= 1, (u, v) is a mixture of (3, 2) and (2,4).,
2u + v =8, 2(u – 1) + (v – 1) =5, 2(u-1)= v - 1=2.5; u=2.25, v=3.5 optimal solution= Nash solution= the arbitration pair.
This arbitration pair is a mixture of given payoffs: (2.25, 3.5) =(3/4)(2, 4) + (1/4)(3.2).
Ann Bob
Ann Bob 1 5
Bob Ann 5 1
------------------
Shapley values 3 3
2 (45 pts). Players: A, B, C
strategies payoffs
1 1 1 3 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 two Pareto optimal triples:
strategies payoffs
1 1 1 3* 3* 2* equilibrium & Pareto optimal
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
|
3 |
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
|
6 |
1 |
|
5 |
3 |
|
0 |
2 |
|
5 |
5*' |
|
0 |
0 |
|
2 |
5 |
v(A,C) = 5= the value of the matrix game
|
5*' |
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 3 4
BAC 4 1 3
BCA 5 1 2
CAB 5 3 0
CBA 5 3 0
---------------
7/2 5/2 2 Shapley values.
The mixed Pareto optimal payoffs = mixtures of (3,3,2) and (2,3,3)..
B gets 3 at every Pareto optimal triple.Nash solution: the optimal solution of
a>= 1, , b =3, c >= 0=v)C), (a-1)c -> max, (a,c) a mixture of (3, 2) and (2,3).
a+c = 5, (a-1) + c = 4, a -1 = 2, c = 2.
So the Nash solution is (3,3,2). To reach it, the players use the strategy profile (1,1,1).