Math 484.1.  April 19, 2006 .   Midterm 3.  
5 problems, 10 pts each. Name______________

 Solve matrix games:
1.
1
 2
 3
 4
 5
3
 3
 4
 5
 4
5
 4*'
 6
 4
 5
5
 1
 4
 3
 2
4
 1
 2
 1
 3
*' saddle point.

2.
1
 2
 3
 4
 5
3
 1
 4
 5
 4
3
 1
 2
 3
 1
5
 1
 4
 3
 2
4
 1
 2
 1
 3
By domination, we get

 c1
 c2
r1
 1
 2
r4
 5
 1
Row player's optimal strategy is (4r1+r4)/5 = [0.8,  0,  0,  0.2, 0]T. Column player's optimal strategy is (c1+4c2)/5 =[0.2, 0.8, 0, 0, 0].
The value of game is 9/5 = 1.8.

3.
2
 0
 2
 -2
2
 -2
 0
 5
4
 2
 -5
 0
The first column goes by domination. Then we get a symmetric game, so its value  is 0.
An optimal strategy for the column player is [0, 5, 2, 2]/9,. Its transpose without the first entry is an optimal strategy for   the row player .

4.
3
 3
 0
 1
 4
 1
 0*'
 1
0
 2
 3
 3
 2
 2
 0
 1
2
 2
 0
 1
 2
 0
 0
 1
*' saddle point.

5.
1
 4
 3
 2
 5
 4
 3
 5
4
 3
 1
 4
 3
 1
 5
 4
5
 2
 5
 6
 3
 5
 6
 4

The columns 4-8 go by domination. Optimal strategies: [9, 6, 7]/22T and  [[3, 7, 1, 0, 0, 0 , 0,0]/11.
The values of game is 34/11.