1. Planning of Barus & Holley Addition
Activity Duration, Days Predecessors
Procurement
215
Site Preparation 60
Pour Foundation 31 SP
Erect Steel 15
P, PF
Roof, Ext. Wall 40 ES
Fabricate Glass 50 ES
Int Walls & Gl. 85
REW
Landscaping 55
REW
Acquire Furn.
280
Install Furn. 15
IWG, AF, FG

2. Forward Pass: Find ES(I)
ES(I) = Earliest Start of Activities emanating from node I 60
270 2 5 PF SP
REW L
IWG P ES FG 9 1 3 4 6 7
370
215
230
280
355 AF IF 8
355

3. Backward Pass: Find LC(I)
LC(I) = Latest Completion
of Activities terminating
at Node I.
184
270 60
270 2 5 PF
REW L SP
IWG P ES FG 9 1 3 4 6 7
370
215
230
280
355
215
230
370
355
355 AF IF 8
355
355

4. Identify Critical Path
184
270 60
270 2 5 PF
REW L SP
IWG P ES FG 9 1 3 4 6 7
370
215
230
280
355
215
370
230
355
355 AF IF 8
355
355

5. Determine Slack Times 184 270 60 270 2 5 SP(124) L (45) PF(124) REW
IWG ES P P
FG(75) 9 1 3 4 6 7
370
215
230
280
355
215
370
230
355
355
AF(75) IF 8
355
355

6. Two-Person, Zero-Sum Game: The Campers
Matrix of Payoffs to Row Player:
Column Player:
Carol (j)
(1) (2) (3) (4)
(1)
(2)
(3)
(4)
Row Player:
Ray (i)

7. Two-Person, Zero-Sum Game: The Campers
Column Player:
Carol (j)
Matrix of Payoffs to Row Player:
Row Minima:
(1) (2) (3) (4)
(1)
(2)
(3)
(4) 1 2 3
Row Player:
Ray (i)
Column Maxima:

8. Two-Person, Zero-Sum Game: The Campers
Column Player:
Carol (j)
Matrix of Payoffs to Row Player:
Row Minima:
(1) (2) (3) (4)
(1)
(2)
(3)
(4) 1 2 3
Row Player:
Ray (i)
MaxiMin
Column Maxima:
Game has a saddle point!
MiniMax

9. Two-Person, Zero-Sum Game: Advertising
Matrix of Payoffs to Row Player:
Column Player:
TV N TVN TV N
TVN
Row Player:

10. Two-Person, Zero-Sum Game: Advertising
Matrix of Payoffs to Row Player:
Column Player:
Row Minima:
TV N TVN TV N
TVN
-.6
-.4
Row Player:
Column Maxima:

11. Two-Person, Zero-Sum Game: Advertising
Matrix of Payoffs to Row Player:
Column Player:
Row Minima:
TV N TVN TV N
TVN
-.6
-.4
Row Player:
MaxiMin
Column Maxima:
MiniMax
Game has a saddle point!

12. Two-Person, Zero-Sum Game: Advertising
Matrix of Payoffs to Row Player:
Column Player:
Row Minima:
TV N TVN TV N
TVN
-.4
-.6
Row Player:
MaxiMin
Column Maxima:
Game has a saddle point!
MiniMax

13. Two-Person, Zero-Sum Game: Advertising
Matrix of Payoffs to Row Player:
Column Player:
Row Minima:
TV N TVN TV N
TVN
.6 0
-.6
Row Player:
MaxiMin
Column Maxima:
.6 0
Game has a saddle point!
MiniMax

14. Two-Person, Zero-Sum Game: Advertising
Matrix of Payoffs to Row Player:
Column Player:
Row Minima:
TV N TVN TV N
TVN
Row Player:
MaxiMin
Column Maxima:
Game has a saddle point!
MiniMax

15. Two-Person, Zero-Sum Game: Mixed Strategies
Column Player:
Matrix of Payoffs to Row Player:
Row Minima:
C C2 R1 R2 -2
Row Player:
Column Maxima:
10 5

16. Two-Person, Zero-Sum Game: Mixed Strategies
Column Player:
Matrix of Payoffs to Row Player:
Row Minima:
C C2 R1 R2 -2
MaxiMin
Row Player:
Column Maxima:
10 5
MiniMax VR VC
No Saddle Point!

17. Two-Person, Zero-Sum Game: Mixed Strategies
Column Player:
Matrix of Payoffs to Row Player:
Row Minima:
Y1 Y2
C C2
X1 R1
X2 R2 -2
MaxiMin
Row Player:
Column Maxima:
10 5
MiniMax
MaxiMin
MiniMax
No Saddle Point!

18. Graphical Solution VR 10 VR < 10(1-X1) VR < -2 +7X1 50/17
Optimal Solution:
X1=12/17, X2=5/17
VRMAX=50/17 1
12/17 X1

19. Graphical Solution VR 10 Y1=1 VR < 10(1-X1) Y1=0 VR < -2 +7X1
50/17
Y1=.25
Optimal Solution:
X1=12/17, X2=5/17
VRMAX=50/17 1
12/17 X1

20. Two-Person, Zero-Sum Games: Summary
Represent outcomes as payoffs to row player
Evaluate row minima and column maxima
If maximin=minimax, players adopt pure strategy corresponding to saddle point; choices are in stable equilibrium -- secrecy not required
If maximin minimax, use linear programming to find optimal mixed strategy; secrecy essential
Number of options to consider can be reduced by using iterative dominance procedure

21. The Minimax Theorem “Every finite, two-person, zero-sum game
has a rational solution in the form
of a pure or mixed strategy.”
John Von Neumann, 1926