1. Game Theory II

2. LP Formulation Recall, and, by maximin theorem,
Further, these must hold for each strategy of player 2 including the pure stategies; e.g.,
y = (1, 0, 0), y = (0, 1, 0), y = (0, 0, 1)

3. LP Formulation Then, for each pure strategy,
If we plug this back into the expected payoff equation, we get

4. LP Formulation
Since,

5. LP Formulation
We now have

6. LP Formulation
The problem is, we don’t have an objective function or knowledge of what is. We solve this by adding an artificial variable, for and maximizing it which, by minimax theorem, will equal

7. LP Formulation
Similarly, player 2 would like to minimize the maximum payoff to player 1. Then, by the same logic,

8. Political Campaign Problem
Let’s reconsider our revised political campaign problem.

9. Political Campaign Problem
For player 2,

10. Political Campaign Problem
For player 1

11. Political Campaign Problem
For player 1
This is the same solution we derived from the graphical and algebraic solution.

12. Political Campaign Problem
For player 2,

13. Political Campaign Problem
For player 2,
This is the same solution we derived from the graphical and algebraic solution.

14. Example
Two companies share the bulk of the market for a particular product. Each is now planning its new marketing plans for the next year in an attempt to wrest some sales away from the other company (industry market is relatively mature). Each company is considering 3 possibilities: 1) better packaging, 2) increased advertising, and 3) price reduction. Costs are roughly equivalent but are sufficiently large that only one strategy will be selected. Estimated impact on increased percentage of sales for company 1 follows:

15. Example
Without eliminating dominated strategies, use minimax (or maximin) to determine the best strategy for each company.

16. Example
Maximin
Minimax

17. Example
Identify and eliminate dominated strategies and resolve.

18. Example Identify and eliminate dominated strategies and resolve.
For player 1, no strategy dominates.
For player 2, strategy 3 dominates strategy 1 and 2 because it has smaller losses

19. Example Identify and eliminate dominated strategies and resolve.
For player 1, no strategy dominates.
For player 2, strategy 3 dominates both strategy 1 and 2 because it has smaller losses

20. Example Identify and eliminate dominated strategies and resolve.
For player 1, no strategy dominates.
For player 2, strategy 3 dominates both strategy 1 and 2 because it has smaller losses
For player 1, strategy 1 now dominates both 2 and 3

21. Example
Formulate and solve as an LP. Do all approaches give the same solution? Why or why not?

22. Example

23. Example

24. Example

25. Example

26. Athletic Competition
The KDK swim team has an important meet with the BDK swim team next week. Each team has a star swimmer (Erin and Katie) who can swim very well in the 100 yard butterfly, backstroke, and breaststroke events. However competition rules limit them to two events each. Therefore, the team coaches must decide how best to use them to maximum advantage. Whoever is used in the third event will have a slower time than any of the 3 swimmers tabled for each team.

27. Athletic Competition
The KDK swim team has an important meet with the BDK swim team next week. Each team has a star swimmer (Erin and Katie) who can swim very well in the 100 yard butterfly, backstroke, and breaststroke events. However competition rules limit them to two events each. Therefore, the team coaches must decide how best to use them to maximum advantage. Whoever is used in the third event will have a slower time than any of the 3 swimmers tabled for each team.
Best time is 5 pts, 2nd place is 3 pts, 3rd place is 1 pt.

28. Athletic Competition
Strategy 1 strong swimmer on butterfly and backstroke
Strategy 2 strong swimmer on butterfly and breaststroke
Strategy 3 strong swimmer on backstroke and breaststroke

29. Athletic Competition

30. Athletic Competition

31. Athletic Competition

32. Athletic Competition

33. Athletic Competition

34. Athletic Competition

35. Athletic Competition

36. Athletic Competition
There is very little chance team KDK can win. As long as team BDK avoids strategy 3 (Katie swims backstroke and breaststroke), team BDK should win.

37. Athletic Competition
Note there is no feasible solution for constraints 1 and 2 unless x4 is neg.

38. Athletic Competition

39. Athletic Competition