1. STRATEGIC DECISION MAKING
GAME THEORY
STRATEGIC DECISION MAKING

2. What is Game Theory?
GT is an analytical tool for social sciences that is used to model strategic interactions or conflict situations.
Strategic interaction: When actions of a player influence payoffs to other players

3. GT: science or art?
GT is the science of rational behavior in interactive situations.
Good strategists mix the science of GT with their own experience.

4. Why is GT important? Facilitates strategic thinking.
Provides a standard taxonomy that is needed for a scientific approach in analyzing strategic interactions.
Helps confirm long held beliefs.
Provides new insights.
“To be literate in the modern age, you need to have a general understanding of GT.” P.Samuelson

5. GT in the News
GT, long an intellectual passtime, came into its own as a business tool. Forbes, July 3, 1995
FCC hired game theorists to construct rules of an auction for new wireless phone systems’ licenses. In response, communications companies hired game theorists first to negotiate with FCC and then help prepare optimal bids given the rules of the auction. Business Week, Mar. 14, 1994.
GT is hot. The Wall Street Journal, Feb. 13, 1995.

6. GT in the News (cont.)
…lately game theorists have focused on real-world issues—how to raise auction proceeds by revealing the bids and how Wal-Mart can coexist with local retailers. Consultants are jumping in. McKinsey & Co. has set up a game theory unit. Forbes, Nov. 7, 1994.

7. Where can we use GT?
Any situation that requires us to anticipate our rival’s response to our action is a potential context for GT.
Games: Checkers, poker, chess, tennis, soccer etc.
Economics: Industrial Organization, Micro/Macro/ International/Labor/Natural resource Economics, and Public Finance
Political science: war/peace (Cuban missile crisis)
Law: Designing laws that work
Biology: animal behavior, evolution
Information systems: System competition/evolution

8. Where can we use GT? (cont.)
Business:
Games against rival firms:
Pricing, advertising, marketing, auctions, R&D, joint ventures, investment, location, quality, take over etc.
Games against other players
Employee/employer, managers/stockholders
Supplier/buyer, producer/distributor, firm/government

9. Some Terminology Strategy Payoffs Rationality
Common knowledge of rules
Equilibrium

10. Strategic (Normal) Form Games
Static Games of Complete and Imperfect Information

11. What is a Normal Form Game?
A normal (strategic) form game consists of:
Players: list of players
Strategies: all actions available to all players
Payoffs: a payoff assigned to every contingency (every possible strategy profile as the outcome of the game)

12. Prisoners’ Dilemma
Two suspects are caught and put in different rooms (no communication). They are offered the following deal:
If both of you confess, you will both get 5 years in prison (-5 payoff)
If one of you confesses whereas the other does not confess, you will get 0 (0 payoff) and 10 (-10 payoff) years in prison respectively.
If neither of you confess, you both will get 2 years in prison (-2 payoff)

13. Easy to Read Format of Prisoner’s Dilemma
Confess
Don’t Confess
-5, -5
0, -10
-10, 0
-2, -2
Prisoner 1

14. Assumptions in Static Normal Form Games
All players are rational.
Rationality is common knowledge.
Players move simultaneously. (They do not know what the other player has chosen).
Players have complete but imperfect information.

15. Solution of a Static Normal Form Game
Equilibrium in strictly dominant strategies
A strictly dominant strategy is the one that yields the highest payoff compared to the payoffs associated with all other strategies.
Rational players will always play their strictly dominant strategies.

16. Solution of a Static Normal Form Game
Iterated elimination of strictly dominated strategies
Rational players will never play their dominated strategies.
Eliminating dominated strategies may solve the game.

17. Solution of a Static Normal Form Game (cont.)
Nash Equilibrium (NE):
In equilibrium neither player has an incentive to deviate from his/her strategy, given the equilibrium strategies of rival players.
Neither player can unilaterally change his/her strategy and increase his/her payoff, given the strategies of other players.

18. Definition of Nash Equilibrium
A strategy profile is a list (s1, s2, …, sn) of the strategies each player is using.
If each strategy is a best response given the other strategies in the profile, the profile is a Nash equilibrium.
Why is this important?
If we assume players are rational, they will play Nash strategies.
Even less-than-rational play will often converge to Nash in repeated settings.

19. An Example of a Nash Equilibrium
Column a b a
1,2
0,1
Row b
1,0
2,1
(b,a) is a Nash equilibrium.
To prove this:
Given that column is playing a, row’s best response is b.
Given that row is playing b, column’s best response is a.

20. Finding Nash Equilibria – Dominated Strategies
What to do when it’s not obvious what the equilibrium is?
In some cases, we can eliminate dominated strategies.
These are strategies that are inferior for every opponent action.
In the previous example, row = a is dominated.

21. Example A 3x3 example: Column a b c a 73,25 57,42 66,32 Row b 80,26
35,12
32,54 c
28,27
63,31
54,29

22. Example A 3x3 example: Column a b c a 73,25 57,42 66,32 Row b 80,26
35,12
32,54 c
28,27
63,31
54,29
c dominates a for the column player

23. Example A 3x3 example: Column a b c a 73,25 57,42 66,32 Row b 80,26
35,12
32,54 c
28,27
63,31
54,29
b is then dominated by both a and c for the row player.

24. Example A 3x3 example: Column a b c a 73,25 57,42 66,32 Row b 80,26
35,12
32,54 c
28,27
63,31
54,29
Given this, b dominates c for the column player –
the column player will always play b.

25. Solution of Prisoners’ Dilemma Dominant Strategy Equilibrium
Confess
Don’t Confess
-5, -5
0, -10
-10, 0
-2, -2
Prisoner 1

26. Solution of Prisoners’ Dilemma Iterated Elimination Procedure
Confess
Don’t Confess
-5, -5
0, -10
-10, 0
-2, -2
Prisoner 1

27. Solution of Prisoners’ Dilemma Cell-by-cell Inspection
Confess
Don’t Confess
-5, -5
0, -10
-10, 0
-2, -2
Prisoner 1

28. NE of Prisoners’ Dilemma
The strategy profile {confess, confess} is the unique pure strategy NE of the game.
In equilibrium both players get a payoff of –5.
Inefficient equilibrium; (don’t confess, don’t confess) yields higher payoffs for both.

29. A Pricing Example 100, 100 -10, 140 140, -10 0, 0 High Price Low Price
Firm 2
High Price
Low Price
100, 100
-10, 140
140, -10
0, 0
Firm 1

30. 3x3 Game Using Iterated Elimination
Player 2
Left
Center
Right
Top
1, 0
1, 3
3, 0
Middle
0, 2
0, 1
Bottom
2, 4
5, 3
Player 1

31. A Coordination Game Battle of the Sexes
Husband
Opera
Movie
2, 1
0, 0
1, 2
Wife

32. Battle of the Sexes: After 30 Years of Marriage
Husband
Opera
Movie
3, 2
0, 0
1, 2
Wife

33. Mixed strategies
Unfortunately, not every game has a pure strategy equilibrium.
Rock-paper-scissors
However, every game has a mixed strategy Nash equilibrium.
Each action is assigned a probability of play.
Player is indifferent between actions, given these probabilities.

34. Mixed Strategies
In many games (such as coordination games) a player might not have a pure strategy.
Instead, optimizing payoff might require a randomized strategy (also called a mixed strategy)
Wife
football
shopping
football
2,1
0,0
Husband
shopping
1,2
0,0

35. A Strictly Competitive Game Matching Pennies
Player 2
Heads
Tails
1, -1
-1, 1
Player 1
No NE in pure strategies

36. Dynamic Games of Complete and Perfect Information
Extensive Form Games
Dynamic Games of Complete and Perfect Information

37. What is a Game Tree? Right Left A B C D P11 P21 P12 P22 P13 P23 P14
Player 1
Right
Left
Player 2
Player 2 A B C D
P11
P21
P12
P22
P13
P23
P14
P24

38. An Advertising Example
Migros
Normal
Aggressive
Wal-Mart
Wal-Mart
Enter
Enter
Stay out
Stay out
680
-50
730
700
400
800

39. Assumptions in Dynamic Extensive Form Games
All players are rational.
Rationality is common knowledge
Players move sequentially. (Therefore, also called sequential games)
Players have complete and perfect information
Players can see the full game tree including the payoffs
Players can observe and recall all previous moves

40. Solution of an Extensive Form Game
Subgame Perfect Equilibrium: For an equilibrium to be subgame perfect, it has to be a NE for all the subgames as well as for the entire game.
A subgame is a decision node from the original game along with the decision nodes and end nodes.
Backward induction is used to find SPE

41. Advertising Example: 3 proper subgames
Migros
Wal-Mart
Wal-Mart
680
-50
730
700
400
800

42. Solution of the Advertising Game
Subgame 1
Subgame 2
Wal-Mart
Wal-Mart
Enter
Enter
Stay out
Stay out
680
-50
730
700
400
800

43. Solution of the Advertising Game (cont.)
Migros
Normal
Aggressive
730
700
400
SPE of the game is the strategy profile: {aggressive, (stay out, enter)}

44. Properties of SPE
The outcome that is selected by the backward induction procedure is always a NE of the game with perfect information.
SPE is a stronger equilibrium concept than NE
SPE eliminates NE that involve incredible threats.

45. Suppose WM threatens to enter no matter what Migros does
Suppose WM threatens to enter no matter what Migros does. Is this a credible threat?
Migros
Normal
Aggressive
Wal-Mart
Wal-Mart
Enter
Enter
Stay out
Stay out
680
-50
730
700
400
800

46. JUST PLAYING!
Repeated Games

47. Game 1 Firm 1 Firm 2 Low Output High Output 40, 40 60, 30 30, 60
50, 50
Firm 1

48. Game 2
Greece
War
Peace
1, 1
3, 0
0, 3
2, 2
Turkiye

49. Repeated Normal Form Games
REPEATED GAMES
Repeated Normal Form Games

50. Prisoners’ Dilemma Revisited
Suppose that the two suspects play the same game every time they get caught.
Can they coordinate their choices in order to get the best outcome for both of them?
Finitely repeated game
Infinitely repeated game

51. Twice-repeated PD: First stage payoff matrix after adding NE payoffs of the second stage
Prisoner 2
Confess
Don’t Confess
-10, -10
-5, -15
-15, -5
-7, -7
Prisoner 1

52. N-times repeated PD
In a finitely repeated (n times repeated game where n  2) PD game, the cooperative outcome (don’t confess, don’t confess) cannot be enforced.
Since in the last stage (nth stage) the NE is (confess, confess) and all players know this, in all previous stages the same NE will prevail.

53. Infinitely Repeated PD
When the game is played infinitely or players do not know when the game is going to end, the backward induction breaks down. Following trigger strategies can enforce the cooperative outcome.
Trigger strategy: A player cooperates as long as the other players cooperate, but any defection from cooperation on the part of the rivals triggers the player to behave noncooperatively for a specified period of time (period of punishment).

54. Trigger Strategies
Grim strategy: A trigger strategy in which the punishment period lasts till the end of the game.
Grim strategy for PD game: Play “don’t confess” in the first period. In period t, play “don’t confess” if the outcome was (don’t confess, don’t confess) in all preceding t-1 periods, and play “confess” otherwise.

55. Trigger Strategies (cont.)
Tit-for-tat (TFT): A trigger strategy in which the punishment period lasts as long as the rival keeps on cheating (returning back to cooperative periods of game play is possible).
TFT strategy for PD game: Play “don’t confess” in the first period. In period t, play “don’t confess” if the rival’s most recent play was (don’t confess, don’t confess), and play “confess” otherwise.