1. Game Theory
Topic 2
Simultaneous Games
“Loretta’s driving because I’m drinking and I’m drinking because she’s driving.”
- The Lockhorns

2. Review Understanding the game Noting if the rules are flexible
Anticipating our opponents’ reactions
Thinking one step ahead
Where does this lead us?
We’ve defined the “game” but not the outcome
Mike Shor

3. Equilibrium
The likely outcome of a game when rational, strategic agents interact
Each player is playing his or her best strategy given the strategy choices of all other players
No player has incentive to change his or her action unilaterally
Outline:
Model interactions as games
Identify the equilibria
Decide when they are likely to occur
Mike Shor

4. Cigarette Advertising on TV
All US tobacco companies advertised heavily on TV
Surgeon General issues official warning
Cigarette smoking may be hazardous
Cigarette companies fear lawsuits
Government may recover healthcare costs
Companies strike agreement
Carry the warning label and cease TV advertising in exchange for immunity from federal lawsuits.
1964
1970
Mike Shor

5. Strategic Interaction
Players: Reynolds and Philip Morris
Strategies: Advertise or Not Advertise
Payoffs: Companies’ Profits
Environment:
Each firm earns $50 million from its customers
Advertising costs a firm $20 million
Advertising captures $30 million from competitor
How to represent this game?
Mike Shor

6. Strategic Form of a Game
PLAYERS
Philip Morris
No Ad Ad
Reynolds
50 , 50
20 , 60
60 , 20
30 , 30
STRATEGIES
PAYOFFS
Mike Shor

7. What to Do?
Philip Morris
No Ad Ad
Reynolds
50 , 50
20 , 60
60 , 20
30 , 30
If you are advising Reynolds, what strategy do you recommend?
Mike Shor

8. Best Replies
A strategy is a best reply to some opponents’ strategy if it does at least as well as any other strategy
si is a best reply to s-i if
for every si’
Not necessarily unique
Mike Shor

9. Solving the Game Best reply for Reynolds: Philip Morris No Ad Ad
50 , 50
20 , 60
60 , 20
30 , 30
Best reply for Reynolds:
If Philip Morris advertises:
If Philip Morris does not advertise:
Mike Shor

10. Dominance
A strategy is dominant if it outperforms all other strategies no matter what opposing players do
Games with dominant strategies are easy to solve
No need for “what if …” thinking
Mike Shor

11. Dominance si strictly dominates si’ if for every s-i
(the payoff is strictly higher for every strategy of the other players)
si weakly dominates si’ if
for every s-i, and
for some s-i
Mike Shor

12. Dominance
A strategy si is strictly dominant if it strictly dominates all other strategies for that player
A strategy si is weakly dominant if it weakly dominates all other strategies for that player
Mike Shor

13. Dominance Example 1 A strictly dominates B & A strictly dominates C
Therefore A is strictly dominant X Y Z A 10 20 30 B 8 18 25 C 5
Mike Shor

14. Dominance Example 2 A strictly dominates B & A weakly dominates C’
Therefore A is weakly dominant X Y Z A 10 20 30 B 8 18 25 C’
Mike Shor

15. Dominance Example 3 A strictly dominates B & A does not dominate C’’
Therefore A is not dominant X Y Z A 10 20 30 B 8 18 25
C’’
Mike Shor

16. Dominance If you have a dominant strategy use it.
(and no ability to agree on an alternate course of action)
use it.
If your opponent has a dominant strategy
then expect her to play it.
Mike Shor

17. Prisoner’s Dilemma Both players have a dominant strategy
Optimal
No Ad Ad
50 , 50
20 , 60
60 , 20
30 , 30
Equilibrium
Both players have a dominant strategy
The equilibrium results in lower payoffs for each player
Mike Shor

18. Prisoner’s Dilemma Both players have a dominant strategy (s1,s1)
u11 , u11
u12 , π21
u21 , u12
u22 , π22
Both players have a dominant strategy (s1,s1)
u11 > u21 u12 > u22
The equilibrium results in lower payoffs for each player
u22 > u11
The above two statements imply:
u12 > u22 > u11 > u21
Mike Shor

19. Cigarette Advertising
After the 1970 agreement:
Cigarette advertising decreased by $63 million
Industry Profits rose by $91 million
Mike Shor

20. Prisoner’s Dilemma
The dominant strategy will be played
Mike Shor

21. Social Behavior in Pigs
Baldwin and Meese (1979), “Social Behavior in Pigs Studied by Means of Operant Conditioning,” Animal Behavior
Two small pigs:
First pig gets 8 units of food, second gets 2
If simultaneous, each gets 5
Pushing the lever costs 1
One small, one big:
If big pig is first, eats all of the food
If small pig is first, it gets 6 units of food
If simultaneous, big pig gets 7
Mike Shor

22. Prisoner’s Dilemma The dominant strategy will be played
An equilibrium is NOT necessarily efficient
Players can be forced to accept mutually bad outcomes
Bad to be playing a prisoner’s dilemma, but good to make others play
Mike Shor

23. How to Win a Bidding War by Bidding Less?
The battle for Federated (1988)
Parent of Bloomingdales
Current share price ≈ $60
Expected post-takeover share price ≈ $60
Macy’s offers $70/share
contingent on receiving 50% of the shares
Do you tender your shares to Macy’s?
Mike Shor

24. How to Win a Bidding War (continued)
Robert Campeau bids $74 per share not contingent on amount acquired
Campeau’s Mixed Scheme:
If less than 50% tender their shares, each receives:
$74 per share
If X>50% tender, each receives:
Mike Shor

25. The Federated Game To whom do you tender your shares?
Majority of Others
Macy’s
Campeau
You
$70
$60
$74
$67+
To whom do you tender your shares?
Mike Shor

26. How to Win a Bidding War
Each player has a dominant strategy: Tender shares to Campeau
Resulting Price:
(½ x 74) + (½ x 60) = $67
BUT: Macy’s offered $70 !
Mike Shor

27. “ The biggest, looniest deal ever. ”
Dominant Strategies
“ The biggest, looniest deal ever. ”
– Fortune Magazine, July 1988
on Campeau’s acquisition of Federated Stores
Mike Shor

28. Prisoner’s Dilemma Examples
Pricing by Firms
High or low prices?
Value menus and loyalty programs
Divorce
Hire attorneys or proceed amicably?
Nuclear Weapons
Build or don’t build weapons?
State governments
Inducements to attract business to a state
Mike Shor

29. Dominated Strategies Two restaurants compete
Can charge price of $30, $50, or $60
Customer base consists of tourists and natives
600 tourists pick randomly
400 natives select the lowest price
Marginal costs are $10
Mike Shor

30. Tourists & Natives Example scenario:
Restaurant 1: $50, Restaurant 2: $60
Restaurant 1 gets:
300 tourists natives
= 700 customers x ($50-$10) = $28K
Restaurant 2 gets:
300 tourists + 0 natives
= 300 customers x ($60-$10) = $15K
Mike Shor

31. Tourists & Natives R. 2 $30 $50 $60 R. 1 10 , 10 14 , 12 14 , 15
12 , 14
20 , 20
28 , 15
15 , 14
15 , 28
25 , 25
in thousands of dollars
Mike Shor

32. Dominance
A strategy si is strictly dominated if some strategy si’ strictly dominates it
A strategy si is weakly dominated if some strategy si’ weakly dominates it
Mike Shor

33. Iterated Deletion of Strictly Dominated Strategies
Does any player have a
(strictly) dominated strategy?
Eliminate the strictly dominated strategy
Reduce the size of the game
Repeat: Iterate the above procedure
Mike Shor

34. Iterated Deletion of Dominated Strategies
$30
$50
$60
R. 1
10 , 10
14 , 12
14 , 15
12 , 14
20 , 20
28 , 15
15 , 14
15 , 28
25 , 25
Mike Shor

35. No Dominated Strategies
Often there are no dominated strategies
Some games may have multiple equilibria
Equilibrium selection becomes an issue
Method:
For each player, find the best response to every strategy of the other player
Mike Shor

36. Equilibrium
An outcome in which every player is playing a best response to the strategies of all other players.
An equilibrium is a strategy profile s such that si is a best reply to s-i for all i.
Mike Shor

37. Equilibrium Illustration
The Lockhorns
Mike Shor

38. Games of Coordination Complements & technology adoption
Two complementing firms
Must use same technology, but each firm has a preferred technology
Equilibrium does not offer a unique prediction
Commit (or go first) to win!
Firm 2 A B
Firm 1
100 , 50
0 , 0
0 , 0
50 , 100
Mike Shor

39. Games of Assurance Joint research ventures
Each firm may invest $50,000 into an R&D project
Project succeeds only if both invest
If successful, each nets $75,000
Firm 2
$50K $0
Firm 1
75 , 75
-50 , 0
0 , -50
0 , 0
Mike Shor

40. Games of Chicken Entry into small markets Firm 2 Stay Swerve Firm 1
-50 , -50
100 , 0
0 , 100
50 , 50
Mike Shor

41. The Right Game to Play Why do we “solve” games?
To know which one to play!
How do internal corporate changes impact the outcome of strategic interaction?
Some games are better than others
Mike Shor

42. Capacity Constraints
Can decreasing others’ added value increase our profits?
Can decreasing total industry value increase our profits?
Mike Shor

43. Multiple Equilibria
What is the predictive power of game theory when there are multiple equilibria?
Sometimes nothing ?
Refinements
Focal points
Efficiency
Evolutionary stability
Fairness
Risk dominance
Mike Shor

44. Summary Games have predictable outcomes Select the right game to play
Notice dominant & dominated strategies
Select the right game to play
Looking ahead:
Sequential Games:
How do games unfold over time?
Mike Shor