1. Principles of Game Theory
Lecture 3: Simultaneous Move Games

2. Administrative Problem sets due by 5pm Quiz 1 is Sunday
Piazza or ~gasper/GT?
Quiz 1 is Sunday
Beginning or end of class?
Questions from last time?

3. Review Simultaneous move situations Backward induction (rollback)
Strategies vs Actions

4. Normal form games Simultaneous move games Two components to the game
Many situations mimic situations of 2+ people acting at the same time
Even if not exactly, then close enough – any situation where the player cannot condition on the history of play.
Referred to as Strategic or Normal form games
Two components to the game
The strategies available to each player
The payoffs to the players
“Simple” games often represented as a matrix of payoffs.

5. Cigarette Advertising example
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

6. Strategic Interaction: Cigarette Advertising
Players?
Reynolds and Philips Morris
Strategies:
Advertise or Not
Payoffs
Companies’ Profits
Strategic Landscape
Firm i can earn $50M from customers
Advertising campaign costs i $20M
Advertising takes $30M away from competitor j

7. Strategic Form Representation
PAYOFFS
PLAYERS
Philip Morris
No Ad Ad
Reynolds
50 , 50
STRATEGIES

8. Strategic Form Representation
PLAYERS
Philip Morris
No Ad Ad
Reynolds
50 , 50
20 , 60
60 , 20
30 , 30
STRATEGIES
PAYOFFS
PAYOFFS

9. What would you suggest?
If you were consulting for Reynolds, what would you suggest?
Think about best responses to PM
If PM advertises?
If PM doesn’t?
Philip Morris
No Ad Ad
Reynolds
50 , 50
20 , 60
60 , 20
30 , 30

10. Nash Equilibrium Equilibrium
Likely outcome of a game when rational strategic agents interact
Each player is playing his/her best strategy given the strategy choices of all other players
No player has an incentive to change his or her strategy unilaterally
Mutual best response.
Not necessarily the best outcome for both players.

11. Dominance
A strategy is (strictly/weakly) dominant if it (strictly/weakly) outperforms all other choices no matter what opposing players do.
Strict >
Weak ≥
Games with dominant strategies are easy to analyze
If you have a dominant strategy, use it.
If your opponent has one, expect her to use it.

12. Solving using dominance
Philip Morris
No Ad Ad
Reynolds
50 , 50
20 , 60
60 , 20
30 , 30
Both players have a dominant strategy
Equilibrium outcome results in lower payoffs for each player
Game of the above form is often called the “Prisoners’ Dilemma”
Optimal
Equilibrium

13. Pricing without Dominant Strategies
Games with dominant strategies are easy to analyze but rarely are we so lucky.
Example:
Two cafés (café 1 and café 2) compete over the price of coffee: $2, $4, or $5
Customer base consists of two groups
6000 Tourists: don’t know anything about the city but want coffee
4000 Locals: caffeine addicted but select the cheapest café
Cafés offer the same coffee and compete over price
Tourists don’t know the price and ½ go to each café

14. Café price competition
Example scenario:
Café 1 charges $4 and café 2 charges $5:
Recall: tourists are dumb and don’t know where to go
Café 1 gets:
3000 tourists locals = 7K customers * $4 = 28K
Café 2 gets
3000 tourists + 0 locals = 3K customers * $5 = 15K
Draw out the 3x3 payoff matrix given
$2, $4, or $5 price selection (simultaneous selection)
6K tourists and 4K locals.

15. Café price competition$2 $4 $5
Café 1
10 , 10
14 , 12
14 , 15
12 , 14
20 , 20
28 , 15
15 , 14
15 , 28
25 , 25
No dominant strategy

16. Dominated Strategies Café 2 $2 $4 $5 Café 1 10 , 10 14 , 12 14 , 15
A player might not have a dominant strategy but may have a dominated strategy
A strategy, s, is dominated if there is some other strategy that always does better than s.
Café 2 $2 $4 $5
Café 1
10 , 10
14 , 12
14 , 15
12 , 14
20 , 20
28 , 15
15 , 14
15 , 28
25 , 25

17. Dominance solvable
If the iterative process of removing dominated strategies results in a unique outcome, then we say that the game is dominance solvable.
We can also use weak dominance to “solve” the game, but be careful
Player 2
Left
Right
Player 1 Up
0,0
1,1
Down

18. Weakly Dominated Strategies
Player 2
Left
Right
Player 1 Up
0,0
1,1
Down
(Down, Right) is an equilibrium profile
But so is (Down, Left) and (Up, Right).
Why?
Recall our notion of equilibrium: No player has an incentive to change his or her strategy unilaterally

19. Fictitious Play Often there are not dominant or dominated strategies.
In such cases, another method for finding an equilibrium involves iterated “what-if..” fictitious play:

20. Best Response Analysis
Similarly you can iterate through each strategy and list the best response for the opponent.
Then repeat for the other player.
Mutual best responses are eq

21. Multiple Equilibria
We’ve said nothing about there always being a unique equilibrium. Often there isn’t just one:

22. Equilibrium Selection
With multiple equilibria we face a very difficult problem of selection:

23. Equilibrium Selection
With multiple equilibria we face a very difficult problem of selection:
Imagine Harry had different preferences:

24. Equilibrium Selection
With multiple equilibria we face a very difficult problem of selection:
Classic issues of coordination:

25. No equilibrium in pure strategies
Nor must there exist an equilibrium in pure strategies
Pure strategies means no randomization (penalty kicks)
We’ll talk about general existence later
Player 2
Rock
Paper
Scissors
Player 1
0,0
-1,1
1,-1

26. Multiple players
While a X b matrixes work fine for two players (with relatively few strategies – a strategies for player 1 and b strategies for player 2), we can have more than two players: a X b X … X z

27. Homework Study for the quiz
Next time: more mathematical introduction to simultaneous move games
Focus on section 1.2 of Gibbons

28. Equilibrium Illustration
The Lockhorns: