1. Finance 510: Microeconomic Analysis
The Basics of Game Theory

2. What is a Game?
Players
Rules
Payoffs

3. Prisoner’s Dilemma…A Classic!
Two prisoners (Jake & Clyde) have been arrested. The DA has enough evidence to convict them both for 1 year, but would like to convict them of a more serious crime.
Jake
Clyde
The DA puts Jake & Clyde in separate rooms and makes each the following offer:
Keep your mouth shut and you both get one year in jail
If you rat on your partner, you get off free while your partner does 8 years
If you both rat, you each get 4 years.

4. Confess Don’t Confess -4 -4 0 -8 -8 0 -1 -1 Strategic (Normal) Form
Jake is choosing rows Clyde is choosing columns
Clyde
Confess
Don’t Confess
Jake

5. Confess Don’t Confess -4 -4 0 -8 -8 0 -1 -1
Suppose that Jake believes that Clyde will confess. What is Jake’s best response?
If Clyde confesses, then Jake’s best strategy is also to confess
Clyde
Confess
Don’t Confess
Jake

6. Confess Don’t Confess -4 -4 0 -8 -8 0 -1 -1
Suppose that Jake believes that Clyde will not confess. What is Jake’s best response?
If Clyde doesn’t confesses, then Jake’s best strategy is still to confess
Clyde
Confess
Don’t Confess
Jake

7. Confess Don’t Confess -4 -4 0 -8 -8 0 -1 -1 Dominant Strategies
Jake’s optimal strategy REGARDLESS OF CLYDE’S DECISION is to confess. Therefore, confess is a dominant strategy for Jake
Clyde
Confess
Don’t Confess
Jake
Note that Clyde’s dominant strategy is also to confess

8. Confess Don’t Confess -4 -4 0 -8 -8 0 -1 -1 Nash Equilibrium
The Nash equilibrium is the outcome (or set of outcomes) where each player is following his/her best response to their opponent’s moves
Clyde
Confess
Don’t Confess
Jake
Here, the Nash equilibrium is both Jake and Clyde confessing

9. Confess Don’t Confess -4 -4 0 -8 -8 0 -1 -1 The Prisoner’s Dilemma
The prisoner’s dilemma game is used to describe circumstances where competition forces sub-optimal outcomes
Clyde
Confess
Don’t Confess
Jake
Note that if Jake and Clyde can collude, they would never confess!

10. Repeated Games
Jake
Clyde
The previous example was a “one shot” game. Would it matter if the game were played over and over?
Suppose that Jake and Clyde were habitual (and very lousy) thieves. After their stay in prison, they immediately commit the same crime and get arrested. Is it possible for them to learn to cooperate? 1 2 3 4 5
Time
Play PD Game
Play PD Game
Play PD Game
Play PD Game
Play PD Game
Play PD Game

11. Repeated Games We can use backward induction to solve this. 1 2 3 4 5
Jake
Clyde
We can use backward induction to solve this. 1 2 3 4 5
Time
Play PD Game
Play PD Game
Play PD Game
Play PD Game
Play PD Game
Play PD Game
Confess Confess
Confess Confess
Confess Confess
Confess Confess
Confess Confess
Confess Confess
Similar arguments take us back to period 0
However, once the equilibrium for period 5 is known, there is no advantage to cooperating in period 4
At time 5 (the last period), this is a one shot game (there is no future). Therefore, we know the equilibrium is for both to confess.

12. Infinitely Repeated Games
Jake
Clyde 1 2
Play PD Game
Play PD Game
Play PD Game
……………
Suppose that Jake knows Clyde is planning on NOT CONFESSING at time 0. If Jake confesses, Clyde never trusts him again and they stay in the non-cooperative equilibrium forever
Lifetime Reward from confessing
Lifetime Reward from not confessing
Not confessing is an equilibrium as long as i < 3 (300%)!!

13. Infinitely Repeated Games
Jake
Clyde 1 2
Play PD Game
Play PD Game
Play PD Game
……………
Suppose that Jake knows Clyde is planning on NOT CONFESSING at time 0. If Jake confesses, Clyde never trusts him again and they stay in the non-cooperative equilibrium forever
The Folk Theorem basically states that if we can “escape” from the prisoner’s dilemma as long as we play the game “enough” times (infinite times) and our discount rate is low enough

14. The Chain Store Paradox
Suppose that McDonalds has an exclusive territory where is earns $100,000 per year, but faces the constant threat of Burger King moving in. If Burger King enters, McDonald's profits fall to $80,000. If it fights, it loses $10,000 today, but creates a reputation that deters future entry.
Should McDonalds fight?
Cost of Entry Deterrence
Present Value of Entry Deterrence

15. The Chain Store Paradox
Accommodate
($80,$10)
Enter
Fight
($-10,-$10)
Accommodate
Accommodate
($100,$0)
($80,$10)
Stay Out
($80,$10) A
Enter
Fight
($60,$0)
Fight F
($-10,-$10)
($-10,-$10)
($100,$0) A
Stay Out A
($60,$0)
At the end game, it is always optimal for McDonalds to Accommodate.

16. The Chain Store Paradox
Accommodate
($80,$10)
Enter
Fight
($-10,-$10)
Accommodate
Accommodate
($100,$0)
($80,$10)
Stay Out
($80,$10) A
Enter
Fight
($60,$0)
Fight F
($-10,-$10)
($-10,-$10)
($100,$0) A
Stay Out A
($60,$0)
However, given McDonald's accommodation, Burger King always enters!

17. The Chain Store Paradox
Accommodate
($80,$10)
Enter
Fight
($-10,-$10)
Accommodate
Accommodate
($100,$0)
($80,$10)
Stay Out
($80,$10) A
Enter
Fight
($60,$0)
Fight F
($-10,-$10)
($-10,-$10)
($100,$0) A
Stay Out A
($60,$0)
However, if entry always occurs, then fighting is not optimal in the prior period!

18. Micro Macro 2 1 0 0 1 2 Choosing Classes! Player B Player A
Suppose that you and a friend are choosing classes for the semester. You want to be in the same class. However, you prefer Microeconomics while your friend prefers Macroeconomics. You both have the same registration time and, therefore, must register simultaneously
Player B
Micro
Macro
2 1
0 0
1 2
What is the equilibrium to this game?
Player A

19. Micro Macro 2 1 0 0 1 2 Choosing Classes! Player B Player A
If Player B chooses Micro, then the best response for Player A is Micro
If Player B chooses Macro, then the best response for Player A is Macro
Player B
Micro
Macro
2 1
0 0
1 2
The Equilibrium for this game will involve mixed strategies!
Player A

20. Choosing Classes!
Suppose that Player A has the following beliefs about Player B’s Strategy
Probabilities of choosing Micro or Macro
Player A’s best response will be his own set of probabilities to maximize expected utility

21. Subject to
Probabilities always have to sum to one
Both classes have a chance of being chosen

22. First Order Necessary Conditions

23. Best Responses
What this says is that if Player A believes that Player B will select Macro with a 2/3 probability, then Player A is willing to randomize between Micro and Macro
Notice that if we 1/3 and 2/3 for the above probabilities, we get
If Player B is following a 1/3, 2/3 strategy, then any strategy yields the same expected utility for player B

24. It’s straightforward to show that there are three possible Nash Equilibrium for this game
Both always choose Micro
Both Randomize between Micro and Macro
Both always choose Macro
Note that the strategies are known with certainty, but the outcome is random!

25. Micro Macro 2 1 0 0 1 2 Player B Player A Sequential Games
In many games of interest, some of the choices are made sequentially. That is, one player may know the opponents choice before she makes her decision.
Player B
Micro
Macro
2 1
0 0
1 2
Consider the previous game, (with three possible equilibria), but now, let Player A choose first.
Player A

26. We can use a decision tree to write out the extensive form of the game
Player A
Player A moves first in stage one.
Micro
Macro
Player B
Player B
The second stage (after the first decision is made) is known as the subgame.
Micro
Macro
Micro
Macro
(2, 1)
(0, 0)
(0, 0)
(1, 2)

27. We can use a decision tree to write out the extensive form of the game
Now, if Player A chooses Micro
Suppose that Player A chooses Macro.
Player A
Micro
Player B should choose Micro
Macro
Player B should choose Macro
Player B
Player B
Micro
Macro
Micro
Macro
(2, 1)
(0, 0)
(0, 0)
(1, 2)

28. In this game, player A has a first mover advantage
Player A knows how player B will respond, and therefore will always choose Micro (and a utility level of 2) over Macro (and a utility level of 1)
Player A
In this game, player A has a first mover advantage
Micro
Macro
Player B
Player B
Micro
Macro
Micro
Macro
(2, 1)
(0, 0)
(0, 0)
(1, 2)

29. What about the Macro/Macro equilibrium?
If player A know that Player B was following a pure strategy of always choosing Macro, then we could get a Macro/Macro result.
Player A
Micro
Macro
But always choosing Macro is not a solution in the subgame. Therefore, Macro/Macro is not subgame perfect
Player B
Player B
Micro
Macro
Micro
Macro
(2, 1)
(0, 0)
(0, 0)
(1, 2)

30. Note: Simultaneous Move Games
Suppose that we assume Player A moves first, but Player B can’t observe Player A’s choice?
Player A
Micro
Macro
Player B
Player B
We are back to the original mixed strategy equilibrium!
Micro
Macro
Micro
Macro
(2, 1)
(0, 0)
(0, 0)
(1, 2)

31. In the third stage, the best response is to kill the hostages (1, -.5)
In the Movie Air Force One, Terrorists hijack Air Force One and take the president hostage. Can we write this as a game?
Terrorists
Take Hostages
Don’t Take Hostages
President
(0, 1)
Negotiate
Don’t Negotiate
In the third stage, the best response is to kill the hostages
(1, -.5)
Terrorists
Given the terrorist response, it is optimal for the president to negotiate in stage 2
Kill
Don’t Kill
Given Stage two, it is optimal for the terrorists to take hostages
(-.5, -1)
(-1, 1)

32. Terrorists
Take Hostages
The equilibrium is always (Take Hostages/Negotiate). How could we change this outcome?
Don’t Take Hostages
President
(0, 1)
Suppose that a constitutional amendment is passed ruling out hostage negotiation (a commitment device)
Negotiate
Don’t Negotiate
(1, -.5)
Terrorists
Kill
Without the possibility of negotiation, the new equilibrium becomes (No Hostages)
Don’t Kill
(-.5, -1)
(-1, 1)

33. Backward Induction…the Centipede game!1 2 3 4 5 6 A B A B A B
$5.00 $5.00
$1.00 $1.00
$0.00 $3.00
$2.50 $2.50
$1.50 $4.50
$3.50 $3.50
$3.00 $6.00
Two players (A and B) make alternating decisions (Right or Down). Note that at each stage in the game, the total reward increases.

34. Backward Induction…the Centipede game!1 2 3 4 5 6 A B A B A B
$5.00 $5.00
$1.00 $1.00
$0.00 $3.00
$2.50 $2.50
$1.50 $4.50
$3.50 $3.50
$3.00 $6.00
In stage 6, B’s best move is down
In stage 5, Given B’s expected move in stage 6, A will choose down ($3.50 vs. $3)
In stage 4, Given A’s move in stage 5, B will choose down ($4.50 vs. $3.50)

35. Backward Induction…the Centipede game!1 2 3 4 5 6 A B A B A B
$5.00 $5.00
$1.00 $1.00
$0.00 $3.00
$2.50 $2.50
$1.50 $4.50
$3.50 $3.50
$3.00 $6.00
In stage 3, Given B’s move in stage 4, A will choose down ($2.50 vs. $1.50)
In stage 2, Given A’s move in stage 3, B will choose down ($3.00 vs. $2.50)
In stage 1, Given B’s move in stage 2, A will choose down ($1.00 vs. $0)

36. Backward Induction…the Centipede game!1 2 3 4 5 6 A B A B A B
$5.00 $5.00
$1.00 $1.00
$0.00 $3.00
$2.50 $2.50
$1.50 $4.50
$3.50 $3.50
$3.00 $6.00
Through backward induction, we find that the equilibrium of this game is A choosing down in the first stage and ending the game!

37. A bargaining example…
Two players have $1 to divide up between them. On day one, Player A makes an offer, on day two player B makes a counteroffer, and on day three player A gets to make a final offer. If no agreement has been made after three days, both players get $0.
Player A discounts future payments at rate
Player B discounts future payments at rate
Player A is more impatient

38. What should Player A offer in Day 3?
Player B
Day 1
What should Player A offer in Day 3?
Accept
Reject
If player A offers $0, Player B is indifferent
Player B
Offer
Player A
Day 2
Accept
Reject
Player A
Offer
Player B
Day 3
Player A = $1, Player B = $0
Accept
Reject
(0,0)

39. What should Player B offer in Day 2?
Player A
Offer
Player B
Day 1
What should Player B offer in Day 2?
Accept
Reject
We know that Player A is indifferent between $1 tomorrow and $
Player B
today
Offer
Player A
Day 2
Player A = $
Player B = $
Accept
Reject
Player A
Offer
Player B
Day 3
Accept
Reject
(0,0)

40. What should Player A offer in Day 1? Player A
Player B
Day 1
Player B = $
Accept
Reject
Player B
Offer
Player A
Day 2
Accept
Reject
What should Player A offer in Day 1?
Player A
Offer
We know that Player B is indifferent between $
Player B
Day 3
today and $
tomorrow
Accept
Reject
(0,0)

41. In this game, player A has a last mover advantage Player B = $0.09
Offer
Player A = $
Player B
Day 1
Player B = $
Accept
Reject
The Nash Equilibrium is Player B accepting Player A’s offer on Day one.
A couple points…
Player A = $0.91
In this game, player A has a last mover advantage
Player B = $0.09
This advantage grows as either A becomes more patient or B becomes less patient
Player A = $1 or
Player B = $0

42. How about this game?
Acme and Allied are introducing a new product to the market and need to set a price. Below are the payoffs for each price combination.
Acme
$.95
$1.30
$1.95
$1.00
3 6
7 3
10 4
$1.35
5 1
8 2
14 7
$1.65
6 0
6 2
8 5
Allied
What is the Nash Equilibrium?

43. Iterative Dominance
Note that Allied would never charge $1 regardless of what Acme charges ($1 is a dominated strategy). Therefore, we can eliminate it from consideration.
Acme
With the $1 Allied Strategy eliminated, Acme’s strategies of both $.95 and $1.30 become dominated.
$.95
$1.30
$1.95
$1.00
3 6
7 3
10 4
$1.35
5 1
8 2
14 7
$1.65
6 0
6 2
8 5
Allied
With Acme’s strategies reduced to $1.95, Allied will respond with $1.35