The Basics of Game Theory Finance 510: Microeconomic Analysis.
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Transcript of The Basics of Game Theory Finance 510: Microeconomic Analysis.
The Basics of Game Theory
Finance 510: Microeconomic Analysis
What is a Game?
Prisoner’s Dilemma…A Classic!
Jake
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.
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.
Strategic (Normal) Form
Jake
Clyde
Confess Don’t Confess
Confess -4 -4 0 -8
Don’t Confess
-8 0 -1 -1
Jake is choosing rows Clyde is choosing columns
Jake
Clyde
Confess Don’t Confess
Confess -4 -4 0 -8
Don’t Confess
-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
Jake
Clyde
Confess Don’t Confess
Confess -4 -4 0 -8
Don’t Confess
-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
Jake
Clyde
Confess Don’t Confess
Confess -4 -4 0 -8
Don’t Confess
-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
Note that Clyde’s dominant strategy is also to confess
Nash Equilibrium
Jake
Clyde
Confess Don’t Confess
Confess -4 -4 0 -8
Don’t Confess
-8 0 -1 -1
The Nash equilibrium is the outcome (or set of outcomes) where each player is following his/her best response to their opponent’s moves
Here, the Nash equilibrium is both Jake and Clyde confessing
The Prisoner’s Dilemma
Jake
Clyde
Confess Don’t Confess
Confess -4 -4 0 -8
Don’t Confess
-8 0 -1 -1
The prisoner’s dilemma game is used to describe circumstances where competition forces sub-optimal outcomes
Note that if Jake and Clyde can collude, they would never confess!
Repeated GamesJake 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?
Time0 1 2 3 4 5
Play PD Game
Play PD Game
Play PD Game
Play PD Game
Play PD Game
Play PD Game
Repeated GamesJake Clyde
Time0 1 2 3 4 5
Play PD Game
Play PD Game
Play PD Game
Play PD Game
Play PD Game
Play PD Game
We can use backward induction to solve this.
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.
Confess Confess
However, once the equilibrium for period 5 is known, there is no advantage to cooperating in period 4
Confess Confess
Confess Confess
Confess Confess
Confess Confess
Confess Confess
Similar arguments take us back to period 0
Infinitely Repeated Games Jake Clyde
0 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
iiiiPDV
4...
)1(
4
)1(
4
)1(
40
32
Lifetime Reward
from confessing
iiiiPDV
11...
)1(
1
)1(
1
)1(
11
32
Lifetime Reward from not confessing
Not confessing is an equilibrium as long as i < 3 (300%)!!
Infinitely Repeated Games Jake Clyde
0 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
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 $10,000 today, but creates a reputation that deters future entry.
Should McDonalds fight?
Niii )1(
000,20$...
)1(
000,20$
)1(
000,20$000,110$
2
Present Value of Entry DeterrenceCost of Entry Deterrence
Accommodate
Fight
Accommodate
Fight
Enter
Stay Out
Accommodate
Fight
($80,$10)
($-10,-$10)
($100,$0)
($60,$0)
The Chain Store Paradox
($80,$10)
($-10,-$10)
Enter
Stay Out
A
F
A
A
($80,$10)
($-10,-$10)
($100,$0)
($60,$0)
At the end game, it is always optimal for McDonalds to Accommodate.
Accommodate
Fight
Accommodate
Fight
Enter
Stay Out
Accommodate
Fight
($80,$10)
($-10,-$10)
($100,$0)
($60,$0)
The Chain Store Paradox
($80,$10)
($-10,-$10)
Enter
Stay Out
A
F
A
A
($80,$10)
($-10,-$10)
($100,$0)
($60,$0)
However, given McDonald's accommodation, Burger King always enters!
Accommodate
Fight
Accommodate
Fight
Enter
Stay Out
Accommodate
Fight
($80,$10)
($-10,-$10)
($100,$0)
($60,$0)
The Chain Store Paradox
($80,$10)
($-10,-$10)
Enter
Stay Out
A
F
A
A
($80,$10)
($-10,-$10)
($100,$0)
($60,$0)
However, if entry always occurs, then fighting is not optimal in the prior period!
Choosing Classes!
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
Micro Macro
Micro 2 1 0 0
Macro 0 0 1 2Pla
yer
A
Player B
What is the equilibrium to this game?
Micro Macro
Micro 2 1 0 0
Macro 0 0 1 2Pla
yer
A
Player B
Choosing Classes!
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
The Equilibrium for this game will involve mixed strategies!
Choosing Classes!
Suppose that Player A has the following beliefs about Player B’s Strategy
Macro
Micro
r
l
Pr
Pr
Probabilities of
choosing Micro or Macro
Player A’s best response will be his own set of probabilities to maximize expected utility
Macrop
Microp
b
t
Pr
Pr
)1()0()0()2(,
rlbrltpp
ppMaxbt
btbtrbltbt pppppppp 2112),,(
)1()0()0()2(,
rlbrltpp
ppMaxbt
Subject to
0
0
1
b
t
bt
p
p
pp Probabilities always have to sum to one
Both classes have a chance of being chosen
btbtrbltbt pppppppp 2112),,(
First Order Necessary Conditions
02 1 l02 r
01 bt pp
01 tp02 bp
02 01 0tp
0bp
0
0
b
t
p
p021
1
2
lr
rl
3
2
3
1 rl
Best Responses
3
2
3
1 rl
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
rbltpp
ppMaxbt
2,
Notice that if we 1/3 and 2/3 for the above probabilities, we get
bt
ppppMax
bt 3
2
3
2,
If Player B is following a 1/3, 2/3 strategy, then any strategy yields the same expected utility for player B
3
2
3
1 rl pp
3
1
3
2 bt pp
0 1 rl pp 0 1 bt pp
1 0 rl pp 1 0 bt pp
It’s straightforward to show that there are three possible Nash Equilibrium for this game
Both always choose Micro
Both always choose Macro
Both Randomize between Micro and Macro
Note that the strategies are known with certainty, but the outcome is random!
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.
Micro Macro
Micro 2 1 0 0
Macro 0 0 1 2Pla
yer
A
Player B
Consider the previous game, (with three possible equilibria), but now, let Player A choose first.
We can use a decision tree to write out the extensive form of the game
Player A
Player B Player B
Mic
ro
Mic
ro
Mic
ro
Macro
Macro
Macro
(2, 1) (0, 0) (0, 0) (1, 2)
The second stage (after the first decision is made) is known as the subgame.
Player A moves first in stage one.
We can use a decision tree to write out the extensive form of the game
Player A
Player B Player B
Mic
ro
Mic
ro
Mic
ro
Macro
Macro
Macro
(2, 1) (0, 0) (0, 0) (1, 2)
Suppose that Player A chooses Macro.
Player B should choose Macro
Now, if Player A chooses Micro
Player B should choose Micro
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
Player B Player B
Mic
ro
Mic
ro
Mic
ro
Macro
Macro
Macro
(2, 1) (0, 0) (0, 0) (1, 2)
In this game, player A has a first mover advantage
Player A
Player B Player B
Mic
ro
Mic
ro
Mic
ro
Macro
Macro
Macro
(2, 1) (0, 0) (0, 0) (1, 2)
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.
But always choosing Macro is not a solution in the subgame. Therefore, Macro/Macro is not subgame perfect
Note: Simultaneous Move Games
Player A
Player B Player B
Mic
ro
Mic
ro
Mic
ro
Macro
Macro
Macro
(2, 1) (0, 0) (0, 0) (1, 2)
Suppose that we assume Player A moves first, but Player B can’t observe Player A’s choice?
We are back to the original mixed strategy equilibrium!
Terrorists
Terrorists
President
Take
H
osta
ges
Neg
otia
te
Kill
Don’t Take
Hostages
Don’t K
ill
Don’t
Negotiate
(1, -.5)
(-.5, -1) (-1, 1)
(0, 1)
In the Movie Air Force One, Terrorists hijack Air Force One and take the president hostage. Can we write this as a game?
In the third stage, the best response is to kill the hostages
Given the terrorist response, it is optimal for the president to negotiate in stage 2
Given Stage two, it is optimal for the terrorists to take hostages
Terrorists
Terrorists
President
Take
H
osta
ges
Neg
otia
te
Kill
Don’t Take
Hostages
Don’t K
ill
Don’t
Negotiate
(1, -.5)
(-.5, -1) (-1, 1)
(0, 1)
The equilibrium is always (Take Hostages/Negotiate). How could we change this outcome?
Suppose that a constitutional amendment is passed ruling out hostage negotiation (a commitment device)
Without the possibility of negotiation, the new equilibrium becomes (No Hostages)
Backward Induction…the Centipede game!
A B A B A B $5.00 $5.00
$3.00 $6.00
$2.50 $2.50
$0.00 $3.00
$1.50 $4.50
$3.50 $3.50
$1.00 $1.00
Two players (A and B) make alternating decisions (Right or Down). Note that at each stage in the game, the total reward increases.
1 2 3 4 5 6
Backward Induction…the Centipede game!
A B A B A B $5.00 $5.00
$3.00 $6.00
$2.50 $2.50
$0.00 $3.00
$1.50 $4.50
$3.50 $3.50
$1.00 $1.00
1 2 3 4 5 6
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)
Backward Induction…the Centipede game!
A B A B A B $5.00 $5.00
$3.00 $6.00
$2.50 $2.50
$0.00 $3.00
$1.50 $4.50
$3.50 $3.50
$1.00 $1.00
1 2 3 4 5 6
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)
Backward Induction…the Centipede game!
A B A B A B $5.00 $5.00
$3.00 $6.00
$2.50 $2.50
$0.00 $3.00
$1.50 $4.50
$3.50 $3.50
$1.00 $1.00
1 2 3 4 5 6
Through backward induction, we find that the equilibrium of this game is A choosing down in the first stage and ending the game!
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
1
1
Player A is more impatient
Player A
Player B
Offer
Accept Reject
Player B
Offer
Player A
Accept Reject
Player A
Offer
Player B
Accept Reject
(0,0)
Day 1
Day 2
Day 3
What should Player A offer in Day 3?
If player A offers $0, Player B is indifferent
Player A = $1, Player B = $0
Player A
Player B
Offer
Accept Reject
Player B
Offer
Player A
Accept Reject
Player A
Offer
Player B
Accept Reject
(0,0)
Day 1
Day 2
Day 3
What should Player B offer in Day 2?
We know that Player A is indifferent between $1 tomorrow and $ today
Player A = $ Player B = $ 1
Player A
Player B
Offer
Accept Reject
Player B
Offer
Player A
Accept Reject
Player A
Offer
Player B
Accept Reject
(0,0)
Day 1
Day 2
Day 3
What should Player A offer in Day 1?
We know that Player B is indifferent between $ )1(
)1( today and $tomorrow
Player A = $ )1(1 Player B = $ 1
Player A
Player B
Offer
Accept Reject
Day 1Player A = $ )1(1 Player B = $ 1
The Nash Equilibrium is Player B accepting Player A’s offer on Day one.
A couple points…
In this game, player A has a last mover advantage 9.
Player A = $0.91
Player B = $0.09
This advantage grows as either A becomes more patient or B becomes less patient
1 Player A = $1
Player B = $0or
0
How about this game?
$.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
Alli
ed
Acme
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.
What is the Nash Equilibrium?
Iterative Dominance
$.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
Alli
ed
Acme
Note that Allied would never charge $1 regardless of what Acme charges ($1 is a dominated strategy). Therefore, we can eliminate it from consideration.
With the $1 Allied Strategy eliminated, Acme’s strategies of both $.95 and $1.30 become dominated.
With Acme’s strategies reduced to $1.95, Allied will respond with $1.35