Finance 30210: Managerial
Economics
The Basics of Game Theory
What is a Game?
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.
Strategic (Normal) Form
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
Clyde
Jake
Note that Clyde’s
dominant strategy is
also to confess
Confess Don’t
Confess
Confess -4 -4
0 -8
Don’t
Confess
-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
Jake
Here, the Nash equilibrium is
both Jake and Clyde
confessing
Clyde
Confess Don’t
Confess
Confess -4 -4
0 -8
Don’t
Confess
-8
0
-1
-1
The Prisoner’s Dilemma
The prisoner’s dilemma game is used to
describe circumstances where
competition forces sub-optimal outcomes
Jake
Note that if Jake and Clyde
can collude, they would
never confess!
Clyde
Confess Don’t
Confess
Confess -4 -4
0 -8
Don’t
Confess
-8
0
-1
-1
“Winston tastes good like a
cigarette should!”
“Us Tareyton smokers would
rather fight than switch!”
Advertise
Don’t
Advertise
Advertise
10 10
30
Don’t
Advertise
5
20 20
30
5
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?
0
1
2
Play
PD Game
Play
PD Game
Play
PD Game
Time
3
Play
PD Game
4
Play
PD Game
5
Play
PD Game
Repeated Games
Jake
Clyde
We can use backward induction to solve this.
0
1
2
Play
PD Game
Play
PD Game
Play
PD Game
Time
Confess
Confess
Confess
Confess
Confess
Confess
3
Play
PD Game
Confess
Confess
4
5
Play
PD Game
Play
PD Game
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.
Infinitely Repeated Games
0
1
2
Play
PD Game
Play
PD Game
Play
PD Game
Jake
Clyde
……………
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 noncooperative equilibrium forever
Lifetime Reward
from confessing
4
4
4
4
PDV  0 


 ...  
2
3
(1  i) (1  i) (1  i)
i
Lifetime Reward
from not
confessing
1
1
1
1
PDV  1 


 ...  1 
2
3
(1  i) (1  i ) (1  i)
i
Not confessing is an equilibrium as long as i < 3 (300%)!!
Infinitely Repeated Games
0
1
2
Play
PD Game
Play
PD Game
Play
PD Game
Jake
Clyde
……………
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 noncooperative 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
Suppose that McDonald’s is currently the only restaurant in
town, but Burger King is considering opening a location. Should
McDonald's fight for it’s territory?
Fight
0
0
IN
2
Cooperate
2
5
Out
1
Now, suppose that this game is played repeatedly. That is, suppose that
McDonald's faces possible entry by burger King in 20 different locations. Can
entry deterrence be a credible strategy?
Enter
Cooperate
Don’t
Fight
Don’t
Fight
Enter
2
Enter
2
Don’t
Fight
2
Total =2*20 = 40
OR
Enter
Fight
Fight
0
Don’t
Enter
Don’t
Enter
5
5
Don’t
Enter
5
Don’t
Enter
5
Total =19*5 = 95
Now, suppose that this game is played repeatedly. That is, suppose that
McDonald's faces possible entry by burger King is 20 different locations. Can
entry deterrence be a credible strategy?
Enter
Fight
Enter
Fight
Enter
Fight
Don’t
Enter
Don’t
Fight
Don’t
Enter
Don’t
Fight
Don’t
Enter
Don’t
Fight
20th
location
What will Burger King
do here?
Does McDonald’s
have an incentive to
fight here?
If there is an “end date” then McDonald's threat loses its credibility!!
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
What is the Nash
Equilibrium?
Allied
$.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
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.
Allied
With Acme’s
strategies reduced
to $1.95, Allied will
respond with $1.35
$.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
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
Player B
What is the
equilibrium to
this game?
Player A
Micro
Macro
Micro
2 1
0
0
Macro
0
1
2
0
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
Player B
There are two types of
equilibria for this game: Pure
strategies and mixed
strategies!
Player A
Micro
Macro
Micro
2 1
0
0
Macro
0
1
2
0
A quick detour: Expected Value
Suppose that I offer you a
lottery ticket: This ticket has a
2/3 chance of winning $100 and
a 1/3 chance of losing $100.
How much is this ticket worth to
you?
Suppose you played this ticket 6 times:
Attempt
Outcome
1
$100
2
$100
3
-$100
4
$100
5
-$100
6
$100
Total Winnings: $200
Attempts: 6
Average Winnings: $200/6 = $33.33
A quick detour: Expected Value
Given a set of probabilities,
Expected Value measures the
average outcome
Expected Value = A weighted average of the possible outcomes where the
weights are the probabilities assigned to each outcome
Suppose that I offer you a
lottery ticket: This ticket has a
2/3 chance of winning $100 and
a 1/3 chance of losing $100.
How much is this ticket worth to
you?
2
1
EV   $100    $100   $33.33
3
 3
Choosing Classes!
Suppose that player B chooses Micro 20% of the time. What should Player A
do?
Micro:
EV  .202  .80  .4
Macro:
EV  .20  .81  .8
Player B
Player A
If player B
chooses Micro
20% of the time,
you are better
off choosing
Macro.
Micro
Macro
Micro
2 1
0
0
Macro
0
1
2
0
Choosing Classes!
Suppose Player B chooses Micro with probability pL
Chooses Macro with probability pR
Player B
EV   pL 2   pR 0
Macro:
EV   pL 0  PR 1
Micro
Player A
Micro:
If you are indifferent…
Micro
2 1
0
0
Macro
0
1
2
2 pL  pL  1
2 pL  pR
pL  pR  1
3 pL  1
1
3
2
PR 
3
pL 
Macro
0
There are three possible Nash Equilibrium for this game
pl  1
pr  0
pt  1
pb  0
Both always choose Micro
1
pl 
3
2
pr 
3
2
pt 
3
1
pb 
3
Both Randomize between
Micro and Macro
pl  0
pr  1
pt  0
pb  1
Both always choose Macro
Note that the strategies are known with certainty, but the outcome is random!
Ever Cheat on your taxes?
In this game you get to
decide whether or not to
cheat on your taxes while
the IRS decides whether or
not to audit you
Cheat
Don’t
Cheat
What is the
equilibrium to
this game?
Don’t
Audit
Audit
5 -5 -25 5
0
0
-1
-1
If the IRS never audited, your best strategy is to cheat (this would only
make sense for the IRS if you never cheated)
If the IRS always audited, your best strategy is to never cheat (this
would only make sense for the IRS if you always cheated)
The Equilibrium for this
game will involve only mixed
strategies!
Cheat
Don’t
Cheat
Don’t
Audit
Audit
5 -5 -25 5
0
0
-1
-1
Cheating on your taxes!
Suppose that the IRS Audits 25% of all returns. What should you do?
Cheat: EV  .755  .25 25  2.5
Don’t Cheat: EV  .750  .251  .25
If the IRS audits
25% of all
returns, you are
better off not
cheating. But if
you never cheat,
they will never
audit, …
Cheat
Don’t
Cheat
Don’t
Audit
Audit
5 -5 -25 5
0
0
-1
-1
The only way this game can work is for you to cheat sometime, but not all the
time. That can only happen if you are indifferent between the two!
Suppose the government audits with probability p A
Doesn’t audit with probability pDA
Cheat:
EV   pDA 5   pA  25
Don’t Cheat:
EV   pDA 0  PA 1
If you are indifferent…
5 p DA  25 p A   p A
5 p DA  24 p A
5
pA 
p DA
24
pA  pDA  1
Don’t Audit
Audit
Cheat
5
-5
-25
Don’t Cheat
0
0
-1
5
p DA  p DA  1
24
29
p DA  1
24
24
p DA 
29 (83%)
pA 
5
-1
5
(17%)
29
We also need for the government to audit sometime, but not all the time. For
this to be the case, they have to be indifferent!
Suppose you cheat with probability pC
Don’t cheat with probability
Audit:
pDC
EV   pDC  1   pC 5
Don’t Audit:
EV   pDC 0  PC  5
If they are indifferent…
5 pC  p DC  5 pC
10 pC  p DC
1
pC 
p DC
10
pC  pDC  1
Don’t Audit
Audit
Cheat
5
-5
-25
Don’t Cheat
0
0
-1
pC 
1
(9%)
11
1
p DC  p DC  1
10
11
p DC  1
10
10
p DC 
(91%)
11
5
-1
Now we have an equilibrium for this game that is sustainable!
The government audits with probability p  17%
A
Doesn’t audit with probability p  83%
DA
Suppose you cheat with probability pC  9%
Don’t cheat with probability pDC  91%
Don’t Audit
Cheat
5
-5
(7.5%)
We can find the odds of any
particular event happening….
You Cheat and get audited:
pC  p A   .09.17  .0153
Don’t Cheat
0
0
(75%)
(1.5%)
Audit
-25
5
(1.5%)
-1
-1
(15%)
The Airline Price Wars
Suppose that American and Delta
face the given demand for flights
to NYC and that the unit cost for
the trip is $200. If they charge the
same fare, they split the market
p
$500
$220
American
180
What will the equilibrium
be?
Q
P = $500
P = $220
P = $500
$9,000
$9,000
$3,600
$0
P = $220
$0
$3,600
$1,800
$1,800
Delta
60
The Airline Price Wars
If American follows a strategy of charging $500 all the time, Delta’s best
response is to also charge $500 all the time
If American follows a strategy of charging $220 all the time, Delta’s best
response is to also charge $220 all the time
American
P = $220
P = $500
$9,000
$9,000
$3,600
$0
P = $220
$0
$3,600
$1,800
$1,800
Delta
This game has multiple
equilibria and the result depends
critically on each company’s
beliefs about the other
company’s strategy
P = $500
The Airline Price Wars: Mixed Strategy Equilibria
Suppose American charges $500 with probability pH
Charges $220 with probability
pL
Charge $500: EV   pH 9000   pL 0
Charge $220:EV   pH 3600  PL 1800
American
P = $500
P = $220
$9,000
$9,000
$3,600
$0
9000 pH  3600 pH  1800 pL
Delta
P = $500
pL  3 pH
(6%)
P = $220
$0
$3,600
(19%)
pL 
3
(75%)
4
pH 
1
(25%)
4
(19%)
$1,800
$1,800
(56%)
Suppose that we make the game
sequential. That is, one side makes
its decision (and that decision is
public) before the other
Don’t Audit
(-25, 5)
(5, -5)(-1, -1)
(0, 0)
Audit
Cheat
5
-5
-25
Don’t Cheat
0
0
-1
5
-1
If the IRS observes you cheating,
their best choice is to Audit
Don’t Audit
(-25, 5)
(5, -5)(-1, -1)
vs
(0, 0)
Audit
Cheat
5
-5
-25
Don’t Cheat
0
0
-1
5
-1
If the IRS observes you not
cheating, their best choice is to not
audit
Don’t Audit
(-25, 5)
(5, -5)(-1, -1)
(0, 0)
vs
Audit
Cheat
5
-5
-25
Don’t Cheat
0
0
-1
5
-1
Knowing how the IRS will respond,
you never cheat and they never
audit!!
Don’t Audit
Cheat
5
-5
(0%)
Don’t Cheat
0
0
(100%)
(-25, 5)
(5, -5)(-1, -1)
vs
(0, 0)
Audit
-25
5
(0%)
-1
-1
(0%)
Now, lets switch
positions…suppose the IRS
chooses first
Don’t Audit
Cheat
5
-5
(0%)
Don’t Cheat
0
0
(0%)
(-25, 5)
(-1, -1)(5, -5)
(0, 0)
Audit
-25
5
(0%)
-1
-1
(100%)
Again, we could play this game
sequentially
P = $500
P = $500
P = $220
$9,000
$9,000
$3,600
$0
(100%)
P = $220
$0
$3,600
(0%)
(9,000, 9,000)
(3,600, 0) (0, 3,600)
(0%)
$1,800
$1,800
(0%)
(1,800, 1,800)
Delta’s reward is on
the left
In the Movie Air Force
One, Terrorists hijack Air
Force One and take the
president hostage. Can
we write this as a game?
(Terrorists payouts on left)
Terrorists
President
(1, -.5)
(0, 1)
In the third stage, the best
response is to kill the hostages
Terrorists
Given the terrorist response, it is
optimal for the president to
negotiate in stage 2
(-.5, -1)
(-1, 1)
Given Stage two, it is optimal
for the terrorists to take
hostages
Terrorists
The equilibrium is always (Take
Hostages/Negotiate). How could we
change this outcome?
President
(1, -.5)
(0, 1)
Suppose that a constitutional
amendment is passed ruling out
hostage negotiation (a commitment
device)
Terrorists
Without the possibility of
negotiation, the new equilibrium
becomes (No Hostages)
(-.5, -1)
(-1, 1)
A bargaining
example…How do
you divide $20?
Player A
Offer
Player B
Accept
Day 1
Reject
Player B
Two players have $20 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.
Offer
Player A
Accept
Day 2
Reject
Player A
Offer
Player B
Accept
Day 3
Reject
(0,0)
Player A
Offer
Player A knows
Day 1 what happens in
day 2 and knows
that player B wants
to avoid that!
Player B
Accept
Reject
Player A: $19.99
Player B: $.01
Player B
Offer
Player B knows
what happens in
Day 2 day 3 and wants to
avoid that!
Player A
Accept
Reject
Player A: $19.99
Player B: $.01
Player A
Offer
Player B
Accept
Reject
(0,0)
If day 3 arrives,
Day 3 player B should
accept any offer –
a rejection pays
out $0!
Player A: $19.99
Player B: $.01
Player A
Lets consider a
variation…
Offer
Player B
Variation : Negotiations take a
lot of time and each player has
an opportunity cost of waiting:
•Player A has an
investment opportunity that
pays 20% per year.
•Player B has an
investment strategy that
pays 10% per year
Accept
Year 1
Reject
Player B
Offer
Player A
Accept
Year 2
Reject
Player A
Offer
Player B
Accept
Year 3
Reject
(0,0)
Player A
If player B rejects,
she gets $3.35 in
Year 1 one year. That’s
worth $3.35/1.10
today
Offer
Player B
Accept
Reject
Player A: $16.95
Player B: $3.05
Player B
Offer
If player A rejects,
she gets $19.99 in
Year 2 one year. That’s
worth $19.99/1.20
today
Player A
Accept
Reject
Player A: $16.65
Player B: $3.35
Player A
Offer
Player B
Accept
Reject
(0,0)
If year 3 arrives,
Year 3 player B should
accept any offer –
a rejection pays
out $0!
Player A: $19.99
Player B: $.01