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.
Jake is choosing rows
Clyde is choosing columns
Clyde
Jake
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
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
Jake
Here, the Nash equilibrium is
both Jake and Clyde
confessing
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
Clyde
Jake
Note that if Jake and Clyde
can collude, they would
never confess!
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
Jake
The previous example was a
“one shot” game. Would it
matter if the game were played
over and over?
Clyde
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
Perhaps the dynamics of the
game changes because the
members are interacting over
time – this brings is a possible
punishment for cheating.
Clyde
Confess
Jake
Jake
-4 -4
Don’t
Confess
-8
0
0
-8
-1
-1
“I plan on not confessing today…if you don’t confess today, I will not
confess tomorrow, but if you confess today, I will confess forever!”
0
1
Play Prisoner’s
Dilemma Game
Play Prisoner’s
Dilemma Game
Time
Confess
Don’t
Confess
2
Play Prisoner’s
Dilemma Game
3
Play Prisoner’s
Dilemma Game
4
5
Play Prisoner’s
Dilemma Game
Play Prisoner’s
Dilemma Game
“I plan on not confessing today…if
you don’t confess today, I will not
confess tomorrow, but if you
confess today, I will confess
forever!”
Jake
Don’t
Confess:
-1
Clyde
0
1
Play Prisoner’s
Dilemma Game
Play Prisoner’s
Dilemma Game
Time
Confess:
-1
0
Don’t Confess: -6
Cheat: -20
-4
Confess
-1
-4
-4 -4
Don’t
Confess
-8
-1
2
Play Prisoner’s
Dilemma Game
Confess
-1
3
Play Prisoner’s
Dilemma Game
-4
0
Don’t
Confess
0
-8
-1
-1
-1
4
5
Play Prisoner’s
Dilemma Game
Play Prisoner’s
Dilemma Game
-4
Clyde shouldn’t confess, right?
-4
We need to use backward
induction to solve this.
Jake
Clyde
0
1
Play Prisoner’s
Dilemma Game
Play Prisoner’s
Dilemma Game
Time
2
Play Prisoner’s
Dilemma Game
3
Play Prisoner’s
Dilemma Game
Clyde
4
5
Play Prisoner’s
Dilemma Game
Play Prisoner’s
Dilemma Game
Confess
Regardless of what took place the
first four time periods, what will
happen in period 5?
Confess
-4 -4
Don’t
Confess
-8
Don’t
Confess
0
What should
Clyde do here?
0
-8
-1
-1
We need to use backward
induction to solve this.
Clyde
0
1
Play Prisoner’s
Dilemma Game
Play Prisoner’s
Dilemma Game
Time
Jake
2
Play Prisoner’s
Dilemma Game
3
Play Prisoner’s
Dilemma Game
Clyde
4
5
Play Prisoner’s
Dilemma Game
Play Prisoner’s
Dilemma Game
Confess
Confess
Given what
happens in period
5, what should
happen in period 4?
Confess
-4 -4
Don’t
Confess
-8
Don’t
Confess
0
What should
Clyde do here?
0
-8
-1
-1
We need to use backward
induction to solve this.
Clyde
Jake
0
1
Play Prisoner’s
Dilemma Game
Play Prisoner’s
Dilemma Game
Play Prisoner’s
Dilemma Game
Confess
Confess
Time
Confess
Confess
-4 -4
Don’t
Confess
-8
0
Knowing the future
prevents credible
promises/threats!
2
3
Clyde
4
5
Play Prisoner’s
Dilemma Game
Play Prisoner’s
Dilemma Game
Play Prisoner’s
Dilemma Game
Confess
Confess
Confess
Don’t
Confess
0
-8
-1
-1
What we would need is for this game
to never end!
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 we value the future
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
The equilibrium will be that
McDonalds cooperates and
Burger King Enters
5
Out
0
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
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
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
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
Player A
Micro
Macro
Micro
2 1
0
0
Macro
0
1
2
0
If both choose Micro, neither
side has an incentive to
deviate – an equilibrium!
Player B
Player A
Micro
A pure strategy equilibrium refers
to an outcome where both sides
make a move with certainty. An
equilibrium can be maintained
because neither side has any
incentive to deviate from their
choice
Macro
Micro
2 1
0
0
Macro
0
1
2
0
If both choose Macro, neither
has an incentive to deviate –
an equilibrium!
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
Mixed strategy equilibria involve players making moves with various
probabilities
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
If player B
chooses Micro
20% of the time,
you are better
off choosing
Macro.
Player A
Micro
Macro
Micro
2 1
0
0
Macro
0
1
2
0
To maintain a mixed strategy equilibrium, neither player can have a superior choice
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…
2 pL  pR
pL  pR  1
Micro
2 1
0
0
Macro
0
1
2
2 pL  pL  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
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
Both always choose Micro
Note that the strategies are known with certainty, but the outcome is random!
The Stag Hunt: Two individuals are out on a hunt. Each must make a decision on
what to hunt without knowledge of the other individual’s choice
Only one hunter is required to
catch a rabbit – a small, sure
reward
Two hunters are required to take
down a stag – a bigger but riskier
reward
Stag
What’s the equilibrium
here?
Rabbit
Stag
4
4
0
1
Rabbit
1
0
1
1
The Stag Hunt: Two individuals are out on a hunt. Each must make a decision on
what to hunt without knowledge of the other individual’s choice
If both hunt the stag, neither
has an incentive to deviate –
an equilibrium!
Stag
Rabbit
Stag
4
4
0
1
Rabbit
1
0
1
1
If both hunt the rabbit, neither
has an incentive to deviate –
an equilibrium!
Suppose that you believed that
your fellow hunter was equally
likely to hunt the stag or the rabbit
what would you do?
50%
50%
Stag
Rabbit
Stag
4
4
0
1
Rabbit
1
0
1
1
If you hunt the rabbit: You are guaranteed a
reward of 1 with certainty
If you hunt the stag: 50% of the time you get 4,
50% of the time you get 0
EV  .504  .500  2
In this example, hunting the stag is reward dominant (better average
payout), while hunting the rabbit is risk dominant (lower risk)
What if we change the odds…?
10%
90%
Stag
Rabbit
Stag
4
4
0
1
Rabbit
1
0
1
1
If you hunt the rabbit: You are guaranteed a
reward of 1 with certainty
If you hunt the stag: 10% of the time you get 2,
90% of the time you get 0
EV  .104  .900  .4
Now, hunting the rabbit is both reward dominant and risk dominant!!
Choosing the stag would never be a good idea here.
Let’s find the odds that make the
stag and rabbit equally attractive on
average…
X%
Y%
Stag
Rabbit
Stag
4
4
0
1
Rabbit
1
0
1
1
If you hunt the rabbit: You are guaranteed a
reward of 1 with certainty
If you hunt the stag:
EV   X 4  Y 0
For them to be
equal on average:
 X 4  Y 0  1
4X  1
X  .25
X = 25%, Y = 75%
Therefore, in this example,
you will only hunt the stag if
your fellow hunter hunts the
stag at least 25% of the
time.
Similarly, your fellow hunter
will only hunt the stag if you
hunt the stag at least 25% of
the time.
25%
75%
Stag
25%
Stag
4
6.25%
Rabbit
75%
4
Rabbit
1
0
18.75%
0
1
18.75%
1
1
56.25%
Therefore, in this case, the
stag hunt has three possible
equilibria:
50%
50%
Stag
Rabbit
Stag
4
4
0
1
Rabbit
1
0
1
1
Equilibrium #1: Both players always
hunt the stag
Equilibrium #2: Both players sometimes
hunt the stag (each player must hunt
the stag at least 25% of the time)
Equilibrium #3: Both players never hunt
the stag
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 pDA  25 p A   p A
5 pDA  24 p A
5
pA 
pDA
24
p A  pDA  1
Don’t Audit
Audit
Cheat
5
-5
-25
Don’t Cheat
0
0
-1
5
pDA  pDA  1
24
29
pDA  1
24
24
pDA 
(83%)
29
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  pDC  5 pC
10 pC  pDC
1
pC 
pDC
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%)
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%)
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
Player A
Lets consider a
variation…
Offer
Player B
The Shrinking Pie Game:
Negotiations are costly. After
each round, the pot gets
reduced by 50%:
Accept
Day 1
$20
Reject
Player B
Offer
Player A
Accept
Day 2
$10
Reject
Player A
Offer
Player B
Accept
Day 3
Reject
(0,0)
$5
Player A
If player B rejects,
she gets $5
Day 1 tomorrow. She will
accept anything
$20
better than $5
Offer
Player B
Accept
Reject
Player A: $14.99
Player B: $5.01
Player B
Offer
Player A
Accept
Day 2
Reject
$10
If player A rejects,
she gets $4.99 in
one year. She will
accept anything
better than $4.99
Player A: $5.00
Player B: $5.00
Player A
Offer
Player B
Accept
Reject
(0,0)
If day 3 arrives,
Day 3 player B should
accept any offer –
a rejection pays
$5
out $0!
Player A: $4.99
Player B: $.01