Finance 30210: Managerial Economics

The Basics of Game Theory

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!

“Winston tastes good like a cigarette should!”

“Us Tareyton smokers would rather fight than switch!”

Advertise Don’t Advertise

Advertise 10 10 30 5

Don’t Advertise

5 30 20 20

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

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?

IN

Out

Fight

Cooperate

0

2

1

5

0

2

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 Enter EnterDon’t Fight

Don’t Fight

Don’t Fight

2

Enter Don’t Enter

Don’t Enter

Fight Don’t Enter

Don’t Enter

0

OR

2 2

5 5 5 5

Total =2*20 = 40

Total =19*5 = 95

Cooperate

Fight

Enter

Don’t Enter

Enter

Don’t Enter

Enter

Don’t Enter

Fight

Don’t Fight

Fight

Don’t Fight

Fight

Don’t Fight

20th location

Does McDonald’s have an incentive to fight here?

What will Burger King do here?

If there is an “end date” then McDonald's threat loses its credibility!!

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?

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

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

There are two types of equilibria for this game: Pure strategies and mixed strategies!

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: $200Attempts: 6Average 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?

33.33$100$3

1100$

3

2

EV

Choosing Classes!

Suppose that player B chooses Micro 20% of the time. What should Player A do?

Micro: 4.08.220. EV

Macro:

8.18.02. EV

If player B chooses Micro 20% of the time, you are better off choosing Macro.

Micro Macro

Micro 2 1 0 0

Macro 0 0 1 2Pla

yer

A

Player B

Micro: 02 RL ppEV

Macro: 10 RL PpEV

Suppose Player B chooses Micro with probability Lp

Chooses Macro with probability Rp

If you are indifferent…

RL pp 21 RL pp

3

23

1

13

12

R

L

L

LL

P

p

p

pp

Choosing Classes!

Micro Macro

Micro 2 1 0 0

Macro 0 0 1 2

Player B

Pla

yer A

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

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!

Don’t Audit

Audit

Cheat 5 -5 -25 5

Don’t Cheat

0 0 -1 -1

What is the equilibrium to this game?

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

If the IRS never audited, your best strategy is to cheat (this would only make sense for the IRS if you never cheated)

The Equilibrium for this game will involve only mixed strategies!

Don’t Audit

Audit

Cheat 5 -5 -25 5

Don’t Cheat

0 0 -1 -1

If the IRS always audited, your best strategy is to never cheat (this would only make sense for the IRS if you always cheated)

Cheating on your taxes!

Don’t Audit

Audit

Cheat 5 -5 -25 5

Don’t Cheat

0 0 -1 -1

Suppose that the IRS Audits 25% of all returns. What should you do?

Cheat: 5.22525.575. EV

Don’t Cheat: 25.125.075. EV

If the IRS audits 25% of all returns, you are better off not cheating. But if you never cheat, they will never audit, …

Don’t Audit Audit

Cheat 5 -5 -25 5

Don’t Cheat 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!

Cheat: 255 ADA ppEV

Don’t Cheat: 10 ADA PpEV

Suppose the government audits with probability Ap

Doesn’t audit with probability DAp

If you are indifferent…

DAA

ADA

AADA

pp

pp

ppp

24

5

245

255

1 DAA pp

29

24

124

29

124

5

DA

DA

DADA

p

p

pp

(83%)29

5Ap (17%)

Don’t Audit Audit

Cheat 5 -5 -25 5

Don’t Cheat 0 0 -1 -1

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!

Audit: 51 CDC ppEV

Don’t Audit: 50 CDC PpEV

Suppose you cheat with probability Cp

Don’t cheat with probability DCp

If they are indifferent…

DCC

DCC

CDCC

pp

pp

ppp

10

1

10

55

1 DCC pp

11

10

110

11

110

1

DC

DC

DCDC

p

p

pp

(91%)11

1Cp (9%)

Don’t Audit Audit

Cheat 5 -5 -25 5

Don’t Cheat 0 0 -1 -1

Now we have an equilibrium for this game that is sustainable!

The government audits with probability %17ApDoesn’t audit with probability %83DAp

Suppose you cheat with probability %9Cp

Don’t cheat with probability %91DCp

We can find the odds of any particular event happening….

You Cheat and get audited: 0153.17.09. AC pp (1.5%)

(1.5%)(7.5%)

(15%)(75%)

The Airline Price Wars

p

Q

$500

$220

60 180

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 P = $220

P = $500 $9,000$9,000

$3,600$0

P = $220 $0$3,600

$1,800$1,800

American

Del

taWhat will the equilibrium be?

The Airline Price Wars

P = $500 P = $220

P = $500 $9,000$9,000

$3,600$0

P = $220 $0$3,600

$1,800$1,800

American

Del

ta

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

This game has multiple equilibria and the result depends critically on each company’s beliefs about the other company’s strategy

The Airline Price Wars: Mixed Strategy Equilibria

P = $500 P = $220

P = $500 $9,000$9,000

$3,600$0

P = $220 $0$3,600

$1,800$1,800

American

Del

ta

Charge $500: 09000 LH ppEV

Charge $220: 18003600 LH PpEV

Suppose American charges $500 with probability Hp

Charges $220 with probability Lp

LHH ppp 180036009000

HL pp 3

4

3Lp

4

1Hp(75%) (25%)

(56%)(19%)

(19%)(6%)

Suppose that we make the game sequential. That is, one side makes its decision (and that decision is public) before the other

Che

at

Aud

it

Aud

it

Don’t

Cheat

Don’t

Audit

Don’t

Audit

(-25, 5) (5, -5)(-1, -1) (0, 0)

Don’t Audit Audit

Cheat 5 -5 -25 5

Don’t Cheat 0 0 -1 -1

If the IRS observes you cheating, their best choice is to Audit

Che

at

Aud

it

Aud

it

Don’t

Cheat

Don’t

Audit

Don’t

Audit

(-25, 5) (5, -5)(-1, -1) (0, 0)

Don’t Audit Audit

Cheat 5 -5 -25 5

Don’t Cheat 0 0 -1 -1

vs

If the IRS observes you not cheating, their best choice is to not audit

Che

at

Aud

it

Aud

it

Don’t

Cheat

Don’t

Audit

Don’t

Audit

(-25, 5) (5, -5)(-1, -1) (0, 0)

Don’t Audit Audit

Cheat 5 -5 -25 5

Don’t Cheat 0 0 -1 -1

vs

Knowing how the IRS will respond, you never cheat and they never audit!!

Che

at

Aud

it

Aud

it

Don’t

Cheat

Don’t

Audit

Don’t

Audit

(-25, 5) (5, -5)(-1, -1) (0, 0)

Don’t Audit Audit

Cheat 5 -5 -25 5

Don’t Cheat 0 0 -1 -1

vs

(0%)(0%)

(0%)(100%)

Now, lets switch positions…suppose the IRS chooses first

Aud

it

Che

at

Che

at

Don’t

Audit

Don’t

Cheat

Don’t

Cheat

(-25, 5) (-1, -1)(5, -5) (0, 0)

Don’t Audit Audit

Cheat 5 -5 -25 5

Don’t Cheat 0 0 -1 -1

(0%)(0%)

(100%)(0%)

Again, we could play this game sequentially

$500

$500

$500

$220

$220

$220

(9,000, 9,000) (3,600, 0) (0, 3,600) (1,800, 1,800)

Delta’s reward is on the left

P = $500 P = $220

P = $500 $9,000$9,000

$3,600$0

P = $220 $0$3,600

$1,800$1,800

(0%)(100%)

(0%)(0%)

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? (Terrorists payouts on left)

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)

A bargaining example…How do you divide $20?

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.

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

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

If day 3 arrives, player B should accept any offer – a rejection pays out $0!

Player A: $19.99Player B: $.01

Player B knows what happens in day 3 and wants to avoid that!

Player A: $19.99Player B: $.01

Player A knows what happens in day 2 and knows that player B wants to avoid that!

Player A: $19.99Player B: $.01

Player A

Player B

Offer

Accept Reject

Player B

Offer

Player A

Accept Reject

Player A

Offer

Player B

Accept Reject

(0,0)

Year 1

Year 2

Year 3

Lets consider a variation…

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

Player A

Player B

Offer

Accept Reject

Player B

Offer

Player A

Accept Reject

Player A

Offer

Player B

Accept Reject

(0,0)

Year 1

Year 2

Year 3

If year 3 arrives, player B should accept any offer – a rejection pays out $0!

Player A: $19.99Player B: $.01

If player A rejects, she gets $19.99 in one year. That’s worth $19.99/1.20 today

Player A: $16.65Player B: $3.35

If player B rejects, she gets $3.35 in one year. That’s worth $3.35/1.10 today

Player A: $16.95Player B: $3.05