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