advanced microeconomics ii game theory 2016 fallpanlijun/advancedmicro/01.pdfwhat is game theory? a...

L I JUN PAN

GRADUATE SCHOOL OF ECONOMICS

NAGOYA UNIVERS ITY

1

Advanced Microeconomics II

Game Theory

2016 Fall

Introduction

What is game theory?A Motivating Example

Friends - S02, Ep05

To celebrate Monica's promotion, everyone decides to go out and eat at a restaurant. This, however, causes financial issues with, Phoebe, Joey, and Rachel who are not as well off as the rest of their friends (Ross, Monica, and Chandler).

3

Poor Rich

What is game theory?We focus on games where:There are at least two rational playersEach player has more than one choicesThe outcome depends on the strategies chosen by all players;

there is strategic interaction

Example: Five people go to a restaurant.Each person pays his/her own meal – a simple decision

problemBefore the meal, every person agrees to split the bill evenly

among them – a game

4

What is game theory?Game theory is a formal way to analyze strategic interaction among a group of rational players (or agents) who behave strategically

Game theory has applicationsEconomics

Politics

Law

International Relations

Sports

5

What is Game Theory?

“No man is an island” – John Donn

Study of rational behavior

in interactive or interdependent situations

Bad news:

Knowing game theory does not guarantee winning

Good news:

Framework for thinking about strategic interaction

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Strategies for Studying Games of Strategy

Two general approaches◦ Case-based

◦ Pro: Relevance, connection of theory to application

◦ Con: Generality

◦ Theory◦ Pro: General principle is clear

◦ Con: Application to reality may not be feasible

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TerminologyStrategies◦ Choices available to each of the players

Payoffs◦ Some numerical representation of the objectives of each

player◦ Could take account of fairness/reputation, etc.

◦ Does not mean players are narrowly selfish

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Standard AssumptionsRationality◦ Players are perfect calculators and implementers of their

desired strategy

Common knowledge of rules◦ All players know the game being played

Equilibrium◦ Players play strategies that are mutual best responses

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The Use of Game TheoryExplanatory◦ A lens through which to view and learn from past

negotiations/conflicts

Predictive◦ With many caveats

Prescriptive◦ The main thing you’ll take out of the course is an ability to think

strategically

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Classification of Games In this course, we classify the games from the perspective of move ordersand information

Complete informationIncompleteinformation

Simultaneous moveStatic game of

complete informationStatic game of

incomplete information

Sequential movesDynamic game of

complete informationDynamic game of

incomplete information

Lecture 1Static (or Simultaneous-Move) Games of Complete Information

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Outline of Static Games of Complete Information

Normal-form (or strategic-form) representation

Iterated elimination of strictly dominated strategies

Nash equilibrium

Applications of Nash equilibrium

Mixed strategy Nash equilibrium

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Agenda ExamplesPrisoner’s dilemma

The battle of the sexes

Matching pennies

Static (or simultaneous-move) games of complete information

Normal-form or strategic-form representation

Dominated strategies


Nash equilibrium

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Classic Example: Prisoners’ Dilemma

Two suspects held in separate cells are charged with a major crime. However, there is not enough evidence.

Both suspects are told the following policy:If neither confesses then both will be convicted of a minor offense and

sentenced to one month in jail.

If both confess then both will be sentenced to jail for six months.

If one confesses but the other does not, then the confessor will be released but the other will be sentenced to jail for nine months.

-1 , -1 -9 , 0

0 , -9 -6 , -6

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Prisoner 1

Prisoner 2

Confess

Mum

Confess

Mum

Example: The battle of the sexes

At the separate workplaces, Chris and Pat must choose to attend either an opera or a prize fight in the evening.

Both Chris and Pat know the following:Both would like to spend the evening together.

But Chris prefers the opera.

Pat prefers the prize fight.

2 , 1 0 , 0

0 , 0 1 , 2

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Chris

Pat

Prize Fight

Opera

Prize Fight

Opera

Example: Matching pennies

Each of the two players has a penny.

Two players must simultaneously choose whether to show the Head or the Tail.

Both players know the following rules:If two pennies match (both heads or both tails) then player 2 wins

player 1’s penny.

Otherwise, player 1 wins player 2’s penny.

-1 , 1 1 , -1

1 , -1 -1 , 1

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Player 1

Player 2

Tail

Head

Tail

Head

Static (or simultaneous-move) games of complete information

A set of players (at least two players)

For each player, a set of strategies/actions

Payoffs received by each player for the combinations of the strategies, or for each player, preferences over the combinations of the strategies

{Player 1, Player 2, ... Player n}

S1 S2 ... Sn

ui(s1, s2, ...sn), for alls1S1, s2S2, ... snSn.

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A static (or simultaneous-move) game consists of:

Static (or simultaneous-move) games of complete informationSimultaneous-moveEach player chooses his/her strategy without knowledge of

others’ choices.

Complete informationEach player’s strategies and payoff functions are common

knowledge among all the players.

Assumptions on the playersRationalityPlayers aim to maximize their payoffsPlayers are perfect calculatorsEach player knows that other players are rational

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Static (or simultaneous-move) games of complete informationThe players cooperate?No. Only noncooperative games

The timingEach player i chooses his/her strategy si without

knowledge of others’ choices.

Then each player i receives his/her payoff ui(s1, s2, ..., sn).

The game ends.

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Definition: normal-form or strategic-form representation

The normal-form (or strategic-form)representation of a game G specifies:

A finite set of players {1, 2, ..., n},

players’ strategy profiles S1 S2 ... Sn and

their payoff functions u1 u2 ... un

where ui : S1× S2× ...× Sn→R.

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Normal-form representation: 2-player game

Player 2

s21 s22

Player 1

s11 u1(s11,s21), u2(s11,s21) u1(s11,s22), u2(s11,s22)

s12 u1(s12,s21), u2(s12,s21) u1(s12,s22), u2(s12,s22)

s13 u1(s13,s21), u2(s13,s21) u1(s13,s22), u2(s13,s22)

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Bi-matrix representation◦ 2 players: Player 1 and Player 2

◦ Each player has a finite number of strategies

Example:S1={s11, s12, s13} S2={s21, s22}

Classic example: Prisoners’ Dilemma:normal-form representation

Set of players: {Prisoner 1, Prisoner 2}

Sets of strategies: S1 = S2 = {Mum, Confess}

Payoff functions:u1(M, M)=-1, u1(M, C)=-9, u1(C, M)=0, u1(C, C)=-6;u2(M, M)=-1, u2(M, C)=0, u2(C, M)=-9, u2(C, C)=-6

-1 , -1 -9 , 0

0 , -9 -6 , -6

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Prisoner 1

Prisoner 2

Confess

Mum

Confess

Mum

Players

Strategies

Payoffs


Normal (or strategic) form representation:

Set of players: { Chris, Pat } (={Player 1, Player 2})

Sets of strategies: S1 = S2 = { Opera, Prize Fight}

Payoff functions:u1(O, O)=2, u1(O, F)=0, u1(F, O)=0, u1(F, O)=1;u2(O, O)=1, u2(O, F)=0, u2(F, O)=0, u2(F, F)=2

2 , 1 0 , 0

0 , 0 1 , 2

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Chris

Pat

Prize Fight

Opera

Prize Fight

Opera


Normal (or strategic) form representation:

Set of players: {Player 1, Player 2}

Sets of strategies: S1 = S2 = { Head, Tail }

Payoff functions:u1(H, H)=-1, u1(H, T)=1, u1(T, H)=1, u1(H, T)=-1;u2(H, H)=1, u2(H, T)=-1, u2(T, H)=-1, u2(T, T)=1

-1 , 1 1 , -1

1 , -1 -1 , 1

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Player 1

Player 2

Tail

Head

Tail

Head

Example: Cournot model of duopolyA product is produced by only two firms: firm 1 and firm 2. The quantities are denoted by q1 and q2, respectively. Each firm chooses the quantity without knowing the other firm has chosen.

The market price is P(Q)=a-Q, where Q=q1+q2.

The cost to firm i of producing quantity qi is Ci(qi)=cqi.

The normal-form representation:Set of players: { Firm 1, Firm 2}

Sets of strategies: S1=[0, +∞), S2=[0, +∞)

Payoff functions: u1(q1, q2)=q1(a-(q1+q2)-c), u2(q1, q2)=q2(a-(q1+q2)-c)

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Static (or simultaneous-move) games of complete information - Recap

A set of players (at least two players)

For each player, a set of strategies/actions

Payoffs received by each player for the combinations of the strategies, or for each player, preferences over the combinations of the strategies

{Player 1, Player 2, ... Player n}

S1 S2 ... Sn

ui(s1, s2, ...sn), for alls1S1, s2S2, ... snSn.

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A static (or simultaneous-move) game consists of:

Solving Prisoners’ Dilemma

Confess always does better whatever the other player chooses

Dominated strategyThere exists another strategy which always does better

regardless of other players’ choices

-1 , -1 -9 , 0

0 , -9 -6 , -6

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Prisoner 1

Prisoner 2

Confess

Mum

Confess

Mum

Players

Strategies

Payoffs

Definition: strictly dominated strategy

-1 , -1 -9 , 0

0 , -9 -6 , -6

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In the normal-form game {S1 , S2 , ..., Sn , u1 , u2 , ...,

un}, let si', si" Si be feasible strategies for player i.

Strategy si' is strictly dominated by strategy si" if

ui(s1, s2, ... si-1, si', si+1, ..., sn)

< ui(s1, s2, ... si-1, si", si+1, ..., sn)

for all s1 S1, s2 S2, ..., si-1Si-1, si+1 Si+1, ..., sn Sn.

Prisoner 1

Prisoner 2

Confess

Mum

Confess

Mum

regardless of other

players’ choices

si” is strictly

better than si’

Example

Philip

No Ad Ad

ReynoldsNo Ad 60 , 60 30 , 70

Ad 70 , 30 40 , 40

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Two firms, Reynolds and Philip, share some market

Each firm earns $60 million from its customers if neither do advertising

Advertising costs a firm $20 million

Advertising captures $30 million from competitor

2-player game with finite strategies

Player 2

s21 s22

Player 1

s11 u1(s11,s21), u2(s11,s21) u1(s11,s22), u2(s11,s22)

s12 u1(s12,s21), u2(s12,s21) u1(s12,s22), u2(s12,s22)

s13 u1(s13,s21), u2(s13,s21) u1(s13,s22), u2(s13,s22)

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S1={s11, s12, s13} S2={s21, s22}

s11 is strictly dominated by s12 if u1(s11,s21)<u1(s12,s21) and u1(s11,s22)<u1(s12,s22).

s21 is strictly dominated by s22 ifu2(s1i,s21) < u2(s1i,s22), for i = 1, 2, 3

Definition: weakly dominated strategy

1 , 1 2 , 0

0 , 2 2 , 2

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In the normal-form game {S1 , S2 , ..., Sn , u1 , u2 , ...,

un}, let si', si" Si be feasible strategies for player i.

Strategy si' is weakly dominated by strategy si" if

ui(s1, s2, ... si-1, si', si+1, ..., sn)

(but not always =) ui(s1, s2, ... si-1, si", si+1, ..., sn)

for all s1 S1, s2 S2, ..., si-1Si-1, si+1 Si+1, ..., sn Sn.

Player 1

Player 2

R

U

B

L

regardless of other

players’ choices

si” is at

least as

good

as si’

Strictly and weakly dominated strategy

A rational player never chooses a strictly dominated strategy. Hence, any strictly dominated strategy can be eliminated.

A rational player may choose a weakly dominated strategy.

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Approach

If a strategy is strictly dominated, eliminate it

The size and complexity of the game is reduced

Eliminate any strictly dominated strategies from the reduced game

Continue doing so successively

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Implicit assumptions

A rational player will never play a strictly dominated strategy

Common knowledge of rationality:The structure of the game and the rationality of the players

are common knowledge among the players

All the players know that each player will never play a strictly dominated strategy => they can effectively ignore those strictly dominated strategies that opponents will never play

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Iterated elimination of strictly dominated strategies: an example

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1 , 0 1 , 2 0 , 1

0 , 3 0 , 1 2 , 0Player 1

Player 2

Middle

Up

Down

Left

1 , 0 1 , 2

0 , 3 0 , 1Player 1

Player 2

Middle

Up

Down

Left

Right

Example: Tourists & Natives

Only two bars (bar 1, bar 2) in a city

Can charge price of $2, $4, or $5

6000 tourists pick a bar randomly

4000 natives select the lowest price bar

Example 1: Both charge $2each gets 5,000 customers and $10,000

Example 2: Bar 1 charges $4, Bar 2 charges $5Bar 1 gets 3000+4000=7,000 customers and $28,000Bar 2 gets 3000 customers and $15,000

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Example: Tourists & Natives

Bar 2

$4 $5

Bar 1$4 20 , 20 28 , 15

$5 15 , 28 25 , 25

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Bar 2

$2 $4 $5

Bar 1

$2 10 , 10 14 , 12 14 , 15

$4 12 , 14 20 , 20 28 , 15

$5 15 , 14 15 , 28 25 , 25

Payoffs are in thousands of dollars

New solution concept: Nash equilibrium

Player 2

L C R

Player 1

T 0 , 4 4 , 0 5 , 3

M 4 , 0 0 , 4 5 , 3

B 3 , 5 3 , 5 6 , 6

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The combination of strategies (B, R) has the

following property:

Player 1 CANNOT do better by choosing a strategy

different from B, given that player 2 chooses R.

Player 2 CANNOT do better by choosing a strategy

different from R, given that player 1 chooses B.

Nash Equilibrium: idea

Nash equilibriumA set of strategies, one for each player, such that each

player’s strategy is best for her, given that all other players are playing their equilibrium strategies

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John NashJohn Nash (June 13, 1928 – May 23, 2015)

A mathematician and an economist

He developed several theories that were relevant in understanding economic interactions

His important contribution was the famous Nash equilibrium

He won the Nobel Memorial Prize in Economic Sciences in 1994

He suffered from mental illness from 1959, and his recovery became the basis for his biography, A Beautiful Mind

He died of a car accident with his wife on May 23, 2015

John NashA Beautiful Mind: a 2001 American biographical drama film based on the life of John Nash

John NashJohn Nash and his wife

In the movie

Definition: Nash Equilibrium

Prisoner 2

Mum Confess

Prisoner

1

Mum -1 , -1 -9 , 0

Confess 0 , -9 -6 , -6

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In the normal-form game {S1 , S2 , ..., Sn , u1 , u2 , ..., un}, a combination

of strategies ),...,( **1 nss is a Nash equilibrium if, for every player i,

),...,,,,...,(

),...,,,,...,(

**1

*1

*1

**1

**1

*1

niiii

niiii

sssssu

sssssu

for all ii Ss . That is, *is solves

Maximize ),...,,,,...,( **1

*1

*1 niiii sssssu

Subject to ii Ss

Given others’

choices, player i

cannot be better-

off if she deviates

from si*

2-player game with finite strategies

Player 2

s21 s22

Player 1

s11 u1(s11,s21), u2(s11,s21) u1(s11,s22), u2(s11,s22)

s12 u1(s12,s21), u2(s12,s21) u1(s12,s22), u2(s12,s22)

s13 u1(s13,s21), u2(s13,s21) u1(s13,s22), u2(s13,s22)

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S1={s11, s12, s13} S2={s21, s22}

(s11, s21)is a Nash equilibrium if u1(s11,s21) u1(s12,s21), u1(s11,s21) u1(s13,s21) andu2(s11,s21) u2(s11,s22).

Finding a Nash equilibrium: cell-by-cell inspection

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1 , 0 1 , 2 0 , 1

0 , 3 0 , 1 2 , 0Player 1

Player 2

Middle

Up

Down

Left

1 , 0 1 , 2

0 , 3 0 , 1Player 1

Player 2

Middle

Up

Down

Left

Right

Using best response function to find Nash equilibrium: example

Player 2

L’ C’ R’

Player 1

T’ 0 , 4 4 , 0 3 , 3

M’ 4 , 0 0 , 4 3 , 3

B’ 3 , 3 3 , 3 3.5 , 3.6

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M’ is Player 1’s best response to Player 2’s strategy L’

T’ is Player 1’s best response to Player 2’s strategy C’

B’ is Player 1’s best response to Player 2’s strategy R’

L’ is Player 2’s best response to Player 1’s strategy T’

C’ is Player 2’s best response to Player 1’s strategy M’

R’ is Player 2’s best response to Player 1’s strategy B’

Excercise: Tourists & Natives

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Bar 2

$2 $4 $5

Bar 1

$2 10 , 10 14 , 12 14 , 15

$4 12 , 14 20 , 20 28 , 15

$5 15 , 14 15 , 28 25 , 25

Payoffs are in thousands of dollars

Use best response function to find the Nash

equilibrium.


Opera is Player 1’s best response to Player 2’s strategy Opera

Opera is Player 2’s best response to Player 1’s strategy OperaHence, (Opera, Opera) is a Nash equilibrium

Fight is Player 1’s best response to Player 2’s strategy Fight

Fight is Player 2’s best response to Player 1’s strategy FightHence, (Fight, Fight) is a Nash equilibrium

2 , 1 0 , 0

0 , 0 1 , 2

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Chris

PatPrize Fight

Opera

Prize Fight

Opera


Head is Player 1’s best response to Player 2’s strategy Tail

Tail is Player 2’s best response to Player 1’s strategy Tail

Tail is Player 1’s best response to Player 2’s strategy Head

Head is Player 2’s best response to Player 1’s strategy Head

Hence, NO Nash equilibrium

-1 , 1 1 , -1

1 , -1 -1 , 1

50

Player 1

Player 2

Tail

Head

Tail

Head

Definition: best response function

51

In the normal-form game

{S1 , S2 , ..., Sn , u1 , u2 , ..., un},

if player 1, 2, ..., i-1, i+1, ..., n choose strategies

nii ssss ,...,,,..., 111 , respectively,

then player i's best response function is defined by

} allfor ),,...,,,,...,(

),...,,,,...,(:{

) ,...,,,..., (

111

111

111

iiniiii

niiiiii

niii

Sssssssu

sssssuSs

ssssB

Player i’s best

response

Given the

strategies

chosen by

other players

Definition: best response function

52

An alternative definition:

Player i's strategy ),...,,...,( 111 niiii ssssBs if and only if

it solves (or it is an optimal solution to)

Maximize ),...,,,,...,( 111 niiii sssssu

Subject to ii Ss

where nii ssss ..., , , ..., , 111 are given.

Player i’s best response to other players’ strategies is an optimal solution to

Using best response function to define Nash equilibriumIn the normal-form game {S1, ..., Sn , u1, ..., un},

a combination of strategies ),...,( **1 nss is a Nash

equilibrium if for every player i,

) ..., , , ,..., ( **1

*1

*1

*niiii ssssBs

53

A set of strategies, one for each player, such that each player’s strategy is best for her, given that all other players are playing their strategies, or

A stable situation that no player would like to deviate if others stick to it

Cournot model of duopolyA product is produced by only two firms: firm 1 and firm 2. The quantities are denoted by q1 and q2, respectively. Each firm chooses the quantity without knowing the other firm has chosen.

The market priced is P(Q)=a-Q, where a is a constant number and Q=q1+q2.


54

Cournot model of duopolyThe normal-form representation:Set of players: { Firm 1, Firm 2}


Payoff functions: u1(q1, q2)=q1(a-(q1+q2)-c)u2(q1, q2)=q2(a-(q1+q2)-c)

55

Cournot model of duopolyHow to find a Nash equilibriumFind the quantity pair (q1*, q2*) such that q1* is firm 1’s best response

to Firm 2’s quantity q2* and q2* is firm 2’s best response to Firm 1’s quantity q1*

That is, q1* solves Max u1(q1, q2*)=q1(a-(q1+q2*)-c)subject to 0 q1 +∞

and q2* solvesMax u2(q1*, q2)=q2(a-(q1*+q2)-c)subject to 0 q2 +∞

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Using best response function to find Nash equilibrium

In a 2-player game, ( s1, s2 ) is a Nash equilibrium if and only if player 1’s strategy s1 is her best response to player 2’s strategy s2, and player 2’s strategy s2 is her best response to player 1’s strategy s1.

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-1 , -1 -9 , 0

0 , -9 -6 , -6Prisoner 1

Prisoner 2

Confess

Mum

Confess

Mum

Cournot model of duopolyHow to find a Nash equilibriumFind the quantity pair (q1*, q2*) such that q1* is firm 1’s best response

to Firm 2’s quantity q2* and q2* is firm 2’s best response to Firm 1’s quantity q1*

That is, q1* solves Max u1(q1, q2*)=q1(a-(q1+q2*)-c)subject to 0 q1 +∞

and q2* solvesMax u2(q1*, q2)=q2(a-(q1*+q2)-c)subject to 0 q2 +∞

58

Cournot model of duopolyHow to find a Nash equilibriumSolve

Max u1(q1, q2*)=q1(a-(q1+q2*)-c)subject to 0 q1 +∞

FOC: a - 2q1 - q2*- c = 0q1 = (a - q2*- c)/2

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Cournot model of duopolyHow to find a Nash equilibriumSolve

Max u2(q1*, q2)=q2(a-(q1*+q2)-c)subject to 0 q2 +∞

FOC: a - 2q2 – q1* – c = 0q2 = (a – q1* – c)/2

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Cournot model of duopolyHow to find a Nash equilibriumThe quantity pair (q1*, q2*) is a Nash equilibrium if

q1* = (a – q2* – c)/2q2* = (a – q1* – c)/2

Solving these two equations gives usq1* = q2* = (a – c)/3

61

Cournot model of duopolyBest response functionFirm 1’s best function to firm 2’s quantity q2:

R1(q2) = (a – q2 – c)/2 if q2 < a– c; 0, otherwise

Firm 2’s best function to firm 1’s quantity q1:R2(q1) = (a – q1 – c)/2 if q1 < a– c; 0, otherwise

62

q1

q2

(a – c)/2

(a – c)/2

a – c

a – c

Nash

equilibrium

Cournot model of oligopolyA product is produced by only n firms: firm 1 to firm n. Firm i’s quantity is denoted by qi. Each firm chooses the quantity without knowing the other firms’ choices.

The market priced is P(Q)=a-Q, where a is a constant number and Q=q1+q2+...+qn.


63

Cournot model of oligopolyThe normal-form representation:Set of players: { Firm 1, ... Firm n}

Sets of strategies: Si=[0, +∞), for i=1, 2, ..., n

Payoff functions: ui(q1 ,..., qn)=qi(a-(q1+q2 +...+qn)-c)for i=1, 2, ..., n

64

Cournot model of oligopolyHow to find a Nash equilibriumFind the quantities (q1*, ... qn*) such that qi* is firm i’s best

response to other firms’ quantities

That is, q1* solves Max u1(q1, q2*, ..., qn*)=q1(a-(q1+q2* +...+qn*)-c)subject to 0 q1 +∞

and q2* solvesMax u2(q1*, q2 , q3*, ..., qn*)=q2(a-(q1*+q2+q3*+ ...+ qn*)-c)subject to 0 q2 +∞

.......

65

Bertrand model of duopoly (differentiated products)Two firms: firm 1 and firm 2.

Each firm chooses the price for its product without knowing the other firm has chosen. The prices are denoted by p1 and p2, respectively.

The quantity that consumers demand from firm 1:

q1(p1, p2) = a – p1 + bp2.

The quantity that consumers demand from firm 2:

q2(p1, p2) = a – p2 + bp1.


66

Bertrand model of duopoly (differentiated products)The normal-form representation:

Set of players: { Firm 1, Firm 2}


Payoff functions: u1(p1, p2)=(a – p1 + bp2 )(p1 – c)u2(p1, p2)=(a – p2 + bp1 )(p2 – c)

67

Bertrand model of duopoly (differentiated products)How to find a Nash equilibriumFind the price pair (p1*, p2*) such that p1* is firm 1’s best

response to Firm 2’s price p2* and p2* is firm 2’s best response to Firm 1’s price p1*

That is, p1* solves Max u1(p1, p2*) = (a – p1 + bp2* )(p1 – c)subject to 0 p1 +∞

and p2* solvesMax u2(p1*, p2) = (a – p2 + bp1* )(p2 – c)subject to 0 p2 +∞

68

Bertrand model of duopoly (differentiated products)How to find a Nash equilibriumSolve firm 1’s maximization problem

Max u1(p1, p2*) = (a – p1 + bp2* )(p1 – c)subject to 0 p1 +∞

FOC: a + c – 2p1 + bp2* = 0p1 = (a + c + bp2*)/2

69

Bertrand model of duopoly (differentiated products)How to find a Nash equilibriumSolve firm 2’s maximization problem

Max u2(p1*, p2)=(a – p2 + bp1* )(p2 – c)subject to 0 p2 +∞

FOC: a + c – 2p2 + bp1* = 0p2 = (a + c + bp1*)/2

70

Bertrand model of duopoly (differentiated products)How to find a Nash equilibriumThe price pair (p1*, p2*) is a Nash equilibrium if

p1* = (a + c + bp2*)/2p2* = (a + c + bp1*)/2

Solving these two equations gives usp1* = p2* = (a + c)/(2 –b)

71

The problems of commonsn farmers in a village. Each summer, all the farmers graze their goats on the village green.

Let gi denote the number of goats owned by farmer i.

The cost of buying and caring for a goat is c, independent of how many goats a farmer owns.

The value of a goat is v(G) per goat, where G = g1 + g2 + ... + gn

There is a maximum number of goats that can be grazed on the green. That is,v(G)>0 if G < Gmax, and v(G)=0 if G Gmax.

Assumptions on v(G): v’(G) < 0 and v”(G) < 0.

Each spring, all the farmers simultaneously choose how many goats to own.

72

The problems of commonsThe normal-form representation:Set of players: { Farmer 1, ... Farmer n}

Sets of strategies: Si=[0, Gmax), for i=1, 2,..., n

Payoff functions: ui(g1, ..., gn)=gi v(g1 + ...+ gn) – c gi

for i = 1, 2, ..., n.

73

The problems of commonsHow to find a Nash equilibriumFind (g1*, g2*, ..., gn*) such that gi* is farmer i’s best response to

other farmers’ choices.

That is, g1* solves Max u1(g1, g2*, ..., gn*)= g1 v(g1 + g2* ...+ gn*) – c g1

subject to 0 g1 < Gmax

and g2* solvesMax u2(g1*, g2 , g3*, ..., gn*)= g2v(g1*+g2+g3*+ ...+ gn*)–cg2

subject to 0 g2 < Gmax

.......

74

The problems of commonsHow to find a Nash equilibrium

and gn* solvesMax un(g1*, ..., gn-1*, gn)= gnv(g1*+...+ gn-1*+ gn)–cgn

subject to 0 gn < Gmax

.......

75

The problems of commons

FOCs:

0)*...*()*...*(

.........

0*)...**(*)...**(

0*)...*(*)...*(

1111

3212321

21121

cgggvggggv

cggggvgggggv

cgggvggggv

nnnnn

nn

nn

76


How to find a Nash equilibrium(g1*, g2*, ..., gn*) is a Nash equilibrium if

0*)*...*(*)*...*(

.........

0*)...***(*)...***(

0*)...**(*)...**(

1111

3212321

21121

cgggvggggv

cggggvgggggv

cgggvggggv

nnnnn

nn

nn

77


Summing over all n farmers’ FOCs and then dividing by nyields

*...*** where

0*)(*1

*)(

21 ngggG

cGvGn

Gv

78


The social problem

0*)*(***)*(

satisfies **solution optimal theHence,

0)()(

:FOC

0 s.t.

)(Max

max

cGvGGv

G

cGvGGv

GG

GcGGv

79


?***

0*)*(***)*(

0*)(*1

*)(

GG

cGvGGv

cGvGn

Gv

80

SummaryNash Equilibrium

Best Response Function

Cournot models of duopoly and oligopoly

Bertrand model of duopoly


Next time◦ Mixed strategy Nash equilibrium

81

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