Game Theory (Microeconomic Theory (IV))

Instructor: Yongqin Wang

Email: yongqin_wang@yahoo.com.cn

School of Economics and CCES, Fudan University

December, 2004

1.Static Game of Complete Information

1.3 Further Discussion on Nash Equilibrium (NE) 1.3.1 NE versus Iterated Elimination of Strict

Dominance Strategies

Proposition A In the -player normal form game

if iterated elimination of strictly dominated strategies

eliminates all but the strategies , then

these strategies are the unique NE of the game.

1 1{ ,..., ; ,..., }n nG S S u un

* *1( ,..., )ns s

A Formal Definition of NE

In the n-player normal form

the strategies are a NE, if for each player

i,

is (at least tied for) player i’s best response to the

strategies

specified for the n-1 other players,

1 1{ ,..., ; ,..., }n nG S S u u* *1( ,..., )ns s

*is

* * * * * * * * *1 1 1 1 1 1( ,..., , , ,..., ) ( ,..., , , ,..., )i i i n i i i i ns s s s s u s s s s s

Cont’d

Proposition B In the -player normal form game

if the strategies are a NE, then they

survive iterated elimination of strictly dominated

strategies.

1 1{ ,..., ; ,..., }n nG S S u u

* *1( ,..., )ns s

n

1.3.2 Existence of NE

Theorem (Nash, 1950): In the -player normal form game

if is finite and is finite for every , then there exist at least one NE, possibly involving mixed strategies.

See Fudenberg and Tirole (1991) for a rigorous proof.

n

1 1{ ,..., ; ,..., }n nG S S u u

n iS i

1.4 Applications 1.4.1 Cournot Model

Two firms A and B quantity compete.

Inverse demand function

They have the same constant marginal cost, and

there is no fixed cost.

, 0P a Q a

Cont’d

Firm A’s problem:

2

2

( )

2 0

2

2 0

A A A A B A A

AA B

A

BA

A

A

Pq cq a q q q cq

da q q c

dq

a q cq

d

dq

Cont’d

By symmetry, firm B’s problem.

Figure Illustration: Response Function, Tatonnement Process

Exercise: what will happens if there are n identical Cournot competing firms? (Convergence to Competitive Equilibrium)

1.4.2 The problem of Commons

David Hume (1739): if people respond only to private

incentives, public goods will be underprovided and

public resources over-utilized.

Hardin(1968) : The Tragedy of Commons

Cont’d

There are farmers in a village. They all graze their

goat on the village green. Denote the number of goats

the farmer owns by , and the total number of

goats in the village by

Buying and caring each goat cost and value to a farmer

of grazing each goat is .

nthi

ig

1 ... nG g g

c( )v G

Cont’d

A maximum number of goats : ,

for but for

Also

The villagers’ problem is simultaneously choosing how

many goats to own (to choose ).

max : ( ) 0G v G

maxG G ( ) 0v G maxG G'( ) 0, ''( ) 0v G v G

ig

Cont’d

His payoff is

(1)

In NE , for each , must maximize (1),

given that other farmers choose

1 1 1( ... ... )i i i i n ig v g g g g g cg * *1( ,..., )ng g *

igi

* * * *1 1 1( ,..., , , )i i ng g g g

Cont’d

First order condition (FOC):

(2)

(where )

Summing up all farmers’ FOC and then dividing by yields

(3)

* *( ) '( ) 0i i i i iv g g g v g g c

* * * * *1 1 1... ...i i i ng g g g g

n n*1

( *) '( *) 0v G G v G cn

Cont’d

In contrast, the social optimum should resolve

FOC: (4)

Comparing (3) and (4), we can see that

Implications for social and economic systems (Coase Theorem)

**G

max ( )Gv G Gc

( **) ** '( **) 0v G G v G c

* **G G

2. Dynamic Games of Complete Information

2.1 Dynamic Games of Complete and Perfect Information

2.1.A Theory: Backward Induction

Example: The Trust Game

General features:

(1) Player 1 chooses an action from the feasible set

.

(2) Player 2 observes and then chooses an action

from the feasible set .

(3) Payoffs are and .

1a 1A

1a 2a2A

1 1 2( , )u a a 2 1 2( , )u a a

Cont’d

Backward Induction:

Then

“People think backwards”

2 2 1 2arg max ( , )a u a a

1 1 1 2 1arg max ( , ( ))a u a R a

2.1.B An example: Stackelberg Model of Duopoly

Two firms quantity compete sequentially.

Timing: (1) Firm 1 chooses a quantity ;

(2) Firm 2 observes and then chooses a quantity

；

(3) The payoff to firm is given by the profit function

is the inverse demand function, ,

and is the constant marginal cost of production (fixed

cost being zero).

1 0q

1q 2 0q

( , ) [ ( ) ]i i j iq q q P Q c

i

( )P Q a Q 1 2Q q q c

Cont’d

We solve this game with backward induction

(provided that ).

2 2 1 2 2 1 2

* 12 2 1

arg max ( , ) ( )

( )2

q q q q a q q c

a q cq R q

1q a c

Cont’d

Now, firm 1’s problem

so, .

1 1 1 2 1 1 1 2 1

*1

arg max ( , ( )) [ ( ) ]

2

q q R q q a q R q c

a cq

*2 4

a cq

Cont’d

Compare with the Cournot model.

Having more information may be a bad thing

Exercise: Extend the analysis to firm case.n

2.2 Two stage games of complete but imperfect information2.2.A Theory: Sub-Game Perfection

Here the information set is not a singleton.

Consider following games (1)Players 1 and 2 simultaneously choose actions

and from feasible sets and , respectively. (2) Players 3 and 4 observe the outcome of the first

stage ( , ) and then simultaneously choose actions and from feasible sets and , respectively.

(3) Payoffs are ,

2a1a

1A 2A

1a 2a3A 4A

1 2 3 4 ( , , , )iu a a a a 1,2,3,4i

An approach similar to Backward Induction

1 and 2 anticipate the second behavior of 3 and 4 will be given by

then the first stage interaction between 1 and 2 amounts to the following simultaneous-move game:

(1)Players 1 and 2 simultaneously choose actions and from feasible sets and respectively.

(2) Payoffs are Sub-game perfect Nash Equilibrium is

1a

* *3 1 2 4 1 2( ( , ), ( , ))a a a a a a

2a

* *1 2 3 1 2 4 1 2( , , ( , ), ( , ))iu a a a a a a a a

* * * *1 2 3 4( , , , )a a a a

1A 2A

2.2B An Example: Banks Runs

Two depositors: each deposits D in a bank, which invest these

deposits in a long-term project.

Early liquidation before the project matures, 2r can be

recovered, where D>r>D/2. If the bank allows the investment

to reach maturity, the project will pay out a total of 2R, where

R>D.

Assume there is no discounting.

Insert Matrixes

Interpretation of The model, good versus bad equilibrium.

Cont’d

Date 1

Date 2

r, r D,2r-D

2r-D, D Next stage

R, R 2R-D, D

D, 2R-D R, R

Cont’d

In Equilibrium

Interpretation of the Model and the Role of law and other institutions

r, r D, 2r-D

2r-D, D R, R

2.3 Repeated Game

2.3A Theory: Two-Stage Repeated Game

Repeated Prisoners’ Dilemma

Stage Game

1,1 5,0

0,5 4,4

2,2 6,1

1,6 5,5

Cont’d

Definition Given a stage game G, let the finitely

repeated game in which G is played T times, with the

outcomes of all preceding plays observed before the

next play begins. The payoff for G(T) are simply the sum

of the payoffs from the stage games.

Proposition If the stage game G has a unique NE, then

for any finite T , the repeated game G(T) has a unique

sub-game perfect outcome: the Nash equilibrium of G is

played in every stage. (The paradox of backward induction)

Some Ways out of the Paradox

Bounded Rationality (Trembles may

matter)

Multiple Nash Equilibrium( An Two-

Period Example)

Uncertainty about other players

Uncertainty about the futures

2.3B Theory: Infinitely Repeated Games

Definition 1 Given the discount factor , the present value of the infinitely repeated sequence of payoffs is

Interpretation of the discount factor.

Definition 2 (Selten, 1965) A Nash Equilibrium is subgame perfect if the players’ strategies constitute a Nash equilibrium

in every subgame.

1 2 3, , ,...

2 11 2 3

1

... tt

t

Existence of SPE

Theorem: Every finite extensive game has a SPE.

Comments: Compare with NE.

Cont’d

Definition3: Given the discounted factor , the average payoff of the infinite sequence of payoffs is

1 2 3, , ,...

1

1

(1 ) tt

t

Cont’d

The Folk Theorm: For every feasible payoff vector v with i iv v for

all players i, there exists a 1 such that for all (,1) there exist a

Nash Equilibrium with payoff v.

(See Fudenberg and Tirole (1991) for a rigorous proof.)

Social Norms versus Laws (Kaushik Basu, 2001): The Core Theorem

Implications of Repeated Games

Reputation-building Collusion Social mobility and social capital Organization theory (Kreps) Exit and Voice

2.4 Dynamic Games with Complete but Imperfect Information

At least some information set is not a

singleton

Sub-game Perfection

Static (or Simultaneous-Move) Games of Incomplete Information

Introduction to Static Bayesian Games

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

All these are common knowledge among all the players.

Static (or simultaneous-move) games of INCOMPLETE information

Payoffs are no longer common knowledge

Incomplete information means that At least one player is uncertain about

some other player’s payoff function.

Static games of incomplete information are also called static Bayesian games

Cournot duopoly model of complete information

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)

All these information is common knowledge

Cont’d

A homogeneous product is produced by only two firms: firm 1 and firm 2. The quantities are denoted by q1 and q2, respectively.

They choose their quantities simultaneously.

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

Firm 1’s cost function: C1(q1)=cq1. All the above are common

knowledge

Cont’d

Firm 2’s marginal cost depends on some factor (e.g. technology) that only firm 2 knows. Its marginal cost can be HIGH: cost function: C2(q2)=cHq2. LOW: cost function: C2(q2)=cLq2.

Before production, firm 2 can observe the factor and know exactly which level of marginal cost is in.

However, firm 1 cannot know exactly firm 2’s cost. Equivalently, it is uncertain about firm 2’s payoff.

Firm 1 believes that firm 2’s cost function is C2(q2)=cHq2 with probability , and C2(q2)=cLq2 with probability 1–.

All the above are common knowledge

Cont’d

A solution for the Cournot duopoly model of incomplete information

Firm 2 knows exactly its marginal cost is high or low.

If its marginal cost is high, i.e. 222 )( qcqC H , then, for any given 1q , it will solve

0 ..

])([

2

212

qts

cqqaqMax H

FOC: )(21

)( 02 1221 HHH cqacqcqqa

)(2 Hcq is firm 2's best response to 1q , if its marginal cost is high.

Cont’d

Firm 2 knows exactly its marginal cost is high or low.

If its marginal cost is low, i.e. 222 )( qcqC L , then, for any given 1q , it will solve

0 ..

])([

2

212

qts

cqqaqMax L

FOC: )(21

)( 02 1221 LLL cqacqcqqa

)(2 Lcq is firm 2's best response to 1q , if its marginal cost is low.

Cont’d

Firm 1 knows exactly its cost function 111 )( cqqC . Firm 1 does not know exactly firm 2's marginal cost is high

or low. But it believes that firm 2's cost function is 222 )( qcqC H

with probability , and 222 )( qcqC L with probability 1 Equivalently, it knows that the probability that firm 2's

quantity is )(2 Hcq is , and the probability that firm 2's quantity is )(2 Lcq is 1 . So it solves

0 ..

]))(([)1(

]))(([

1

211

211

qts

ccqqaq

ccqqaqMax

L

H

Cont’d

Firm 1's problem:

0 ..

]))(([)1(

]))(([

1

211

211

qts

ccqqaq

ccqqaqMax

L

H

FOC:

0])(2[)1(])(2[ 2121 ccqqaccqqa LH

Hence, 2

])([)1(])([ 221

ccqaccqaq LH

1q is firm 1's best response to the belief that firm 2 chooses )(2 Hcq with probability , and )(2 Lcq with probability 1

Cont’d

Now we have

)(21

)( 12 HH cqacq

)(21

)( 12 LL cqacq

2])([)1(])([ 22

1ccqaccqa

q LH

We have three equations and three unknowns. Solving these gives us

Cont’d

)(6

1)2(

31

)(*2 LHHH ccccacq

)(6

)2(31

)(*2 LHLL ccccacq

3)1(2*

1LH ccca

q

Firm 1 chooses *1q

Firm 2 chooses )(*2 Hcq if its marginal cost is high, or )(*

2 Lcq if its marginal cost is low.

This can be written as ( *1q , ( )(*

2 Hcq , )(*2 Lcq ))

One is the best response to the other

A Nash equilibrium solution called Bayesian Nash equilibrium.

3. Static Games of Incomplete Information3.1 Theory: Static Bayesian Games and Bayesian NE

3.1.A: An Example: Cournot Competition under Asymmetric Information

The basic Set-up:

1 2

( )P Q a Q

Q q q

Cont’d

Introduction of asymmetric information:

with probability

and with probability

1 1 1( )C q cq1 1 1( )C q cq

2 2 2( ) HC q c q

2 2 2( ) LC q c q (1- )

Cont’d

Firm 2 knows its cost functions and firm 1’s, but firm 1 only knows its own function and that firm 2’s Marginal cost is with Probability ,and with probability

All of this is common knowledge.

Hc Lc(1 )L

Cont’d

* *2 1 2 2

* *2 1 2 2

( ) arg max ( )

( ) arg max ( )

H H

L L

q c a q q c q

q c a q q c q

* * *1 1 2 1 1 2 1arg max ( ( ) (1 ) ( )H Lq a q q c c q a q q c c q

Cont’d

The FOC:*

* 12

** 12

( )2

( )2

HH

LL

a q cq c

a q cq c

* *2 2*

1

( ) (1 ) ( )

2H La q c c a q c c

q

cont’d

Solutions:

Comparison with the Complete-Information version

*2

*2

*1

2 1( ) ( )

3 62

( ) ( )3 6

2 (1 )

3

HH H L

LL H L

H L

a c cq c c c

a c cq c c c

a c c cq

3.1.B Normal Form Representation of Static Bayesian Games

Definition: The normal form representation of an n-playe

r static Bayesian game specifies the player’s action s

paces , their type space , their beliefs

, and their payoff functions . Player ’s typ

e , privately known by player , determines player

’s payoff function , and is a member of th

e set of possible types .

1,..., nA A1,..., nT T

1,..., np p 1,..., nu u

iit i

1( ,..., ; )i n iu a a ti

iT

Cont’d

Player ’s belief describes ’s unc

ertainty about the other players’ poss

ible types , given ’s own type . We den

ote this game by

( | )i i ip t ti1n

i

it iti

1 1 1 1,..., ; ,..., ; ,..., ; ,...,n n n nG A A T T p p u u

cont’d

Harsanyi Transformation

Time of a static Bayesian game Nature draws a type vector , ; Nature reveals to player , but not to any other player; The players simultaneously choose actions, player choosing

;

Payoffs are received.

Some remarks

1( ,..., )nt t ti it T

iti

i ia A

i

1( ,..., ; )i n iu a a t

3.1C Definition of Bayesian Nash Equilibrium (BNE)

Definition 1 In the game of static Bayesian game

, a strategy for player

is a function , where for each type in ,

specifies the action from the feasible set that

type would choose if drawn by nature.

1 1 1 1,..., ; ,..., ; ,..., ; ,...,n n n nG A A T T p p u u

( )i is tiit iT ( )i is t

iA it

3.2A Mixed Strategies Revisited

Battle of Sexes (Chris and Pat)

2+ct,1 0,0

0,0 1,2+Pt

The battle of sexes with incomplete information

2,1 0,0 0,0 6 1,2

Cont’d

C hris’s expected payoffs from playing O pera

and from P laying F ight are respectively

2 1 0 2

0 1 1 1

c c

p p pt t

x x x

p p p

x x x

Cont’d

Playing opera is optimal if and only if

3c

xt c

p

(1)

Cont’d

Sim ilarly, Pat’s expected payoffs from playing Flight

and from playing O pera are

1 0 2 2

1 1 0 0 1

p p

c c ct t

x x x

c c c c

x x x x

Cont’d

Thus playing Fight is optimal if and only if

3p

xt p

c

(2)

Cont’d

S o l v i n g ( 1 ) a n d ( 2 ) y i e l d s

2 3 0

p c

p p x

3 9 4 2

12 3

x c x p x

x x x

a s 0x

First-price sealed-bid auction

A single good is for sale.

Two bidders, 1 and 2, simultaneously submit their bids.

Let 1b denote bidder 1's bid and 2b denote bidder 2's bid

The higher bidder wins the good and pays the price she bids

The other bidder gets and pays nothing

In case of a tie, the winner is determined by a flip of a coin

Bidder i has a valuation ]1 ,0[iv for the good. 1v and 2v are independent.

Bidder 1 and 2's payoff functions:

12

1222

1222

2212

21

2111

2111

1211

if0

if2

if

);,(

if0

if2

if

);,(

bb

bbbv

bbbv

vbbu

bb

bbbv

bbbv

vbbu

cont’d

Normal form representation:

Two bidders, 1 and 2

Action sets (bid sets): ) ,0[1 A , ) ,0[2 A

Type sets (valuations sets): ]1 ,0[1 T , ]1 ,0[2 T

Beliefs: Bidder 1 believes that 2v is uniformly distributed on ]1 ,0[ . Bidder 2 believes that 1v is uniformly distributed on ]1 ,0[ . 1v and 2v are independent.

Bidder 1 and 2's payoff functions:

12

1222

1222

2212

21

2111

2111

1211

if0

if2

if

);,(

if0

if2

if

);,(

bb

bbbv

bbbv

vbbu

bb

bbbv

bbbv

vbbu

Cont’d

A strategy for bidder 1 is a function )( 11 vb , for all ]1 ,0[1 v .

A strategy for bidder 2 is a function )( 22 vb , for all ]1 ,0[2 v .

Given bidder 1's belief on bidder 2, for each ]1 ,0[1 v , bidder 1 solves

)}({Prob)(21

)}({Prob)( 221112211101

vbbbvvbbbvMaxb

Given bidder 2's belief on bidder 1, for each ]1 ,0[2 v , bidder 2 solves

)}({Prob)(21

)}({Prob)( 112221122202

vbbbvvbbbvMaxb

cont’d

Check whether

2)( ,

2)( 2

2*2

11

*1

vvb

vvb is Bayesian Nash equilibrium.

Given bidder 1's belief on bidder 2, for each ]1 ,0[1 v , bidder 1's best

response to )( 2*2 vb solves

)}({Prob)(21

)}({Prob)( 2*21112

*2111

01vbbbvvbbbvMax

b

}2

{Prob)(21

}2

{Prob)( 2111

2111

01

vbbv

vbbvMax

b

}2{Prob)(21

}2{Prob)( 1211121101

bvbvbvbvMaxb

11101

2)( bbvMaxb

FOC: 042 11 bv 2

)( 111

vvb

Cont’d

H e n c e , f o r e a c h ]1 ,0[1 v , 2

)( 11

*1

vvb i s b i d d e r 1 ' s b e s t r e s p o n s e t o b i d d e r

2 ' s 2

)( 22

*2

vvb .

B y s y m m e t r y , f o r e a c h ]1 ,0[2 v , 2

)( 22

*2

vvb i s b i d d e r 2 ' s b e s t r e s p o n s e

t o b i d d e r 1 ' s 2

)( 11

*1

vvb .

T h e r e f o r e ,

2)( ,

2)( 2

2*2

11

*1

vvb

vvb i s B a y e s i a n N a s h e q u i l i b r i u m .

( T h i n k o v e r : w h a t w o u l d b e t h e B N E i n s e c o n d - p r i c e a u c t i o n ? )

Relation with Information Economics

Bayesian Game and Mechanism

Design(Adverse Selection)

Dynamic Bayesian Games and Signaling

Moral Hazard

Dynamics

Equilibrium Concepts in Game Theory

NE, SPE, BNE, PBNE Embarrassment of richness(merits

and demerits) and Refinements Evolutionary Game Theory Behavior Game Theory

Concluding Remarks

Taking Stock Further Reading Gibbons (1992)