game theory (microeconomic theory (iv)) instructor: yongqin wang email: yongqin_wang@yahoo.com.cn...
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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)
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