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© 2008 Institute of Information Management National Chiao Tung University
Game theory
• The study of multiperson decisions• Four types of games
• Static games of complete information• Dynamic games of complete information• Static games of incomplete information• Dynamic games of incomplete information
• Static v. dynamic – Simultaneously v. sequentially
• Complete v incomplete information– Players’ payoffs are public known or private
information
© 2008 Institute of Information Management National Chiao Tung University
Concept of Game Equilibrium
• Nash equilibrium (NE)– Static games of complete information
• Subgame perfect Nash equilibrium (SPNE)– Dynamic games of complete information
• Bayesian Nash equilibrium (BNE) – Static games of incomplete information
• Perfect Bayesian equilibrium (PBE)– Dynamic games of incomplete information
© 2008 Institute of Information Management National Chiao Tung University
Lecture Notes II-1 Static Games of Complete Information
• Normal form game
• The prisoner’s dilemma
• Definition and derivation of Nash equilibrium
• Cournot and Bertrand models of duopoly
• Pure and mixed strategies
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Static Games of Complete Information
• First the players simultaneously choose actions; then the players receive payoffs that depend on the combination of actions just chosen
• The player’s payoff function is common knowledge among all the players
© 2008 Institute of Information Management National Chiao Tung University
Normal- Form Representation
• The normal-form representation of a games specifies– (1) the players in the game– (2) the strategies available to each player– (3) the payoff received by each player for
each combination of strategies that could be chosen by the players
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Game definition • Denotation
– n player game – Strategy space Si : the set of strategies available to
player i– : si is a member of the set of strategies Si
– Player i’s payoff function ui(s1,….,sn): the payoff to player i if players choose strategies (s1,….,sn)
• Definition– The normal-form representation of an n-player game
specifies the players’ strategy spaces S1,….,Sn and their playoff functions u1,….,un. We denote game by G={S1,….,Sn ; u1,….,un}
i is S
© 2008 Institute of Information Management National Chiao Tung University
Example: The Prisoner’s Dilemma
-1,-1 -9,0
0,-9 -6,-6
Mum (silent) Fink (confess)
Mum
Fink
Prisoner 2
Prisoner 1
Strategy setsS1=S2={Mum, Fink }Payoff functionsu1(Mum, Mum)= -1, u1(Mum, Fink)= -9, u1(Fink, Mum)=0, u1(Fink, Fink)= -6u2(Mum, Mum)= -1, u2(Mum, Fink)=0, u2(Fink, Mum)= -9, u2(Fink, Fink)= -6
© 2008 Institute of Information Management National Chiao Tung University
Example: The Prisoner’s Dilemma
-1,-1
(R,R)
-9,0
(S,T)
0,-9
(T,S)
-6,-6
(P,P)
Mum (silent) Fink (confess)
Mum
Fink
Prisoner 2
Prisoner 1
T (temptation)>R (reward)>P (punishment)>S (suckers)(Fink, Fink) would be the outcome
© 2008 Institute of Information Management National Chiao Tung University
Strictly Dominated Strategies
• Definition– In the normal-form game G={S1,….,Sn ; u1,….,un}, let si’
and si’’ be feasible strategies for player i . Strategies si’ is strictly dominated by strategy si” if for each feasible combination of the other plays’ strategies, i’s payoff from playing si’ is strictly less than i’s payoff from paying si’’ :
ui(si,….,si-1, si’,si+1,….,sn)< ui(si,….,si-1, si”,si+1,….,sn)
for each (si,….,si-1,si+1,….,sn) that can be constructed from the other players’ strategy spaces S1,….,Si-1,….,Sn
© 2008 Institute of Information Management National Chiao Tung University
Iterated Elimination of Dominated Strategies
1,0 1,2 0,1
0,3 0,1 2,0
1,0 1,2 0,1
0,3 0,1 2,0
1,0 1,2 0,1
0,3 0,1 2,0
1,0 1,2 0,1
0,3 0,1 2,0
L M R
U
D
Outcome =(U,M)
L M R
L M RL M R
U
D
U
D
U
D
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Weakness of Iterated Elimination
• Assume it is common knowledge that the players are rational– All players are rational and all players know that all
players know that all players are rational.• The process often produces a very imprecise prediction
about the play of the game• Example
0,4 4,0 5,3
4,0 0,4 5,3
3,5 3,5 6,6
C RL
T
M
B
No strictly dominatedstrategy was eliminated
© 2008 Institute of Information Management National Chiao Tung University
Concept of Nash Equilibrium
• Each player’s predicted strategy must be that player’s best response to the predicated strategies of the other players
• Strategically stable or self–enforcing– No single wants to deviate from his or her
predicated strategy• A unique solution to a game theoretic problem,
then the solution must be a Nash equilibrium
© 2008 Institute of Information Management National Chiao Tung University
Definition of Nash Equilibrium
• Definition– In the n-player normal-form game G={S1,….,Sn ; u1,
….,un}, the strategies (s1*,….,sn*) are a Nash equilibrium if, for each play i, si* is player i’s best response to the strategies specified for the n-1 other players, (s1*,…, si-1* , si+1*,….,sn*)
for every feasible strategy si in Si ; that is si* solves
1 -1 1 1 -1 1( *, , *, * , *, , *) ( *, , *, , *, , *)i i i i n i i i i nu s s s s s u s s s s s
1 -1 1max ( *, , *, , *, , *)i i
i i i i ns S
u s s s s s
© 2008 Institute of Information Management National Chiao Tung University
Examples of Nash Equilibrium
0,4 4,0 5,3
4,0 0,4 5,3
3,5 3,5 6,6
C RL
T
M
B
-1,-1 -9,0
0,-9 -6,-6
Mum Fink
Mum
Fink
2,1 0,0
0,0 1,2
Opera
Fight
Opera Fight
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Examples of Nash Equilibrium
0,4 4,0 5,3
4,0 0,4 5,3
3,5 3,5 6,6
C RL
T
M
B
BR1(L)=M BR1(C)=T
BR1(R)=B
BR2(T)=L BR2(M)=C
BR2(B)=R
Player 2 ‘s strategy(best response function)
Player 1‘s strategy(best response function)
© 2008 Institute of Information Management National Chiao Tung University
Application 1Cournot Model of Duopoly
• q1,q2 denote the quantities (of a homogeneous product) produced by firm 1 and 2
• Demand function P(Q)=a-Q
– Q=q1+q2
• Cost function Ci(qi)=cqi
• Strategy space• Payoff function
),0[ iS
])([])([),( cqqaqcqqPqqq jiijiijii
])([max),(max *
0
*
0cqqaqqq jii
qjii
q ii
cqaq ji *
2
1
cqaq *2
*1 2
1 cqaq *1
*2 2
13
*2
*1
caqq
First order condition
Firm i ‘s decision
© 2008 Institute of Information Management National Chiao Tung University
Cournot Model of Duopoly (cont’)
• Best response functions
cqaqR 112 2
1)( cqaqR 221 2
1)(
((a-c)/2,0) (a-c,0)
(0,a-c)
(0,(a-c)/2)
R1(q2)
R2(q1)
(q1*,q2*)
Nash equilibrium
q2
q1
© 2008 Institute of Information Management National Chiao Tung University
Application 2Bertrand Model of Duopoly
• Firm 1 and 2 choose prices p1 and p2 for differentiated products
• Quantity that customers demand from firm i is
• Pay off (profit) functions
• Firm i’s decision
jijii bppappq )( ,
]][[])[,()( , cpbppacpppqpp ijiijiijii
]][[max)(max *
0
*,
0cpbppapp iji
pjii
p ii
cbpap ji **
2
1First order condition
cbpap *2
*1 2
1
cbpap *1
*2 2
1 b
capp
2
*2
*1
© 2008 Institute of Information Management National Chiao Tung University
Application 3The problem of Commons
• n farmers in a village
• gi :The number of goats owns by farmer i
• The total numbers of goats G= g1+…+gn
• The value of a goat = v(G)
• v’(G)<0, v”(G)<0
G
v
Gmax
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The Problem of Commons (cont’)
iiiiiii cgggvggg )(),( **
* * *( , ) / ( ) '( ) 0, {1,..., }i i i i i i i i ig g g v g g g v g g c i n
0)('1
)( *** cGvGn
Gv
0)(')( ****** cGvGGv
0max ( )
GGv G Gc
Firm i’s decision
Fist order condition
Social optimum
*** GG Implication : (over grows)
Summarize all equations
© 2008 Institute of Information Management National Chiao Tung University
Application 4 Final-Offer Arbitration
• The firm and union make wage offers wf and wu and then the arbitrator choose that offer that is close to his ideal settlement x
• Arbitrator knows x but the parties do not– The parties believe that x is a randomly distribu
ted according to a CDF F(x) and PDF f(x)
© 2008 Institute of Information Management National Chiao Tung University
Application 4 Final-Offer Arbitration
• Expected wage settlement
• Union and firm’s decisions
22
Pr ufuff
wwF
wwxob is chosenwprob
21 uf
u
wwF is chosenwprob
( , ) 12 2
f u f uf u f u
w w w ww w w w F w F
* ** *min ( , ) 1
2 2f
f u f uf u f u
w
w w w ww w w w F w F
* ** *max ( , ) 1
2 2u
f u f uf u f u
w
w w w ww w w w F w F
© 2008 Institute of Information Management National Chiao Tung University
Final-Offer Arbitration (cont’)
* * * ** * 1
( )2 2 2
f u f uu f
w w w ww w f F
21
22
1)(
****** ufuf
fu
wwF
wwfww
Solve FOC simultaneously
2
1
2
**
uf wwF
* ** *
1
2
u ff u
w ww w
f
The information of wf * +wu * and wu * -wf * with respect to
CDF and PDF
© 2008 Institute of Information Management National Chiao Tung University
Final-Offer Arbitration (cont’)
If2( ) ~ ( , )f x N m
mww fu 2**
2
22
1 1( ) exp{ }
22f x x m
* * 22u fw w
2*
2uw m
2*
2Fw m
Implication: higher uncertainty, more aggressive offer
Special distribution function of x
© 2008 Institute of Information Management National Chiao Tung University
Mixed Strategies
• Matching pennies
• In any game in which each player would like to outguess the other(s), there is no pure strategy Nash equilibrium– E.g. poker, baseball, battle– The solution of such a game necessarily involves unc
ertainty about what the players will do– Solution : mixed strategy
-1,1 1,-1
1,-1 -1,1
Head
Tails
Head Tails
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Definition of Mixed Strategies• Definition
– In the normal-form game G={S1,….,Sn ; u1,….,un}, suppose Si={si1,…,siK}. Then the mixed strategy for player i is a probability distribution pi=(pi1,…,pik), where for k=1,…,K and pi1+,…,+piK=1
• Example– In penny matching game, a mixed strategy for playe
r i is the probability distribution (q,1-q), where q is the probability of playing Heads, 1-q is the probability of playing Trail, and
10 ikp
10 q
© 2008 Institute of Information Management National Chiao Tung University
Mixed strategy in Nash Equilibrium• Strategy set S1={s11,…,s1j}, S2={s21,…,s2k}• Player 1 believes that player 2 will play the strategies (s21,…,s2k) with
probabilities (p21,…,p2k), then player 1’s expected payoff from playing the pure strategy s1j is
• Player 1’s expected payoff from paying the mixed strategy p1=(p11,…,p1j) is
• Definition – In the two player normal-form game G={S1,S2;u1,u2}, the mixed
strategies (p1*,p2*) are a Nash equilibrium if each player’s mixed strategy is a best response to the other player’s mixed strategy. That is
*)(*)*,( 2,11211 ppvppv 2 1 2 2 1 2( *, *) ( *, )v p p v p p
),( 211
12 kj
k
kk ssup
J
jkj
k
kkj ssuppppv
121
1121211 ),(,
J
jkj
k
kkj ssuppppv
121
1221212 ),(,
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Mixed Strategy
-1,1 1,-1
1,-1 -1,1
Head
Tails
Head Tails
q 1-q
r
1-r
Player 1
Player 2
Player 1’s expected playoff =q(-1)+(1-q)(1)=1-2q when he play Head =q(1)+(1-q)(-1)=2q-1 when he play TailCompare 1-2q and 2q-1If q<1/2, then player 1 plays HeadIf q>1/2, then play 1 plays TailIf q=1/2, player 1 is indifferent in Head and Tail
© 2008 Institute of Information Management National Chiao Tung University
Mixed strategy in Nash Equilibrium: example
-1,1 1,-1
1,-1 -1,1
Head
Tails
Head Tailsq 1-q
r
1-r
Player 1
Player 2
Player 1’s expected playoff =rq*(-1)+r(1-q*)(1)+ (1-r)q*(1)+(1-r)(1-q*)(-1) =(2q*-1)+r(2-4q*)Player 2’s expected playoff =qr*(1)+q(1-r*)(-1)+ (1-q)r*(-1)+(1-q)(1-r*)(1) =(2r*-1)+q(2-4r*)r*=1 if q*<1/2r*=0 if q*>1/2r*= any number in (0,1) if q*=1/2
q*=1 if r*<1/2q*=0 if r*>1/2q*= any number in (0,1) if r*=1/2
© 2008 Institute of Information Management National Chiao Tung University
Mixed Strategy in Nash Equilibrium: example (cont’)
(Heads)
(Tails)1
1
Tails Heads
1/2
r*(q)
q*(r)
(r*,q*)=(1/2,1/2)
1/2
Mixed strategy Nash equilibrium
r
q
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Example 2
2,1 0,0
0,0 1,2
Opera
Fight
Opera Fight
q 1-q
r
1-r
Player 1’s expected playoff =rq*(2)+r(1-q*)(0)+ (1-r)q*(0)+(1-r)(1-q*)(1) = r(3q*-1)-(1+q*) Player 2’s expected playoff =qr*(1)+q(1-r*)(0)+ (1-q)r*(0)+(1-q)(1-r*)(2) =q(3r*-2)+1-r*
© 2008 Institute of Information Management National Chiao Tung University
Example 2
(Opera)
(Fight)1
1
Fight Opera
r*(q)
q*(r)
(r*,1-r*)=(2/3,1/3)
2/3
r
q1/3
(q*,1-q*)=(1/3,2/3)
Payer 1’s mixed strategies
Payer 2’s mixed strategies
(r*,q*)=(2/3,1/3)
(r*,q*)=(1,1)
(r*,q*)=(0,0)
pure strategies (r*,1-r*)=(1,0),(0,1)
pure strategies(q*,1-q*)=(1,0),(0,1)
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Theorem: Existence of Nash Equilibrium
• (Nash 1950): In the n-player normal-form game G={S1,….,Sn ; u1,….,un}, if n is finite and Si is finite for every I then there exists at least one Nash equilibrium, possibly involving mixed strategies
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Homework #1
• Problem set– 1.3, 1.5, 1.6, 1.7, 1.8, 1.13(from Gibbons)
• Due date– two weeks from current class meeting
• Bonus credit– Propose new applications in the context of IT/I
S or potential extensions from Application 1-4