Lecture Note 2 F 2 - . . FF

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Lecture Note 2 2

..

!"#$%& Nash (How to Find Nash Equilibrium)

1. Dominant Strategy # $%$&'()*+, Nash ) '()*')0 12345, Dominant Strategy $%$6&)(789$ Best-response +, Dominant Strategy

7"8& 1: Prisoner-dilemma

Prisoner-dilemma '()*+,''(-. /0+1. +)+2 34+ '(56.+'27+ Dominant strategy <+ +2+ Nash equilibrium 34+

strategy profile '(3,13. Dominant strategy <56.+'2 2 7+

E./56.+7+'( Dominant strategy 7F< Nash F++ GF< Nash 34+ strategy profile '(3,.

dominant strategy <56.+7+/+H( ,'I '(34+ best-response strategy <

dominant strategy ++2+ )13+2

7"8& 2

Player 2

a n

A 50,60 15,95

player

1 N 40,40 44,55

')75,-'+< *+'( 2 /0+1.56.+7+'( 2 '+2+'( dominant strategy 7J n -7F< Nash F 6 7J K/,56.+7+--. 'I '(34+ best-response <'I n 7J'I N +2+F< Nash *+ *+'( 2 7J (N,n)

2. >?&@0$A$B')(789$ strictly dominated strategy ,,F86@@1% # (Iterated Strictly Dominated Strategy Process) B')(7>K dominate ,,F'?$5, )(7),)L')0L$%$ &'()*+, Nash )'()*')045,)(7 5,,)KL

'( 4 *+,''(-. /0+1.E*.,+*+'I '(34+ strictly dominated strategy 13-,,Q2K R +/J strategy profile +

(2)

7J (T,L) +2+ strategy profile H34+F< Nash +*+

3. >?FL >AN?07B+?, 1 5, 2 A$'()*+, Nash )'()*')0F'?

A?))B')(7A? ('O')AN?&@0$A$B')(789$ strictly dominated strategy ,,F86@@1% # 6?04L,))(789$ best-response

//: )I+23G+ *+'( <+*/V GW)(J(./

F< Nash -,,5 (Mixed strategy Nash equilibrium)

7"8& 3

L C R

T 2,0 1,0 4,2

M 3,4 1,2 2,3

B 1,3 0,0 1,0

/'I '(34+ strictly dominated strategy 13-. 63/J+2

Player 2

L R

T 2,0 4,2

player

1 M 3,4 2,3

E/ pure strategy Nash equilibrium 1.+2 Best-response <56.+7+- 7J

BR(L) = M BR( R) = T

Best-response <56.+7+'( 2 7J BR(T) = R

BR(M) = L

+2+E/ pure strategy Nash equilibrium 1.7J strategy profile (T,R) - (M,L)

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7"8& 4 (?@@AB7CD7E"):

*/./ Pure strategy Nash equilibrium < 13+2 1.

Firm 2

Advertise Donat advertise Advertise

25K, 27K 25K, 30K Firm 1

Donat

advertise 26K, 27K 20K, 20K

2.

X Y

A 3,5 0,4

B 2,4 2,6

C 1,6 1,5

Mixed Strategy Nash Equilibrium

'(5+1.kHEHF-,, Nash *+'(34+ Pure strategy -. 1 0 / R 1E/F-,, Nash '(34+ Pure strategy 1.

+2+H./F-,, Nash '(34+ Mixed Strategy

'(1.kH13*+,''(-. Mixed strategy 34+*/. probability distribution ,'I *+Q0'I <56.+-7+

*/.) 13+2 7"8& 5: Matching Pennies

Player 2

Head Tail Head

1,-1 -1,1 Player

1 Tail -1,1 1,-1

/0+1. Matching pennies 1 pure strategy Nash equilibrium -6. *+ R /+H(. Nash equilibrium +2+)('(./

(4)

+2 7J Mixed strategy Nash equilibrium +(+ QH('KkH*+++2 E/ Mixed strategy Nash equilibrium 1.1

Mixed strategy Nash equilibrium (LMN 2x2) )I'( 1: randomized strategy

*+<56.+'2 2 7+ <+ mix / 2 'I 7J Head , Tail 0 J( Head - Tail */.5,-'+'+,< +2+ J(+1<< mixed strategy 07J

) ( )

( ₁

1 H u T

u = (1)

) ( )

( ₂

2 h u t

u = (2)

)*/. probability '(56.+7+'( 1 J Head 7', σ1⁽H⁾

probability '(56.+7+'( 1 J Tail 7', 1-σ1⁽H⁾*+'K++ )*/.

probability '(56.+7+'( 2 J head 7', σ2⁽h⁾ - probability '(56.+7+'( 2 J tail 7', 1-σ2⁽h⁾

-'+7 probability /+2<.13*+'( (1) - (2) 1.

)) ( 1 ( 1 )) ( )(

1 ( )) ( 1 )(

1 ( )) ( (

1σ₂ h + − −σ₂ h = − σ₂ h + −σ₂ h (3)

)) ( 1 )(

1 ( )) ( )(

1 ( )) ( 1 )(

1 ( )) ( )(

1

(− σ₁ H + −σ₁ H = σ₁ H + − −σ₁ H (4)

(3) - (4) E-./ probability < mixed strategy Nash equilibrium 1.7Jσ1⁽H⁾=1/2 - σ2⁽h⁾=1/2 +(+7J mixed strategy Nash equilibrium *+

matching pennies 7J (1/2H + 1/2T, 1/2h + 1/2t )

)I'( 2: 63J(/ best-response

E63J(/'I '(34+ best-response <56.+7+-1.+2

-1

1

-1 h=1/2

h=1 h=0

H T

1

(5)

-+2<63-'+5,-'+ +-+++-'+ probability distribution <56.+7+

'( 2 '(+'I h <.+7J'(5,-'+<56.+7+-'+ +(+7J '( σ2⁽h⁾=1/2

E/ best-response strategy set <56.+7+-1.+2 E. σ2⁽h⁾> ½ best-response strategy <56.+7+'( 1 7J + H

E. σ2⁽h⁾= ½ best-response strategy <56.+7+'( 1 7J + H,T, /J mixture 10 1./ H - T

E. σ2⁽h⁾< ½ best-response strategy <56.+7+'( 1 7J + T

*+<+ E63*++K/,56.+7+'( 2 1.

(insert figure)

E/ best-response strategy set <56.+7+'( 2 1.++ 7J E. σ1⁽H⁾> ½ best-response strategy <56.+7+'( 2 7J + h

E. σ1⁽H⁾= ½ best-response strategy <56.+7+'( 2 7J + h,t /J mixture 101.

/ h - t

E. σ1⁽H⁾< ½ best-response strategy <56.+7+'( 2 7J + t

+2+ E/ Nash equilibrium QH(34+ strategy profile '(34+ best- response strategy <56.+'2 2 +.+1. 7J (1/2H + 1/2T, 1/2h + 1/2t )

//: *+/ Mixed strategy Nash equilibrium 'K/J+,'(/ Pure strategy Nash equilibrium 7J 'I '(34+ strictly dominated

-1

1

-1 H=1/2

H=1 H=0

t h

(6)

strategy */.'(+ J(K'I '(+K5J(/34+F-,, Nash QH(E)1.13+2

7"8& 6:

Left Middle Right

Up 0,4 5,6 8,7

Down 2,4 6,5 5,1

*/./F-,, Nash '2-,, pure strategy - mixed strategy *+ .+.+