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"# ก$%ก& $&'%(ก)ก& A. H. Copeland (1945) ก

Posterity may regard this book as one of the major scientific achievements of the first half of the twentieth century. This will undoubtedly be the case if the authors have succeeded in establishing a new exact science -- the science of economics. The foundation which they have laid is extremely promising.

กก %%ก$%% ก % (players) %O P Q)ก&ก" (&RS) R #& ( Q)#R# )

% (กP$ก (strategies) (ก)กก

กTกUR%# &'#&VW (outcome) $ก

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UQ& (&VW\'% &$ (#) ก$% กOOกก (payoff)

$% ( ]' &$Q^ ก&VW% (O(\'%Q)

กก_ก#Wก$%กQ P (rational player) ก)% (VกQ%\'%&VW$ กP'O(\\'%

%T Pก\'% Oกก)กPW#Wก &VW

$ก\\'%Tก &'#Oกก)กกQ_ก'$%V' &VWก#'Oกก$Pก UQ%ก#'$&'%()ก&

$&'%ก#'Oกก%\'%&VW ก& (ก)ก&ก#'Oก%

\'%(]Rก&

ก(%กQ P ก

V#O"ก)% I (% II

กก^% I ก 3 ก) A, B, C (% II ก 2 ก) 1, 2 '&&^กO(&VW&^Q' 6 \\'%

e_$'%#กe$' 3 × 2

กก)%&^$ก ก('U\% )&^$fOQ%

g'ก('$_^V%ก& OOกก\'%OกR%&^)ก

&$ &กOOกก$% I ( &$ &OOกก

$% II

II II

1 2

I A

I B

I C

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(2)

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ก'&ก$OOกก ก) OU#% I \'%ก&

OU#% II (OU% II \'%ก& OU% I

'&&^O$&^O( $%ก& กก&ก"(^ ก (Zero-sum game)

ก^ก'T_$&'%ก#'$_^R&'O(Q%

กกUQ'OOกก VกUQ'OOกก$% I \'%กfVV V(OOกก$% I \'%กf)OOกก$ II

II II

1 2

I A

I B

I C

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⁵²⁰ ³¹⁰²

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# % #% I #กก

% I O##'$ %กOOกกP' Oก ) 10 e_T C (Q&ก B ]'Q&% II O()ก 2

O#T%% II % I O()ก C %% II O(R%ก 1 UQ%% II \'% 5 Q

ก"% I %% II )ก 1 % I O()ก A V)Q%\'%&VW 2

T%% II ^ $ 2

%% I กfO()ก C ^\\%O

&ก)

\'%)&ก) (movement diagram) ]'$กOก%T

\&กP'T ($กOกก\Q%P'Q&ก

ก'&กV ก) กO(\QP''Q_ ก)กQ%

Q PO(ก]e ก&\ e_กก'&ก%&^O(R%กPWก (mixed strategies)

II II

1 2

I A

I B

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ก\

%&^U\' &VW''P') (1, 1) e_ &)กU'&

$%&^

ก)T%% II %% I )ก A $T)ก 2 e_Q%O Oกก$_^

U'ก&T%% I %% II )ก 1 $T)ก B e_Q%OOกกก

T%&^R%Q P'&ก&VW\'%) (-5, -5) e_&VWP'$&^

II II

1 2

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I B

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2,2 5,5^{1, 1}^ ^2,2

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(3)

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V#O"ก"%&^TV'P #' ก&\'% T%% II (ก$)ก 2 % % I U\? T%% I ]ก\'%)กกTก"

$$O('ก^\?

ก_กก'&ก O(V#O"ก&\ (Two-person general sum game) ก)(\)ก& ]'Oก #' )ก (arbitrator) (U$% กP #WUQ& &^kl ()%กก ก#ก)ก&กP% ก N N > 2 O(\%ก' %ก)ก&$กP% (ก

#\'%Q%R#กกP

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!$ก (normal form) Q) กPW (strategic form)

$ก กUQ']'#&'& (X, Y, A) )

X e \$กPW$% I

Y e \$กPW$% II

A opกR&OUO## X ×Y

ก^ก%)กกPWกV%ก& % I )ก x ∈ X

(% II )ก y ∈ Y ]'&^\ ก)ก$กklQ_ Q&Oก

%&^)กกPWก% %ก\'%) A(x, y)

e_QR%&ก# กQT_%ก\'% |A(x, y)| Q QT_%ก |A(x, y)| Q

ก'

V#O"ก

$%&ก P ก)กก 3 $% II \O V(ก 2 Q%OOกก%กPกก" กก 2 $% II 'ก (dominate) ก 3 $% II Q)ก 3 '%กก 2 (Is dominated by 2)

ก #กก %Q PO(\)กก'%ก

II II II II

1 2 3 4

I A I B I C

I D

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Q&กก'

# ก ก S 'กก T T%Pก s $&VW S OOกก'กQ)ก& $&VW T (%Q_ S 'ก T

Q&กก' %Q P\)กก'%กก) (T%% %)กO()กก'ก&^)

Q&กก'&กTR%กUO&'ก$ก#กe\'%

(4)

&ก)

'&ก)$ก\'%

$%&ก P ก)กก 3 $% II \O V(ก 2 Q%OOกก%กPกก" กก 2 $% II 'ก (dominate) ก 3 $% II Q)ก 3 '%กก 2 (Is dominated by 2)

ก #กก %Q PO(\)กก'%ก

II II II II

1 2 3 4

I A I B I C

I D

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ก #กe

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$ A $#กe

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&$ก#กe)OOกก$% I

A=

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âââ^^{m 1}¹¹^{2 1} âââ^^{m 2}^{1 2}^{2 2} ^^^^ âââ^^{m n}^{1 n}^{2 n}

]

^{, a}^{i j}^=A^^xⁱ^{, y}^j^

OP'%

กUQ'ก ! (X, Y, A) Q%กOP' a_ij OP'%

(saddle point) T%

a_ijก& UP'T i

a_ijก& P'Q&ก j

Q&กกOP'% T%ก#กeOP'% %%&^O()กก U\$กก& OP'%'&ก

# $ก v )% I & (ก&$R(\%ก v (UQ&

% II ก(ก&% I R(\ก# v

Oก &$ก#กeOP'%)OP' (3, 2) ]'OOกก 2

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& #$OP'%

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UQ& T i %Q&ก Row min($%P'T i

UQ& Q&ก j %T Column max ($กP'Q&ก j

QP'Q&ก Row min ( UP'T Column maxT%&^

ก& %ก^OP'%'&ก (\'%OกTQ&ก ^{Row min} ก& Q&กT Column max

(5)

&กQOP'%

Oก %P'T Column max ) ²(กP'Q&ก ^{Row min} ) 2

'&&^ก^OP'% = 2 e_ UQ (C, 2) II II II II

1 2 3 4

Row min I A

I B I C

I D

[

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²⁰¹⁶¹²

Column max 12 2 7 16

OP' ก$%ก& OP'%

T% a_ij $OP'%ก#กe% T i $ a_ij O(TQ%

maximin (Q&ก j $ a_ij O(Q&กQ% minimax (

maximin = a_ij = minimax

T% maximin = minimax %OP' &'$TQ% maximin ก& Q&กQ%

minimax O(OP'%

\กf' ก กT$ maximin ก& Q&ก$ minimax \ก& &) ก\OP'% e_ %กQกPW'P'ก^'%

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kuกQ&' 2.1

1.OQก'$ก#กe

2. T(Pก ก'\ Q&กกก'&'&

(Principle of Higher Order Dominance) #Wก)Q&Oก &'ก '%ก ก% ก#กe\'%O($'fก TR%Q&กก ก'ก& กP$กfก (กUO&'ก'%'&กก U^\) s Oก(&\T'ก#กe\'%ก

Oกก$% %OQก#กe\'%Oกก(Pก ก'&'&

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⁴³² ⁶¹³ ⁵²⁰ ⁴¹⁴

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kuกQ&' 2.1

3.O'&ก)(OP'%$ก#กe \^ T%

4. OV#O T%ก i $% I 'กก j $% I

(ก j OP'% a_ij % $ a_kj Q&ก'ก&

O(OP'%'%

5. Oก&ก) &ก PกQ%Q P&กO($%OP'% \O(#

Q%Q POก UQ'กf ก#กe O#Q)\PกกOP'%O(

'%ก& T(ก"^

3.1

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^{, 3.2}

[

²²³ ¹⁰¹ ²⁴³

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^{, 3.3}

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(6)

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V#O"ก#กe \^

ก'&ก\OP'% %&^Q PO(\)กก'ก Q_]'vV( V(%kl $%O(กPWกUQ%ก' s '

\Q' #WกO_UQ%kl $%\TO& Q&ก ก\'% ก)% %กP e_TU"O(

ก)กก\'% ก ก (Mixed strategy)

$"(ก\กPก ก #PW#x (Pure strategy)

II II 1 2

Row min I 1

I 2

I 3

[

²⁰⁵ ³¹⁰²

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³⁰⁵

Column max 2 10

กR%'(("OOกก

# '( (expected value) $OOกก a₁, a₂, ..., a_n

'%O( p₁, p₂, ..., p_n U'& )

a₁p₁ + a₂p₂ + ... + a_n p_n

ก)'($กP$OOกก)$T^UQ&ก"ก&

OOกก )T^UQ&ก)O(%O()กก&^

T%% II R%ก]QS กก#กe $)กก 1 '%O( ½ (ก 2 '%O( ½

'($% I ก)กก 1 ) 2 × ½ + (-3)× ½ = -½ '($% I ก)กก 2 ) 0 × ½ + 2 × ½ = 2 '($% I ก)กก 3 ) -5 × ½ + 10× ½ = 2½ '&&^% I )ก 3 e_Q%'(OOกกกP'

Q&กกR%'(

Q&กกR%'( T%% %กklR%ก (O(R%

ก'&ก\O(ก#'(\$_^กf %Q PO()กกQ%

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O(ก&) ½ % I O()ก 3 UQ%% II %O 2½

'&&^V)Q%ก$% II %P' % II %ก)กก UQ%'(ก)กก$% I UP'O(\\'%

Q)O)กQ%% I \\'% Q) ก)กก

ก-Q)-

V#O"ก-Q)- T%% I ( II (ก$ & Q_Q) % I )

%\'% $ &$T%OU Oก&

T%OU $"(% II )%\'% ก&

T%OU Oก& T%OU

ก ก #$\'% X = {1, 2}, Y = {1, 2} (#กe A

ก^%Q_\'% %&^)

II II

1 2

I 1

I 2

[

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(7)

%&^\ %ก)ก(ก &$ 1Q) 2UQ& กPก

'&&^Q&กกQ P)กP T%% I P]'กQ_

3/5 &^]'v(ก 2/5 &^]'v

T%% II กQ_ P$% I 2Q 3/5 &^(\'% ³Q 2/5 &^ '&&^'(% I \'% -2(3/5) + 3(2/5) = 0 ก))\

Q s ก% I O(\\'% Q) (break even)

T%% II ก P$% I \'% 3Q 3/5&^( 4Q 2/5&^ '&&^'(% I \'% 3(3/5) - 4(2/5) = 1/5ก))\

Q s ก (ก% I \'%(" 1/5

'&&^% ^I %กQกQ%'(กP'

#'ก#(Q #'กPWกUQ%ก&

Q% p O(% I กQ_( 1 - p O(% I

ก

T%% II กQ_ '(% I \'%) -2p + 3(1 – p) = 3 - 5p

T%% II ก '(% I \'%) 3p – 4(1 – p) = 7p - 4

V)Q%% II \ก#' ก (QกกQ_Q) % I %กQ p

UQ% 3 – 5p = 7p – 4 \'% p = 7/12

ก)T%% I กQ_'%O( ^7/12O(\'%\% II O(ก Q_Q) '(% I O(\'%) 3 – 5×7/12 = 1/12 ( 7×

7/12 – 4 = (49 – 48)/12 = 1/12O(

#''&กก กPWกUQ%ก& (equalizing strategy)

#'กPWกUQ%ก&P%กkl

\\'%Q)\% I กกQ%'(กก^ U )

\\\'%T%% II #'ก&ก"('ก&

Q% q O( II กQ_( 1 - q O( II ก

T%% I กQ_ '(% II \'%) -2q + 3(1 – q) = 3 - 5q

T%% I ก '(% II \'%) 3q – 4(1 – q) = 7q - 4

V)Q%% I \ก#' ก (QกกQ_Q) % II %กQ q

UQ% 3 – 5q = 7q – 4 \'% q = 7/12

ก)T%% II กQ_'%O( 7/12O(\'%\% I O(ก Q_Q) '(% II O(\'%) 3 – 5×7/12 = 1/12 ( 7×

7/12 – 4 = (49 – 48)/12 = 1/12

#''&กก กPWQ(P' (optimal strategy or minimax strategy) Q)กPW##กe (minimax strategy)

กPW #PW#(กPW

กUQ'Q% X, Y )e $ก #PW#$% I (% II U'&

กกPWกPOกกP$ก #PW# ก

'&&^ก- $% I )ก'%O( 7/12, 5/12

TV#O"ก #PW#ก'%

O(กO) 1

กOก^ #%O(OV'(\'% ]'Q'('P'

&^)%กO(\T ก ก (Qก\'% 5 % ]'\ % P ก& ]กก\'% 10 % ½ (\\'%(\ ½ e_กV#O"'&ก

$_^ก& ก\'%e_OOกก$## (utility theory) ก ก)ก$%O('%ก& opกR&## (% &'#&VW v$## กUQ'

(8)

##กe

ก (X, Y, A) ก& (T%e X ( Y

e & (

##กe UQ& ก & (

O( V ก$ก

O(กPW$% I (% II e_% I O(\'%%v V

\% II O(\กf

O(กPW$% II (% I e_% II O(v\ก\ก V

\% I O(\กf

T% V กก^P #W T% V > 0 กก^% I

\'% T% V < 0 กก^% II \'%

kuกQ&' 2.2

1. V#O"ก-Q)- T%OOกกกU &$&^"ก&

O#(Qก'&ก (Pก\'%QกP #WQ)\

2. % ^I T) 'U('' % ^IIT) '('UOf' %&^)ก\VQ_

)V%ก&('Q%\Vก T%\V'Q)ก& % I R(

(\'%OOกกก& \V%R(T) (% II Oก& '&ก) (T%\V' ก&% II R(ก& \V%R(T) (% I O) OQ$ก(กPWQ(P'

3. “I know a good game,” says Alex. “We point fingers at each other;

either one finger or two fingers. If we match with one finger, you buy me one Daiquiri. If we match with two fingers, you buy me two Daiquiris. If we don't match I let you off with a payment of a dime. It'll help pass the time.”