The prisoners' dilemma, congestion games and correlation

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The prisoners' dilemma,

congestion games and correlation

by Ferenc Forgó

C O R VI N U S E C O N O M IC S W O R K IN G P A PE R S

http://unipub.lib.uni-corvinus.hu/2187

CEWP 7 /201 6

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# $ & n% P D, : & & 2% &

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$M V* & $EV*(

F G∈ Γ 0 # & > # P(G) 0 %

0 & 0 # >& G M(G) 0 0 &

0 # 0 : 7 / & 0 S(G) SCE, (

F SW(p)0 & - & $ : & & *

0 0 & 0 p( > & SCE G

M V(G) = maxp∈S(G)SW(p) max_p∈M(G)SW(p)

&

EV(G) = maxp∈P(G)SW(p) max_p∈S(G)SW(p).

9 & M V & EV SCE

& # Γ >

MV = sup

G∈Γ

M V(G), EV = sup

G∈Γ

EV(G).

; - # MV B0 B - & EV B- B

( M V 0 0& & ∞ EV 0 0& &

1(

G

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; #1 $2 ?* M V EV + E n% & 2% & & %

# & # # - (

9 4 : 0&

7 0 & M V M V P EV

2 ∞ ∞ 1

3 ⁴₃ ⁴₃ 1

4 ? ≥⁴₃ 1.007478...

... ... ... ...

n ? ≥_4(nⁿ₋₁₎ ≤ ⁴₃

9 & MV P - > M V &

NE, ( 9 . && '

& 0& 0 0& n% & 2% & : &

& # # & P D# (

' ( ) * ) ) +

)

# n% & 2% & & : &

# #

F1 =C F2 =D a₁= 0 b₁=y+ (n−1)z a₂=x b₂=y+ (n−2)z a₃= 2x b₃=y+ (n−3)z

... ...

a_n= (n−1)x b_n =y .

- x, y, z >0( ; , ,

0 0& .

F1 =C F2 =D

C(1) = 0 D(n−1) =y+ (n−1)z C(2) =x D(n−2) =y+ (n−2)z C(3) = 2x D(n−3) =y+ (n−3)z

... ...

C(n) = (n−1)x D(0) =y

.

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& : & # # - b_n < a₁$ (#( .

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# SCE $ #1 2 ?*

(−y+ (n−1)(x−z))q0+1 n

n−1

t=1

(ty−t(n−t)(x−z)+

(n−t)((n−t−1)(x−z)−y))qt+yqn≥0. (2)

9 . > 0& : 8

0 $ # *(

@ 0 0 & 0 # 1

n

t=0

q_t = 1, (3)

qt ≥ 0, t= 0,1, ..., n.

C - : 4 SW - & , : 8

SW =n(n−1)xq₀+

n−1

t=1

(t(y+ (n−t)z)+ (n−t−1)(n−t)x)q_t+nyq_n ( (4)

; # n% P D Q1 Q2

0 > ( ; t & & D n−t & C( Q1 a_t+1≤bn−t

t= 1, ..., n−1 &

y+tz > tx, t= 1, ..., n( (5)

3&& / & & 0 #& / &

y+ (n−1)z >(n−1)x. (6)

9 . # Q2 - # C(n) = (n−1)x > D(0) =y( +

2% & & : & # # n% P D

x, y, z 0< 1

n−1y < x < 1

n−1y+z n≥2. (7)

#M V / & n≥2.; 0

0& (

& ' n% P D & MV =∞.

<

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@ 9 . & x= 1, y=ε, z= 1 ε >0( 9 q₀=q_n =

1

2, q_i= 0, i= 1, ..., n−1 SCE & 0 > 0 0 #

(2). 3& x, y, z (7) && n ≥ 2( 9 SW n% P D

- 0 0 0 # (4) 0

1

2n(n−1) +¹₂nε( 9 & N E n% P D - && &

SW nε( 9 MV

1

2n(n−1) +¹₂nε nε

- # ∞ ε→0(

+ P D 2% & & : & # # && -

@ G MV =∞ - & # - &&( 7 -

- # EV & 2% & & : &

# # (

& , 2% & & : & # # %

& EV ≤2(

@ ( > & 0 t∈[0, n] / W 0

W(t) =

n

t=0

(t(y+ (n−t)z) + (n−t−1)(n−t)x). (8)

9 : W^∗ W(t) [0, n] 0 0 & : %

SW - # [0, n]( 9 H

/ (8) x−z( # # x−z - #

- (

( x≥z( ;

W^∗= max{W(0), W(n)}= ny x≤_n₋¹₁y n(n−1)x _n₋¹₁y < x .

9 0 0 & q_t= 0, t= 0,1, ..., n−1, qn = 1 &- SCE

y >0 - && EV = 1 W^∗=ny( -

0 - W^∗=n(n−1)x.9 0 0 & q₀=q_n= ¹₂, q_t= 0, t=

0, n & 0 SCE0 0 # (2) # #

1

2(−y+ (n−1)(x−z)) +1

2y= (n−1)(x−z)≥0(

9 SW SCE

1

2n(n−1)x+1 2ny(

9 - #

EV ≤ n(n−1)x

1

2n(n−1)x+¹₂ny <2(

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2( x < z( 3 > n ( 3 SCE 0 0 0

# 0 0 & q = 1, qi= 0, i=ⁿ₂.9 > (2)0

n 2y−n

2(n−n

2)(x−z)+(n−n

2)((n−n

2−1)(x−z)−y) = n

2(z−x)>0( (9)

9 SW 0 & # # SCE

W(n 2) =n

2(y+n 2z) +n

2(n

2−1)x) = n 2(y+n

2z+n 2x−x)(

; / W(t) 0 & : %

r= ^y+nz_2(z⁻⁽²ⁿ₋_x)⁻^1)x( ; r > n = - EV = 1(

; r <0 W(0)> W(t) &&t∈[0, n] -

EV ≤ W(0)

W(ⁿ₂) = n(n−1)x

n

2(y+ⁿ₂z+ⁿ₂x−x)= 4 (n−1)x

2y+nz+ (n−2)x = 4 (n−1)x

2y+nz+ (n−2)x<2(

- & & z > x(

- r∈(0, n)( C & H qt (2)

n

2 ≤t ≤n( 9 0 t

0& H qt / t

H x−z / # ( 9 H y q_n

0 H q 0 (9)( 9

& & && 0 0& #

& (

9 : r W && &

[ⁿ₂, n], # : t^∗$0 # # 0 # #

r*( 9 H q_t∗ 0 # q_t∗ = 1, qt = 0, t =t^∗ SCE

EV = 1. 9 - & - 0≤ r < ⁿ₂( 9

: 0 0 0

W^∗=W(r) =n(n−1)x+r²(z−x)< n(n−1)x+n² 4(z−x) 0 r < ⁿ₂ z−x >0( EV -

EV < W^∗

W(ⁿ₂) = n(n−1)x+ⁿ₄ (z−x)

n

2(y+ⁿ₂z+ⁿ₂x−x) =4(n−1)x+n(z−x) 2y+nz+ (n−2)x <2(

7 - - - n n≥3( C &

H qt / & (2) ⁿ⁺¹₂ ≤t ≤n

# : t^∗ W && & EV = 1( 9

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# & - - 0 ≤r < ⁿ⁺¹₂ ( SCE = q − =¹₂, q = ¹₂, q_i= 0, i= ⁿ⁻₂¹,ⁿ⁺¹₂ .9 SW SCE

W = 1

2W(n−1 2 ) +1

2W(n+ 1 2 ) = 1

2(n−1

2 (y+n+ 1

2 z) +n−1 2

n+ 1

2 x+n+ 1

2 (y+n−1

2 z) +n−1 2

n−3 2 x).

9 -

EV ≤ n(n−1)x+r²(z−x)

1

2W(ⁿ⁻₂¹) +¹₂W(ⁿ⁺¹₂ ) ≤

n(n−1)x+⁽ⁿ⁺¹⁾₄ (z−x)

1

2(ⁿ⁻₂¹(y+ⁿ⁺¹₂ z) +ⁿ⁻₂¹ⁿ⁺¹₂ x+ⁿ⁺¹₂ (y+ⁿ⁻₂¹z) +ⁿ⁻₂¹ⁿ⁻₂³x).

3 & & # 0 4 & #

- #

EV ≤ 4n(n−1)x+ (n+ 1)²(z−x)

(n−1)(n+ 1)z+ ⁽ⁿ⁻^1)(n+1)₂ x+⁽ⁿ⁻¹⁾⁽ⁿ₂ ⁻³⁾x.

C - & & . 2( 9

&& - # / & &

4n(n−1)x+ (n+ 1)²z−(n+ 1)²x≤2(n²−1)z+ 2(n−1)²x.

# # - #

(n−3)(n+ 1)x≤(n−3)(n+ 1)z

- 0 & & 0 x < z( 9 - 2% &

& : & # # EV ≤2(

# 0 & P D # 0

0 0 # EV( 3 0 & & P D,

- x = z( # - & , = # &

: # #

- 0 > ( 9 # && B 0& B

: 0 # : & " 0 # $ <!*(

& - n% 0& P D & EV =

2.

@ ; @ ? - & 2% & & :

& # # EV ≤ 2 x ≥ z $ * & x = z(

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9 & / - ' 0 # I n%

0& P D - x= 1, y =ε, z = 1. 9

(7) ε && # ( 0 # (2) (3) - #

&& - #LP - & SW : 4 #SCE:

maxn(n−1)q₀+

n−1

t=1

(tε+ (n−1)(n−t))qt+nεq_n.

0= nq₀−

n−1

t=1

(n−2t)qt−nq_n ≤ 0,

q₀, q₁, ..., qn ≥ 0, q₀+q₁+, ...,+qn= 1.

3 n ( 9 q = 1, qj = 0, j = ⁿ₂ 0& &

u = ¹₂(n−1)− ^ε₂, v = ⁿ₂(n−1) + ⁿ₂ε 0& & & -

0= & ⁿ₂(n−1) +ⁿ₂ε( 9 0 & : SW

n(n−1).9 EV

n(n−1)

n

2(n−1) + ⁿ₂ε = 2 n−1 n−1 +ε

- # 2 ε→0(

9 & n & ( ; - 0&

& q − = ¹₂, q =¹₂, q_j = 0, j=ⁿ⁻₂¹,ⁿ⁺¹₂ .

2% & & : & # # %

& EV = 2.

" n% P D & EV = 2.

; - - -n% , & # 0

2% & & : & # # ( 9

& / & 0 @ % #

7 ( 9 & - ∞ &

- 0 2 n( 9 & & 2 0&

n% , & # ( & & #

# & 4 n & & 0 0

& (

. 9 # 9J3 22? # %

&& . -& # ( .

/ )

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3 & # ; 5 9 &4 5 $2 * & & % ( K & 3 > & ; && # !!'G<G%6

3 K $ <?* + 0= & 4 # (

K & 5 & E '6<% 6

& K C $ * ; 7% & , & # @ & %

& + G!' ? %? G

#1 $2 * 3 # & 4 & / & 0 ' 3 - &(

5 & + & + 6 ' 6%

#1 $2 ?* 5 # - & / & 0 2%

& & % # & # # & E K %

& 22 $ *' ! % 6G

" 0 # " $ <!* 7% , & ( K & 5 &

+ & # !' 2<%?

5 & " A & K%@ $ < * + # && 4 % # ' &

# - & & : / & 0 0 ( ; %

& K & L 9 <'2 %22

0 5 K 0 3 $ 6* 3 # ( 9 5;9

@ 0 # 53

& C $ <!* 3 & # # % # 7 / %

& 0 ( ; & K & L 9 2'6G%6<

+4 & # 5 7 $2 !* 3 # 7% , & %

& : ?' GG% <?

+4 4. +4 & # 5 7 $2 2* 3 3 & & + 7%

@ @ , & + - K & @ 3 & 5 %

$ & & * 2'22%!

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