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Munich Personal RePEc Archive

Minority-proof cheap-talk protocol

Heller, Yuval

Tel-Aviv University

19 August 2005

Online at

https://mpra.ub.uni-muenchen.de/54906/

(2)

Minority Proof Cheap Talk Protocol

Yuval Heller

1

! " #

! $ %& ' (

) * # + ( ( (

, * % !-!$. -&$ % ! ! & /&

0* % !-!$. ! - %

/ " #(

, ( ) 1 ( ' 2 3 3 , ( 1

( # , (

) # 4

(3)

Abstract

#5 6 4

# ! 3 ( 7 ! 3

2 # 2 2 6

8 6 0 2 ! 3( 2

! 3 # (

JEL classification: C72

Keywords: ! ! 3 6 6 !

(4)

-1. Introduction

2 6 / %. ( ' #

3 2

# ! 3* ! # ! !

# ( ( 9 2 : / & ; / ; < # / ( 9 ! 3

4 ( / 8

6 # 2 (

: 7 / $ * 6 (

' !2 2 # 6

2 ! # (-= 0

* 3 2 0 4

2 5 4 2

2 0 *

• !> , / - 2 2

# 2 0 (

• 4 3 ) # # ' / %- = 2 (

7 2 2 :.

2 6 # ( 2 !

3 ! 6 ! 8 6

0 ( 5 0 < !, / & -; ? .

# @/( A 2 2

6 6 8 6 (

7 2 ( 7 # #

0 # B # # 2

2 # 4 # B C ( (. ( = 0 ! 3

# D E # ! 6 2

2 ! ! 8 6 0 2 ! 3(

2 0 2 E (

> / $ > # / $ / & ) # , / <

C (

-0 # # ! (

A 2 # # (

.

(5)

.

5 2 * (

-( . 2 2 ≥ (

0 ( 7 $(

2. Model and Definitions

∈ ∈

= 2 =F/ (( G !

# # ∈ # B ! #

# B # !

=

( ! 0

# ( (

! # ( ? ⊆

=

− =

{

∈ H ∉

}

# ( ( ?

∈ ∈ 2 H ∈ H − −

− ∈

=

# 2

* = H ( 7 # ∈ #

/

(((

= ∈ /

(((

= ⋅ ⋅ ( # ⊆ 2 # ∈

# ∈

=

( 1 ⊂

# ⊂ ! # ( ?

∈ 2 2 = − 2 * ∈ − ∈ − (

Definition 2.1: # ∈ #

⊆ # ≤ # ! # ∈

0 # ∈ ≥ − ( = /! 8 6 8

6 8 / / ! 8 6 8 6 < (

/ &% # I/ ! 8 6 ! 8 6 (

Definition 2.2: 1 ⊆ ( * → (

# ∈ # ∈ 0

∈ 2 =

− ⋅ H ( 1 ⊆

# ! #

# #

2 # # (

Definition 2.3: ∈ ! 6 # ⊆

(6)

! 6 # 2 2

# 4 2 3 # ( =

/! 6 6 ! 6

6 : 7 / $ # I/ ! 6 !

( 0 6 2 #

# * # 3 # # (

Definition 2.4:1 ∈ # ( # ɶ∈

# ⊆ # ≤ #

! ɶ∈ ɶ ɶ 0 ∈ > ɶ ( #

0 ! # (

' ! 3 # #

2 # ! # ( # ɶ

( - * 2 2

− (

Definition 2.5: 1 ⊆ ! # 2 !

ɶ( * 0

(

)

∈ ∈

= −

ɶ

ɶ ɶ

ɶ (

*

(

)

⊆ ≤

=

ɶ ɶ ( * ! 0

(

)

∈ ∈

= − (

/

λ

≤ ɶ < J / 6 /−λ ɶ ⋅! =λ ɶ⋅ ɶ(

? 2 # ! ɶ ɶ

(

)

0

∈ − >

ɶ 2 ɶ( ! ɶ >

! ɶ∈ ɶ ɶ >

! 2 # ( 0 ! 0

# # 2 (

# ! # ! #

2 # λ>λ ɶ( ' ! ! # #

K K ! K K ( ' 0

# # ɶ( ' !

! ( C ( ( # 0 ! (

2 ! 8 6 < ( / &% !

(7)

$

Definition 2.6: 1 = "

φ( =

( )

" 2 2 2 *

3 ($ #

# ! # ∉ ( 3

"( ' # # # (

1

LF G

# "

φ < ∈ ∉ ⊆

=

∏∏ ∏

! # !

# * # # # #

# 1 # #

# ( ! # % = 2 *

• / /

* # " −−

N

! # B !

# (

• *# → ! # B # (

7 # ! / (((

= (

? ∈# 2 / /

((( $ (

Definition 2.7: 1 ! 3 0 ( 0

M 2 0 2 2 B

M

> =φ( = # # 2 2 #

( ! M(

Definition 2.8: 1 ! 3 0 ( # ∈

6 # 2 # # #

( 1 ∈ # ( 7 # = (

Definition 2.9: 1 ! 3 0 (

# ∉ = (

3. Minority Proof Cheap Talk Protocol

Theorem 3: 1 2 # < ∈ !

6 2 ! 2 ( ( ∀ ∈ (

0 " 2 0 ! 3

( )

"

$

7 ! 3 : A - 2 ! # 2 #

3 6 0 32 (

% # #

(8)

%

! 8 6 (

Proof of Theorem 3: 7

! 8 6 ( # 2 = ⋅ +/(

3 (

) < != ( / && 2

2 ! ( # 2 D E

Z Z 2 #

(

/ (((

)

∈ ! ( 2 # # 2 #

# ( # #

/ ((( 2 6 # *

( )

% % %

(

% / (((

)

% %

# / ((( (

! # # # 2 #

( )

% %

(

% / (((

)

%∈−

( )

% %∈−

#

(

% / (((

)

%

( )

% % (

6 # #

(

/ (((

)

( # 2

2 * # 2 ! I/ # #

∉ (

1 ∈N

{

}

N # * >

(

)

/E / λ

> − ɶ ∀ ∈ > " =Z ∪φ

(

)

# Z 2 # 2 * #

{

/ ((

}

Z N

(

)

(

/ /

)

, ((( = ((( = ((( 2

%

%

%! # %;

{

/ ((( /

}

∈ + − / =(((= = (

# # ∈Ζ

( # !

# # = /+(((+ 2

(9)

&

# 0

Z ( '

# =

0 ɶ 2 ɶ ! (

2 4 # < !, / & ( ) #

3 2 = ( ' # 2 0

2 #

2 # ( = 2 # 4 #*& 2

#λ ɶ <λ</ 0 2 # /!λ

3 ( ( # # #

( ' ( (

2 #

3 Z( # #

ɶ( = # 2

Proving the protocol is a strong Nash equilibrium:

1 2 # ( 7 2

( ( # ! ! # ∈ ≤ (

! ! 2 3 * ! #

2 ! ! ( 7 2

3 (

& * ' #

# ∈Z ( #

! # # 2 # #

(

* ' 2 2 2

' 2 * #

#

( P P 2 #

&

4 # # # # # Z #

( # # /IQI ( ' # ( (

(10)

(/ A 2 ! # #

(

%

)

%∈−

( )

% %∈− (

! $ * # # #

* # ! * → 2 O

( A 2 ! 6 =

# (

* 7 ! ! ! 2 # 2

2 ! # − ! # %∉ −

% (

7 # λ 0 (

− # ! (

2 ! # // #

! 3 2 # ! # %

% ( # ! 2 # λ>λ ɶ C ( (

# ! # (/ QED

4. Non Existence of a Cheap Talk ( /2) Proof Protocol

' 2 2 ( # 0 . 2 # )

0 2 R # ! 6 ∈ 2

2 ! !

8 6 (

Example 4:1 2 R # * F /* ) +/ * + G ) # @F /* G

2 * F / /G # +@F+/)*) + G 2 *F G( #

*

• ' # 2 # # ( ( # / 2 % # / # 2

# + # # (

• = 2 # # # .(/(

/

0 # (

//7 # # ! ( '

# (

/ = I/ ! 8 6 # 2 I/ #

(11)

/

1 ! 6 2 ! 2

/ /

((( ((( .( ( # #

# +(

Table 4.1: payoff Table 4.2: strong correlated equilibrium

+ + ((( ((( ((( ((( / -/ -/ / ((( / /

F ((( G

. /

F+ ((( + G

- # / / ((( / -/ / ((( # . /

F ((( G

/ /

F+ ((( + G

/ /

(((

# ! 3 2 M @

! 8 6 ( 1 # #

2 +( # # +

# # # # +

# # + # ↔+#∞ ↔+

+

# 2 6 +( ?

O + ## O O + # + # # # ↔ ∈ ↔

=

( 1

+

↔ #

# 2 # 2 #++ +

+++ #( ? ↔+ +( +++ +

↔ ( 1 # +

#+#∞ ↔++ +

+

# 2 + # ( 9 #

, # # =, # #+ = ( 2 # #

*

(

{

/ /

}

)

(

{

(

/ ((( /

) (

/ ((( /

)

}

)

+

↔ ∈ = +

(

{

}

)

+

↔ ∈ ( 7 2 2 2

# /

+

+

+

↔ *

• 1 # +# + 2

+

+

+

↔ 0 ( 9 # , # # =

2

(

/

)

> (

• 1 # +

# +# + 2

+

+

+

↔ 0 2 +

+

↔ ↔+ (

' # ∈# # +

# #

+ * +B B

/ + # / <O B / + # / ( , # # = (

1 #+#∞ ↔+ 2

/

+

↔ = +

+

(12)

//

@/E- /

/E

-= ( 7 #

* # # # # # (/-QED

5. An Example for Applicability – a 5 Player “Chicken” Game

' 2 # ! # S 3 T 2 P #! P

# ( 1 2 *

• ) # 2 * P 2 P P P

• # *

' # # # .(

' # # 2 # 2 (

' # # #

2 # 2 # *

• 7 # -E&* # # (

• / )% ( ( ≠ % 2 # /E/$ )% # 2 #

= # *

• ! 6 2 # # -( 2

! 6 # # (

• ! 2 # ɶ= (

• # # # # 0 8 6

-( # % 2 # / U # 8

6 2 )% # # (

7 2 2 0 / ; < #

/ ; < !, / & -; ? . 2 # ! 3(

# /! 8 6 ( P2 3P

2 # 4 # #

# ( 9 # P P ! 8

6 ( 0 # # (

6. Concluding Remarks:

/ /2 privacy and /3 resiliency*< != ( / &&

/-+ M +( ' ! #

2 3 # ! 0

(13)

/

# ( - 2 2 *

• E ! # ! ' # 2 2 E #

# # (

• E-! # ! 8 2 E- #

# (

# # < != (

# E-! 6 E-! 8 6 0 2

! 3( 2 E !

6 E !

2 < !, / & (

Simultaneousness and private channels:

# 0 #

( ' !2 ( ' A & 2 0

! # 6

# ( #

! # # (

- Two possible extensions of our protocol are:

• ' ! ! 6 ! ! 8 6

< ( / &% 0 ! 3 (

• ' 3! ! ! 6 < # (

4) Cryptographic protocols: ' 2 # #

0 S !2 #T #

! 6 ! 8 6 2 < E *

? / &; " V ; 1 3 ( .; ( $ (

References:

'( C C( ? >( A W( $( C #*

# ( , ( 9 -!$ (

>( / ( ! X A(7( 1 >(C(

) ( 9 # 'V( , " # , 8(W ( &%!- .(

>( / %.( 4 # 5 ( W( ( ) ( / $%! $(

(14)

/-< # '( / ( 2 # (

= > ( /% - !-. (

< != ( ? 2 ( 7 ( / &&( 9 ! # !

0 ( , ( =9 9 /!/ (

< !, )( / &( 9 2 * 0 6 #

S T ! # ( W( ) ( # & / &!/ (

< !, )( -( 9 3 2 ( W( ) ( # / & . !%/(

< <( , <( 7 ( / &%( 9 ! 8 6 ( W( ) ( # . /!/ (

< ( C <( ( 9 6 (

) ( < ( $$ % /!% &(

9 2 V( W( / & ( ( ) % ! .(

) # )( , <( / ( 9 6 ( ' * 9 7 : )

7( < A( ( 8( ( 9 8 2!Y 3 (

) 5 X( / ( ( > 2 ) V ( $ - & !$/ (

( / ( " ( ) & /-./!/-$.(

? C( .( " 2 ( W(

) ( # //. / .!/-/(

? ( / &( 2 ( W( ) ( #

&-$ !& (

A Y( & #! ! 3 ! 0

*EE222( ( ( EK # E #! 0 (

1 3 ( ( , 3 9( ( .( 9 # ) ! 3(

, ( - 9 /!/ (

,( > W( / $( 9 ! 2 #

? ) ( < ( /% //-!/ &(

C( 7 W( / $( 9 ! 6 ? ) ( < ( /% & !//-(

!> , / - ( # 3 # , A / %

*EE222( ( E E E/ - ( 8 & (

8 W( ( / /( 8 ! ( ( ( . &$! (

> # '( / $( 9 ! 6 * ( ? ) ( < ( /% $!% (

> # '( / &( 9 6 ( > ( ) ( C - %! $ (

" ( V W() ( 9 0 # * 2 ! #

References

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