Minority-proofcheap-talkprotocol Heller,Yuval MunichPersonalRePEcArchive

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Minority-proof cheap-talk protocol

Heller, Yuval

Tel-Aviv University

19 August 2005

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

MPRA Paper No. 54906, posted 31 Mar 2014 15:26 UTC

Minority Proof Cheap Talk Protocol

Yuval Heller

! " #

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) * # + ( ( (

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0* % !-!$. ! - %

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

( # , (

) # 4

# # 2 (

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 !

6 ! (

- 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 # # (

. 2 ) 5B / ! ! 8 )6 (

.

5 2 * ( -

( . 2 2 ≥ (

0 ( 7 $(

2. Model and Definitions

∈ ∈

= 2 =F/ (( G !

# # ∈ # B ! #

# B # !

=

∏

# ( (

! # ( ? ⊆

=

∏

^{

^}

# ( ( ?

∈ ∈ 2 _H ∈ _H − −

−

=

∑

* = _H ( 7 # ∈ #

/ (((

= ∈ = ^/ ⋅ ⋅((( ( # ⊆ 2 # ∈

# ∈

=

∏

# ⊂ ! # ( ?

∈ 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 =

∑

# ! #

# #

2 # # (

Definition 2.3: ∈ ! 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( * ^{ɶ =} _{ɶ∈ ɶ ∈}⁰

(

)

* ^{ɶ =} _{⊆ ≤}

(

)

(

)

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 < ( / &% !

6 : 7 / $ (

$

Definition 2.6: 1 = _{∈ ∈} "

φ( ⁼

( )

3 (^$ #

# ! # ∉ ( 3

"( ' # # # (

1

LF G

# "

φ

< ∈ ∉ ⊆

=

∏∏ ∏

# * # # # #

# 1 #_∞ #_∞

# ( ! # ^% = 2 *

• * # " ^−/−/

∈_N →

∪

# (

• *#_∞ → ! # 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 (

% # # # #_∞ #_∞ # σ! # (

%

! 8 6 (

Proof of Theorem 3: 7

! 8 6 ( # 2 = ⋅ +/(

3 (

) < != ( / && 2

2 ! ( # 2 D E

∈Z Z 2 #

(

)

# ( # #

/ ((( 2 6 # *

( )

(

)

%∈ %≠

# _/ ((( (

! # # # 2 #

( )

(

)

( )

#

(

)

( )

6 # #

(

)

∈ ( # 2

2 * # 2 ! I/ # #

∉ (

1 ∈N

{ }

( )

/E / λ

> − ɶ ∀ ∈ > " =Z ∪φ

( )

# Z 2 # 2 * #

{

}

( )

(

)

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

%

∈ % _%! # %;

{

}

∈ + − _/ =(((= = (

# # ∈Ζ

( # !

# # = _/+(((+ 2

O # * = ! ( '

&

# 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 ≠( ' # ( (

A & (

(^/ 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/ #

! 2 (

/

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

+ # +

# # #

↔ ∈ ↔

=

∪

# 2 # 2 # _↔+

+

↔ +

+

↔

+

↔ + #( ? _↔+

+( ↔+

↔+ +

+

↔ ( 1 # #⁺ _↔+∈#_{∞ ↔+}

↔+ +

+

↔

# 2 + # ( 9 #

, # # =, # #⁺ = ( 2 # #

* ^↔+∈

( {

} )

( { ⁽

^{) (}

⁾ } )

( ^{{ }} )

# / ↔+ +

+

↔ *

• 1 # _↔+∈#_{∞ ↔+} 2

+

↔ +

+

↔ 0 ( 9 # , # # =

2

(

)

• 1 # #⁺ _↔+∈#_{∞ ↔+} 2

+

↔ +

+

↔ 0 2

+

+

↔ ↔+ (

' # ∈# # ∈# #⁺ + * +B B

/ + # ^/ <O B

/ + # ^/ ( , # # = (

1 # _↔+∈#_{∞ ↔+} 2 ^/

↔+ =

+

+

↔ = ( = , # #

//

@/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

3 # # (

/

# ( - 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( ( ) ( / $%! $(

>( A ( -( 1 3( ) %/ $ /$/ !/$$ (

/-

< # '( / ( 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 ! #

( ) % /& -!/ %(