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/
Minority Proof Cheap Talk Protocol
Yuval Heller
1! " #
! $ %& ' (
) * # + ( ( (
, * % !-!$. -&$ % ! ! & /&
0* % !-!$. ! - %
/ " #(
, ( ) 1 ( ' 2 3 3 , ( 1
( # , (
) # 4
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 !
-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 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 # 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 < ( / &% !
$
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 6 (
Proof of Theorem 3: 7
! 8 6 ( # 2 = ⋅ +/(
3 (
) < != ( / && 2
2 ! ( # 2 D E
∈Z Z 2 #
(
/ ((()
∈ ! ( 2 # # 2 ## ( # #
/ ((( 2 6 # *
( )
% %∈ %≠(
% / ((()
%∈ %≠# / ((( (
! # # # 2 #
( )
% %∈(
% / ((()
%∈−( )
% %∈−#
(
% / ((()
%∈( )
% %∈ (6 # #
(
/ ((()
∈ ( # 22 * # 2 ! I/ # #
∉ (
1 ∈N
{
}
∈ ∈N # * >(
)
/E / λ
> − ɶ ∀ ∈ > " =Z ∪φ
(
)
∈# Z 2 # 2 * #
{
/ ((}
⊆Z N(
)
(
/ /)
, ((( = ((( = ((( 2
%
%
∈ %! # %;
{
/ ((( /}
∈ + − / =(((= = (
# # ∈Ζ
( # !
# # = /+(((+ 2
&
# 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 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/ #
/
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
/
+
↔ = +
+
//
@/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
/
# ( - 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( ( ) ( / $%! $(
/-< # '( / ( 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 ! #