Revelation Principle

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2013
金融システム論：学部冬学期基本科目４
第 4 講：誘因整合性（Incentive Compatibility）,
表明原理（Revelation Principle）, 同値定理（Equivalence Theorems）
経済セミナー（2013 年 8, 9 月号）：連載「オークションとマーケットデザイン」第 6 回
メカニズムデザインをゲーム理論的に分析のための
一般的な基本定理を説明する:
Revelation Principle
Equivalence Theorems
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4.1. 直接メカニズム（Direct Mechanism）
自身のタイプを（直接的に）表明する
Direct (revelation) mechanism ( g, x )
Message spaces
Allocation Rule
Payment Rule
Mi  i
g :  A
x :   Rn
Honest (Truthful, Sincere) Strategy in direct mechanism:
si* ( i )   i
for all
i   i
si* :  i   i
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4.2. 誘因整合性（Incentive Compatibility）
直接メカニズムにおいて
正直戦略プロファイルが均衡になっていることを要求する
Incentive Compatibility in Dominant Strategies (DIC)
Direct Mechanism ( g , x ) is said to be
incentive compatible in dominant strategies
if honest strategy profile s*  ( si* )iN is dominant strategy profile
in associated incomplete information game:
for every i  N , every    , and every     ,
U i ( g ( i ,   i ), xi ( i ,   i ),  )  U i ( g (  ), xi (  ),  )
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Incentive Compatibility in Ex Post Equilibrium (EPIC)
Direct Mechanism ( g , x ) is said to be
incentive compatible in Ex Post Equilibrium
if s*  ( si* )iN is ex post equilibrium
in associated incomplete information game:
for every i  N , every    , and every  i   i ,
U i ( g ( ), xi ( ),  )  U i ( g ( i,   i ), xi ( i,   i ),  )
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Incentive Compatibility in Bayesian Nash Equilibrium (BIC)
Direct Mechanism ( g , x ) is said to be
incentive compatible in Bayesian Nash Equilibrium
if s*  ( si* )iN is BNE in associated Bayesian game:
for every i  N , every  i   i , and every  i   i ,
E[U i ( g ( ), xi ( ),  ) |  i ]  E[U i ( g ( i,   i ), xi ( i,   i ),  ) |  i ]
・DIC ⇒ EPIC ⇒ BIC
・With Private Values: DIC ⇔ EPIC
(Why?)
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4.3. 表明原理（Revelation Principle）
任意の（indirect）mechanism と任意の均衡戦略プロファイルが
達成する配分と支払いは
何らかの Incentive compatible な直接メカニズム（direct mechanism）
によっても達成可能である
∴ ゲーム理論分析において
Incentive compatible な direct mechanisms
だけに
分析を集中してよい！
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Revelation Principle in Dominant strategies
For every indirect mechanism ( M , gˆ , xˆ ) and every dominant strategy profile
in associated incomplete information game,
direct mechanism
g( )  gˆ ( sˆ ( ))
and
( g , x ) specified by
x( )  xˆ ( sˆ ( ))
for all
 
is incentive compatible in dominant strategies.
sˆ
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Revelation Principle in Dominant strategies: Proof:
Note sˆ is dominant strategy profile in ( M , gˆ , xˆ ) :
for every i  N , every    , and every     ,
U i ( gˆ ( sˆ ( i ,   i )), xˆ i ( sˆ ( i ,   i ),  )  U i ( gˆ ( sˆ (  )), xˆ i ( sˆ (  )),  ) .
Since g ( )  gˆ ( sˆ ( )) and x ( )  xˆ ( sˆ ( )) , it follows
U i ( g ( i ,   i )), xi ( i ,   i ),  )  U i ( g (  ), xi (  ),  ) .
Hence, ( g , x ) satisfies DIC.
Q.E.D.
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Revelation Principle in Ex Post Equilibrium
For every indirect Mechanism ( M , gˆ , xˆ ) and every ex post equilibrium sˆ
in associated incomplete information game,
direct mechanism
g( )  gˆ ( sˆ ( ))
and
( g , x ) specified by
x( )  xˆ ( sˆ ( ))
for all
 
is incentive compatible in in ex post equilibrium.
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Revelation Principle in BNE
For every indirect Mechanism ( M , gˆ , xˆ ) and every BNE sˆ
in associated Bayesian game,
direct mechanism
g( )  gˆ ( sˆ ( ))
and
( g , x ) specified by
x( )  xˆ ( sˆ ( ))
for all
 
is incentive compatible in BNE．
以下、（続き：同値定理）へ

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