Non-Truthful Implementation -- Weighted Arithmetic Mean as an ...

Introduction

Implementation

WAM

Non-Truthful Implementation – Weighted Arithmetic Mean as an Example – Norimasa Kobayashi

September 26, 2006

Single-Peaked

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Introduction

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Basic Questions:

Desirability of a Voting Rule What is a desirable voting rule? What if voters do not vote truthfully? Procedural vs. Consequential

WAM

Single-Peaked

Introduction

Implementation

WAM

Single-Peaked

Social Choice and Mechanism Design

Awful lot of impossibility results are known for social choice. Revelation Principle Gibbard-Satterthwaite Theorem We have to discard some desiable conditions to realize satisfactory social choice: Restricted Domains Different equilibrium concepts

Introduction

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WAM

Single-Peaked

Voting Scheme agents N = {1, ..., n} alternatives A ⊂ ui (a) for some i ∈ N.

Introduction

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Direct Revelation Mechanism

Definition (Direct Revelation Mechanism) A direct revelation mechanism of a voting scheme aN is a triple (N, S, U) where: S = ×i S i , where S i = A is i’s set of strategies and U = ×i U i , where U i ((ai )i∈N ) = u i (aN ) is i’s payoff Partiularly, we focus on two modes of the play of the mechanism: Naive Each agent i reports her own ideal point a∗ ∈ A. Nash Agents play some Nash equilibrium of the complete information game of the mechanism.

Introduction

Implementation

WAM

Single-Peaked

Implementation

Definition (Truthful Implementation) A voting scheme is truthfully implementable iff there exists an equilibrium of the direct revelation mechanism that is aN∗ = aN (a1∗ , ..., an∗ ). Definition (Non-truthful Implementation) A voting scheme is non-truthfully implementable iff an equilibrium of the direct revelation mechanism aN = aN (a1 , ..., an ) is efficient.

Introduction

Implementation

WAM

Single-Peaked

Weighted Arithmetic Mean (WAM)

Definition (WAM) WAM is a voting scheme on a convex subset of linear space satisfying X aN (a1 , ..., an ) = w i ai i

Theorem (Anonimity) Voting scheme aN satisfies anonymity iff symmetrial in a1 , ..., an . WAM with equal weights w i = 1/n satisfies anonimity.

Introduction

Implementation

WAM

Applications (Interpretations) of WAM

public facility location choice