Title Introduction VCG Mechanisms

VCG Mechanisms

Theodoros Lykouris

National Technical University of Athens

June 18, 2012

VCG Mechanisms National Technical University of Athens

Title Introduction VCG Mechanisms

Algorithmic Game Theory

Contents

Title

IntroductionAlgorithmic Game TheoryMechanism design with money

VCG MechanismsAuctionsVCG Mechanisms

VCG Mechanisms National Technical University of Athens

Title Introduction VCG Mechanisms

Algorithmic Game Theory

Main research areas of AGT

I Computing equilibria in games

I Quantifying inefficiency of equilibria

I Algorithmic mechanism design

VCG Mechanisms National Technical University of Athens

Title Introduction VCG Mechanisms

Algorithmic Game Theory

Mechanism design

I Mechanism design without moneyI ElectionsI Government policy

I Mechanism design with moneyI AuctionsI Markets

VCG Mechanisms National Technical University of Athens

Title Introduction VCG Mechanisms

Mechanism design with money

Contents

Title

IntroductionAlgorithmic Game TheoryMechanism design with money

VCG MechanismsAuctionsVCG Mechanisms

VCG Mechanisms National Technical University of Athens

Title Introduction VCG Mechanisms

Mechanism design with money

Example

I Let’s go to Rome!!!

VCG Mechanisms National Technical University of Athens

Title Introduction VCG Mechanisms

Mechanism design with money

Setting

I set of players I = {1, 2, . . . , n}I set of alternative outcomes AI valuation function υi : A→ R

I Money measures how much player i values each outcome

I set of possible valuations of player i : Vi = {vi : A→ R}

VCG Mechanisms National Technical University of Athens

Title Introduction VCG Mechanisms

Mechanism design with money

Mechanisms

I outcome function χ : V1 × V2 × · · · × Vn → AI payment function p = (p1, p2, . . . , pn)

I pi : V1 × V2 × · · · × Vn → R

VCG Mechanisms National Technical University of Athens

Title Introduction VCG Mechanisms

Mechanism design with money

Induced Game

I Strategies of player i : Vi

I Utilitiesui (υ1, υ2, . . . , υn) = υi (χ(υ1, υ2, . . . , υn))− pi (υ1, υ2, . . . , υn)

I Goal of player i: Maximize their utility

I Thus, player i may lie for a utilityui (υ−i , υ

′i ) = υi (χ(υ−i , υ

′i ))− pi (υ−i , υ

′i )

VCG Mechanisms National Technical University of Athens

Title Introduction VCG Mechanisms

Mechanism design with money

Truthfulness

I Mechanism (χ, p1, . . . , pn) truthful/incentivecompatible/strategyproof iff

I ∀ player i, ∀ strategy profile υ−iI ∀υ′i ∈ Vi

I u(υ−i , υi ) ≥ u(υ−i , υ′i )

VCG Mechanisms National Technical University of Athens

Title Introduction VCG Mechanisms

Auctions

Contents

Title

IntroductionAlgorithmic Game TheoryMechanism design with money

VCG MechanismsAuctionsVCG Mechanisms

VCG Mechanisms National Technical University of Athens

Title Introduction VCG Mechanisms

Auctions

Example

I Stop traveling! Back to my watch!

VCG Mechanisms National Technical University of Athens

Title Introduction VCG Mechanisms

Auctions

Social welfare

I Maximize combined happiness∑n

i=1 vi (α)

I valuation functions declared (b1, b2, . . . , bn)

I α = χ(b1, b2, . . . , bn)

VCG Mechanisms National Technical University of Athens

Title Introduction VCG Mechanisms

Auctions

Single-item sealed-bid auctions

I n players, 1 itemI valuation function of player i:

I υi if he gets the itemI 0 otherwise

I declares bid bi

VCG Mechanisms National Technical University of Athens

Title Introduction VCG Mechanisms

Auctions

Bad ideas for truthfulness

I No payment!I Everybody would exaggerate their bids

I First-price auctionI The winner will lower their bid to pay less

VCG Mechanisms National Technical University of Athens

Title Introduction VCG Mechanisms

Auctions

Second-price(Vickrey) auction

I Allocation: Player with highest bid biI i = arg(maxi=1,...,n(bi ))

I Payment: Pays second highest bidI p∗i = maxj 6=i (bj)

I Payment doesn’t depend on their bid: truthful

VCG Mechanisms National Technical University of Athens

Title Introduction VCG Mechanisms

VCG Mechanisms

Contents

Title

IntroductionAlgorithmic Game TheoryMechanism design with money

VCG MechanismsAuctionsVCG Mechanisms

VCG Mechanisms National Technical University of Athens

Title Introduction VCG Mechanisms

VCG Mechanisms

Groves Auctions

I Allocation : χ(υ1, υ2, . . . , υn) ∈ arg maxα∈A∑n

i=1(υi (α))

I Payments :pi (υ1, υ2, . . . , υn) = h(υ−i )−

∑j 6=i υj(χ(υ1, υ2, . . . , υn))

I Truthful: ui (χ(υ1, υ2, . . . , υn)) = υi (χ(υ1, υ2, . . . , υn))−pi (χ(υ1, υ2, . . . , υn)) =

∑ni=1(υi (χ(υ1, υ2, . . . , υn))) + h(υ−i )

VCG Mechanisms National Technical University of Athens

Title Introduction VCG Mechanisms

VCG Mechanisms

Clarke Pivot rule

I hi (υ−i ) = maxb∈A∑

i 6=j υi (b)

I maximum welfare when i does not participate

I do not punish honest players

I Each player pays the damage they cause to others by theirpresence

VCG Mechanisms National Technical University of Athens

Title Introduction VCG Mechanisms

VCG Mechanisms

Exercise (1)

I k units to sell to n players

I players have same valuation for all units

I implement VCG!

I when units are diversified: combinatorial auctions

VCG Mechanisms National Technical University of Athens

Title Introduction VCG Mechanisms

VCG Mechanisms

Exercise (2)

I 2-connected graph

I players are the edges

I edges’ weight is their valuation

I we pay players to buy their edge

I implement VCG so that all the vertices are pairwise connected!

VCG Mechanisms National Technical University of Athens