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Title Introduction VCG Mechanisms
VCG Mechanisms
Theodoros Lykouris
National Technical University of Athens
June 18, 2012
VCG Mechanisms National Technical University of Athens
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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
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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
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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
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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
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Title Introduction VCG Mechanisms
Mechanism design with money
Example
I Let’s go to Rome!!!
VCG Mechanisms National Technical University of Athens
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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
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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
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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
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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
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Title Introduction VCG Mechanisms
Auctions
Contents
Title
IntroductionAlgorithmic Game TheoryMechanism design with money
VCG MechanismsAuctionsVCG Mechanisms
VCG Mechanisms National Technical University of Athens
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Title Introduction VCG Mechanisms
Auctions
Example
I Stop traveling! Back to my watch!
VCG Mechanisms National Technical University of Athens
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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
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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
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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
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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
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Title Introduction VCG Mechanisms
VCG Mechanisms
Contents
Title
IntroductionAlgorithmic Game TheoryMechanism design with money
VCG MechanismsAuctionsVCG Mechanisms
VCG Mechanisms National Technical University of Athens
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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
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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
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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
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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