Parkes CS 700 1'&$%CS 700: Computational MechanismDesignDavid C. [email protected] 2Fall, 2008 EPFLParkes CS 700 2'&$%Lesson Plan• VCG mechanism• Impossibility Results• Monotonicity, Price-Based Characterization• Combinatorial auctions• Single-minded CAsParkes CS 700 3'&$%Reading[IMD] Introduction to mechanism design (for computerscientists), by N. Nisan. In N. Nisan, T. Roughgarden,E. Tardos, and V. Vazirani, editors, Algorithmic GameTheory, chapter 9. Cambridge University Press, 2007.[CAs] Combinatorial Auctions, by L. Blumrosen and N.Nisan. In N. Nisan, T. Roughgarden, E. Tardos, and V.Vazirani, editors, Algorithmic Game Theory, chapter11. Cambridge University Press, 2007.Parkes CS 700 4'&$%Vickrey-Clarke-Groves Mechanism[Vickrey61, Clarke71, Groves73]Def. [VCG mechanism] Outcome functionf(v) ∈ arg maxa∈Aivi(a)and payment functionpi(v)=j =ivj(f(v−i)) −j =ivj(f(vi,v−i))This is the marginal externality imposed on the otheragents. (In computing f(v−i) adjust the feasiblealternatives A as necessary; e.g., if i is selling a goodthen can no longer allocate.)Thm. The VCG mechanism is strategyproof andefficient.Parkes CS 700 5'&$%[Note 1:]Each agent’s equilibrium utility is:πvick,i= vi(a) − [vi(a) − V (N)+V (N \ i)]= V (N ) − V (N \ i),where a = f(v),andV (N)=maxa∈A ivi(a) andV (N \ i)=maxa∈A j =ivj(a).The payoff to agent i is its marginal contribution to thewelfare of the system. (⇒ ex post IR, or participation.)[Note 2:] Can also write as:pi(v)=vi(a) − [V (N) − V (N \ i)]where a = f(v); second term is “VCG discount.”[Note 3:] VCG may still run at a deficit, but only ifsome agents have positive marginal externality. [e.g., aseller.]Parkes CS 700 6'&$%Example: Vickrey auctionConsider the Vickrey auction for a single item.Let b1denote the highest bid, and b2thesecond-highest bid.Every agent pays marginal externality imposed on restof system.For agent 1 (with highest bid), this is:p1= b2− 0=b2For all other agents, this is 0 (because agent 1 winswith or without them in the system.)⇒ VCG mechanism reduces to sealed-bid secondprice auctionParkes CS 700 7'&$%Aside: History• Second price auction used by stamp collectors formail-in auctions since the mid nineteenth century.• German writer Goethe (1749-1832), wrote on Jan16, 1797 to his publisher“I am inclined to offer Mr. Vieweg... an epic poem...Concerning the royality we will proceed as follows: Iwill hand over to Mr. Counsel Bottiger a sealed notewhich contains my demand, and I wait for what Mr.Vieweg will suggest to offer for my work. If his offer islower than my demand, then I take my note back,unopened and the negotiation is broken. If, however, hisoffer is higher, then I will not ask for more than what iswritten in the note to be opened by Mr. Bottiger.”• In a letter to Boisseree, dated Jan 12, 1828“Let me... name the main evil. It is this: the publisheralways knows the profit to himself and his family,whereas the author is totally in the dark.”Parkes CS 700 8'&$%Example: Shortest Path.[Nisan 99]40ST3060102050204030Biconnected graph, G =(N, E),costcl≥ 0 per edgel ∈ E, edges srategic. Assume large value V to sendmessage.Goal: route packets along the lowest-cost path from Sto T .VCG Payment edge e:pvick,l= −cl−(V − dG) − (V − dG/l)= −cl− (dG/l− dG)Parkes CS 700 9'&$%Example: Multi-unit Auctionm units of a homogeneous item. First, consider thespecial case in which each bidder demands a singleunit.Letvi≥ 0 denote the value of bidder i.Def. The VCG auction for this special case sells theitems to them highest bidders, each pays the m +1sthighest bid price.pi= bi−⎛⎝j≤mbj−j≤m+1,j =ibj⎞⎠= bm+1Parkes CS 700 10'&$%Applications of VCG• Double auction (one buyer with value vb, one sellerwith value vs) [Exercise]• Multi-unit auction with m units to sell of an identicalgood and n buyers, each with private value viforexactly qiunits• Reverse, single item auction. (One buyer that must buythe item, and n>1 sellers each with private value vifor the item.)• Public project of cost C>0 and n agents, each withprivate value vifor the project. Want to build theproject if and only if in the public interest.Parkes CS 700 11'&$%From positive to...• Median rule: Single-peaked preferences, Paretooptimal, Strategyproof, No payments• Groves and VCG mechanism: Quasi-linearpreferences, Efficient, Strategyproof, payments... lots of impossibility resutls as well!Parkes CS 700 12'&$%Gibbard-SatterthwaiteImpossibility[Arrow 51, Gibbard & Satterthwaite 73, 75]Consider SCF, f(θ), an outcome space O, and a typespace Θithat allows all possible strict preferenceorderings overO ,foreveryi.Def. [Dictatorial] Agent i is a dictator in SCF f if forall θ1,...,θn, f(θ) is always agent i’s most preferredoutcome. SCF f is dictorial if some i is a dictator.[Gibbard-Satterthwaite Impossibility] ASCFf(θ)onto O where |O| ≥ 3 is implementable in dominantstrategies (strategyproof) if and only if it is dictatorial.Proof: as a corollary Arrow’s theorem (see p.214-215in Nisan IMD chapter.)Parkes CS 700 13'&$%Implications• Voting systems: If at least 3 outcomes, then allinteresting voting protocols will be manipulable.– the majority vote rule is an obviouscounterexample for 2 outcomes⇒ Consequences for design of computational systemsto promote group decision makingWhat to do?Parkes CS 700 14'&$%Two circumventions• Median-voting rule• VCG mechanismBoth results circumvent GS by providing additionalstructure on the type space.General/No-transfer ⊂ Quasi-linear/TransferParkes CS 700 15'&$%Roberts’ Theorem• Def. ASCFfw,cis an affine maximizer if forsome agent weights w1,...,wn∈ R≥0and someoutcome weights ca∈ R for every a ∈ A,thatf(v) ∈ arg maxa∈A(ca+ iwivi(a)).• Def. A weighted VCG mechanism selectsoutcome fw,c(v) and collects paymentpi(v)=hi(v−i) − j=i(wj/wi)vj(a) − ca/wifrom every agent i,wherehiis an arbitraryfunction that does not depend on vi.Claim: Weighted-VCG mechanism is strategyproof.• Thm. If |A|≥3, f is onto, Vi= RAfor every i,and (f,p1,...,pn) is strategyproof then f is anaffine maximizer.[Proof in Chapter 12 of AGT book]Parkes CS 700 16'&$%Centrality of Groves mechanisms• Thm. Roberts’79: if unconstrained valuationdomain Vi= RAand SP, must be Groves.• Thm. Holmstrom’80: if smoothly connectedpreferences, efficient and SP, must be Grovesmechanism.• Thm. Krishna and