Failures of the VCG Mechanism in Combinatorial Auctions

Failures of the VCG Mechanism in Combinatorial Auctions and Exchanges Vincent Conitzer and Tuomas Sandholm Computer Science Department Carnegie Mellon University

VCG and some of its problems • VCG is canonical mechanism for motivating bidders to bid truthfully in auctions & exchanges – In general enough settings, under certain requirements, it is the only mechanism that accomplishes this [Green & Laffont 1977; Lavi, Mu’Alem & Nisan 2003; Yokoo 2003] • Vulnerable to collusion – E. g. in 1 -item second-price auction, highest and second highest bidder can collude to reduce the winner’s price • Does not give highest possible revenue – E. g. Myerson’s auction [1981] gives higher expected revenue • We will show: In combinatorial auctions and exchanges, these two problems get much worse (we will show much)

Relationship between collusion and revenue failures • Informally, we will study the following problem: • Given some of the bids, how bad can the remaining bidders make the outcome? • “Bad” means: remaining bidders pay too little (are paid too much) for goods they receive (provide) • Has clear implications both for: – Revenue guarantees to auctioneer – Collusion possibilities for the remaining bidders

Complexity • Deciding how bad remaining bids can make outcome is computational question – Could be computationally hard • If it is hard, is this hardness good or bad? • Bad: hard for auctioneer to have revenue guarantee • Good: hard for colluders to manipulate

Outline • For each of: – combinatorial auctions – combinatorial reverse auctions – combinatorial exchanges, • we give: – example worst-case scenario for VCG mechanism – characterization of when colluders can make worst-case scenario happen – computational complexity for colluders of deciding whether worst-case scenario is possible

VCG (Clarke) mechanism • Use the allocation a* that maximizes Σi vi(a*)) • Each bidder i pays Σj≠i vj(a*-i)) - Σj≠i vj(a*)) – where a*-i is the allocation that would be optimal if agent i did not exist

Combinatorial auction: Example • • For sale: 2 items, A and B (free disposal) Suppose the bids are: n v 1({A, B}) = n ; v 2({A, B}) = n A One of the bidders wins, pays n Now two more bids arrive: n v 3({A}) = n+1 ; v 4({B}) = n+1 A 3 and 4 win, and don’t pay anything! n+1 n B n+1 • Can show: first-price auction generates Θ(n) revenue in any equilibrium for this example

Characterization for combinatorial auctions • Thrm. The colluders can win all the items and pay nothing iff it is possible to divide the items across colluders so that… • … for every bid by a noncolluder, the items in that bid are divided over at least two colluders • Thrm. NP-complete to decide even with 2 colluders

Combinatorial reverse auction: Example • To be procured: m items, A 1, A 2, …, Am (free disposal) n • v 1({A 1, A 2, …, Am}) = n ; n A 1 A 2 … Am v 2({A 1, A 2, …, Am}) = n • One of the bidders wins, receives n n n • Now m more bids arrive: A 1 A 2 … Am • vi+2({Ai}) = 0 0 • Now the last m bidders win, and each receives n • Can show: first-price auction requires at most n total payment in any pure-strategy equilibrium – Although there are (strange) mixed-strategy equilibria with arbitrarily large expected payments!

Characterization for combinatorial reverse auctions • Let n be the sum of the values of the accepted bids when the colluders do not participate • Thrm. The k colluders can receive a payment of n each iff… • …it is possible to partition the items into k subsets such that none of these subsets can be covered with a set of noncolluder bids with cost < n • Thrm. NP-hard to decide even with 2 colluders – Even in special case where condition holds iff items can be partitioned so that every noncolluder’s bid is split over all subsets in the partition (NP-complete in this case)

What if no free disposal? Example • • • For sale: 2 items, A and B (free disposal) n v 1({A, B}) = n ; v 2({A, B}) = n A One of the bidders wins, pays n Now two more bids arrive: n v 3({A}) = n+m ; v 4({B}) = n+m A 3 and 4 win, and each receives m n+m n B n+m • Can show: first-price auction generates n revenue in any pure-strategy equilibrium – Although there are mixed-strategy equilibria with arbitrarily large expected payments by the auctioneer!

Characterization without free disposal • Thrm. The k colluders can receive a payment of m each (for arbitrary m), iff… • …it is possible to partition the items into k subsets so that no subset can be covered exactly with noncolluder bids • Thrm. NP-complete to decide even with 2 colluders – Even in special case where all but two items have a singleton bid, and there is another bid on this pair

Characterization without free disposal… • Alternatively, • Thrm. Two (or more) colluders can receive a combined payment of m (for arbitrary m), iff… • … there is at least one item with no singleton bid placed on it by a noncolluder • Trivial to decide

Combinatorial exchanges: Example & characterization • Suppose there are (at least) two items, A and B • Let the colluders bid: • v 1(q. A units of A, -q. B units of B) = m ; v 2(-q. A units of A, q. B units of B) = m B A • If m is large enough, both bids will win • If q. A and q. B are large enough, neither bid can without the other • Thus by making m very large, the colluders can receive arbitrarily large payments… • …while their net contribution in goods is nothing 0

Conclusions • We gave worst-case scenarios for VCG in combinatorial auctions, combinatorial reverse auctions, combinatorial exchanges – Colluders cause bad outcome in spite of good other bids – First-price mechanisms seem to fare better • We characterized when colluders can make worstcase scenario happen • We studied the complexity of deciding whether colluders can make worst-case scenario happen – Hardness bad for revenue guarantees, good against collusion

Future research • Are there reasonable restrictions on preferences that prevent such examples? • Are there other mechanism that avoid these pitfalls?