Optimal Shill Bidding in the VCG Mechanism

Transcription

1 Itai Sher University of Minnesota June 5, 2009

2 An Auction for Many Goods Finite collection N of goods. Finite collection I of bidders. A package Z is a subset of N. v i (Z) = bidder i s value for Z. Free Disposal/Monotonicity: Y Z v i (Y ) v i (Z)

3 VCG Auction Package Bob Ann Use the VCG Mechanism: essentially unique efficient auction with truthful bidding a dominant strategy.

4 VCG Auction Package Bob Ann Efficient Allocation Ann Bid = Value for all packages. Ann s payment = total utility in economy without Ann }{{} I minus utility to everyone but Ann in economy with Ann. }{{} II = Bob s utility when Ann absent minus Bob s utility when Ann present = 34 0 = 34 Bob s payment = 0

5 Shill Bidding in VCG Auction Package Bob Carol 10 Dan 45 Efficient Allocation Carol Dan

6 Shill Bidding VCG Auction Package Bob Carol 10 Dan 45 Efficient Allocation Carol Dan Without Carol Bob Dan Carol s payment = total utility in economy without Carol }{{} I minus utility to everyone but Carol in economy with Carol. }{{} II = (5 + 45) (0 + 45) = 5 }{{}}{{} I II

7 Shill Bidding in VCG Auction Package Bob Carol 10 Dan 45 Efficient Allocation Carol Dan Without Dan Bob Dan s payment = = 24 Total payment = = 29 < }{{} 34 Payment under 1 identity When Dan releases {1, 2} to the economy, Bob receives {1, 2} Bob s marginal value for 3 moves up from 5 to 15. Packages {1, 2} and {3} are complements for Bob.

8 Shill Bidding in VCG Auction Bidder decision-maker. VCG mechanism would like to charge externality to decision-maker. Shill bidding cannot identify the decision-maker.

9 Related Papers Yokoo, Sakurai, and Matsubara (2004) Sangvhi and Parkes (2004) Lehmann, Lehmann, and Nisan (2006) Ausubel and Milgrom (2006) Conitzer and Sandholm (2006)

10 Optimal Shill Bidding This paper: Optimal Shill Bidding What is a bidder s optimal use of shills in the extreme case in which she knows or correctly guesses her opponents bids?

11 Ann & Bob Ann = Character who shill bids. Bob = Aggregate Opponent. Ann has many opponents. v b (Z) := utility from allocating Z efficiently to Ann s opponents. Ann s payment and allocation depends only on aggregate bid of others.

12 Shill Bid Cost Minimization Problem (CMP) Ann knows v b. Ann wants to win Z. What is the cheapest way for Ann to win Z using shills?

13 Solving Ann s Overall Problem CMP yields package prices p Shill Z for each Z N. Then Ann s overall problem is: max{v a (Z) p Shill Z : Z N}

14 Properties of Optimal Shill Bidding 1 Each shill bidder should bid single-mindedly: Single-minded bid for P : I value only package P at value r. 2 Different shills should not compete: Shills should bid for non-overlapping packages

15 Shill Bidding in VCG Auction Package Bob Carol 10 Dan 45 Efficient Allocation Carol Dan Ann must win 123.

16 Shill Bidding in VCG Auction Package Bob Carol 10 Dan 45 Efficient Allocation Carol Dan Without Dan Bob Dan s payment = = 24 Total payment = = 29 < }{{} 34 Payment under 1 identity When Dan releases {1, 2} to the economy, Bob receives {1, 2} Bob s marginal value for 3 moves up from 5 to 15. Packages {1, 2} and {3} are complements for Bob. When Dan is removed, Carol s item is re-allocated to Bob, increasing Dan s payment.

17 Shill Bidding in VCG Auction Package Bob Carol 10 Dan 45 Efficient Allocation Carol Dan Without Dan Bob Dan s payment = total utility in economy without Dan }{{} I minus utility to everyone but Dan in economy with Dan. }{{} II = If Carol increases her bid slightly, she will increase II, and leave I unchanged.

18 Shill Bidding in VCG Auction Package Bob Carol 17 Dan 45 Efficient Allocation Carol Dan Previous low payment = 29

19 Shill Bidding in VCG Auction Package Bob Carol 17 Dan 45 Efficient Allocation Carol Dan Without Carol Bob Dan Previous low payment = 29 Carol s payment = 5 Same as before.

20 Shill Bidding in VCG Auction Package Bob Carol 17 Dan Efficient Allocation Carol Dan Without Dan Carol Bob Previous low payment = 29 Carol s payment = 5 Dan s payment = 19 Total payment = = 24

21 Properties of Optimal Shill Bidding 1 Each shill bidder should bid single-mindedly: Single-minded bid for P : I value only package P at value r. 2 Different shills should not compete: Shills should bid for non-overlapping packages 3 Each shill bidder should place a sufficiently high bid on her package so that she still wins the package if some other shill bidder is excluded. Corollary At an optimal shill bid, the payment of any shill bidder for her package is equal to Bob s value for that package.

22 Still Winning When Another Shill is Excluded Corollary At an optimal shill bid, the payment of any shill bidder for her package is equal to Bob s value for that package. Corollary fails at suboptimal shill bids where Carol does not bid sufficiently high to win when Dan is excluded. Package Bob Carol 10 Dan 45 Efficient Allocation Carol Dan Without Dan Bob Dan s payment = = 24 Dan s payment Bob s value for {1, 2} = 19

23 Still Winning When Another Shill is Excluded Corollary At an optimal shill bid, the payment of any shill bidder for her package is equal to Bob s value for that package. Corollary holds when Carol raises her bid Package Bob Carol 17 Dan Efficient Allocation Carol Dan Without Dan Carol Bob Dan s payment = Bob s value for {1, 2} = 19

24 Partitions P 1 P 1 P 2 P 3 P 2 P 1 P 2 P 3 P 1 P 2 P 4 P 1 P 2 P 5 P 1

25 Optimal Shill Bidding Payment Bob P 1 P 1 P 2 P = 24 P 2 P 1 P = 42 P 3 P 1 P = 21 P 4 P 1 P = 24 P 5 P 1 34 = 34

26 Optimal Shill Bidding Payment Bob P 1 P 1 P 2 P = 24 P 2 P 1 P = 42 P 3 P 1 P = 21 P 4 P 1 P = 24 P 5 P 1 34 = 34

27 Winning a Proper Subset Z of N If Ann must win N, she chooses partition P to minimize: v b (P) P P If Ann must win some subset Z of N, she chooses partition P of Z to minimize: v b (P N Z) P P v b (P N Z) := v b (P (N Z)) v b (N Z) = Bob s value for P given that Bob already has everything other than Z.

28 Submodularity at the Top (Submodularity at the Top): For all Z N and all partitions P of Z: v b (Z N Z) P P v b (P N Z) Z is less than the sum of its parts, given that Bob has N Z. SubTop = substitutes property

29 Characterization of Incentive to Shill Theorem 1 ( Submodularity at the Top): For all Z N and all partitions P of Z: v b (Z N Z) v b (P N Z) P P 2 No profitable shill bid for Ann in VCG auction for N. ( valuations for Ann).

30 Related Results: No Incentive to Shill Against Substitutes Ausubel and Milgrom (2002) Lehmann, Lehmann, and Nisan (2006)

31 Winner Determination Problem Theorem The CMP is equivalent to the WDP. Optimal Shill Bidding is equivalent to the Winner Determination Problem (WDP) WDP = Problem of Efficiently Allocating Goods in a Combinatorial Auction Computationally Intractable

32 Pure Complements Supermodularity = Increasing Marginal Utility v b (1) v b ( ) = 5 0 = 5 < v b (12) v b (2) = = 15 < v b (123) v b (23) = = Payment Bob P 1 P 1 P 2 P = 30 P 2 P 1 P = 35 P 3 P 1 P = 45 P 4 P 1 P = 40 P 5 P 1 50 = 50 Supermodularity 1 Shill Per Item Optimal

33 Substitutes vs. Complements Substitutes Truthful Bidding 1 Identity Complements Incentive to Disintegrate 1 Identity per Item Won Mix of Complements Partial Incentive 1 Identity And Substitutes to Disintegrate per Package

34 Equilibrium Theorem Consider two bidder VCG auction with perfect information where both bidders may use shills. Assume: Bidder 1 has strictly supermodular valuation. Bidder 2 wins at least two items at efficient allocation. Then exists Nash equilibrium in undominated strategies where: 1 Bidder 1 bids truthfully. 2 Bidder 2 wins at least two items; Sponsors one single-minded shill per item won. 3 Bidder 2 s equilibrium payoff strictly higher than under truthful bidding.

35 Proof: Nash Equilibrium Bidder 1 bids truthfully one shill per item a best reply. Bidder 2 uses one shill per item Bidder 1 faces additive aggregate valuation. SubTop Truthful bidding a best reply.

36 Proof: Bidder s Strategies Undominated Lemma No shill bidding strategy dominates truthful bidding. So Player 1 s Strategy Undominated.

37 Proof: Bidder s Strategies Undominated Lemma No shill bidding strategy dominates truthful bidding. So Player 1 s Strategy Undominated. For player 2: 1 For each item x must find value v x at which shill for x bids for x such that overall strategy is undominated. 2 Have very little information to work with. 3 Many possible shill bidding strategies to compare against.

38 Equilibrium Corollary Consider two bidder VCG auction with perfect information where both bidders may use shills. Assume Then: Both bidders have strictly supermodular valuations. Both bidders win at least two items at efficient allocation. The VCG mechanism has multiple Nash equilibira in undominated strategies which are not equivalent in terms of payoffs.

39 Summary Optimal Shill Bidding: CMP + Favorite Pacakge Given shill Prices Characterization of Incentive to Shill: CMP WDP. Submodularity at the Top Substitutes Truthful Bidding. Complements Totally Disintegrate. Mix of Substitutes/Complements Partially Disintegrate. Shill Bidding Occurs in Equilibrium. Shill Bidding Neither Dominates Nor Dominated By Truthful Bidding. Consequences for Collusion.