On Ascending Vickrey Auctions for Heterogeneous Objects

Transcription

1 On Ascending Vickrey Auctions for Heterogeneous Objects Sven de Vries James Schummer Rakesh Vohra (Zentrum Mathematik, Munich) (Kellogg School of Management, Northwestern) (Kellogg School of Management, Northwestern) Iowa, April 2004 p. 1/36

2 Overview Main Point Illustrate the use of a recipe" for ascending-auction design. Plan Specify the example environment. Overview of recipe." Execute recipe. Example of resulting auction. Some results. Iowa, April 2004 p. 2/36

3 Combinatorial Auctions Major Examples frequency bandwidth auctions (FCC, etc.) procurement auctions [electricity (Calif., etc.); online procurement (G.M., etc.)] Iowa, April 2004 p. 3/36

4 Combinatorial Auctions Major Examples frequency bandwidth auctions (FCC, etc.) procurement auctions [electricity (Calif., etc.); online procurement (G.M., etc.)] Characteristics non-additive bidder valuations efficiency requires XOR" bids (i.e., bids on bundles) Iowa, April 2004 p. 3/36

5 The VCG (Direct) Mechanism A strategy-proof and efficient" rule must be a Groves mechanism (see Holmström, or Green and Laffont). The addition of individual rationality characterizes the Vickrey Clarke Groves mechanism: Allocate objects to maximize the sum of valuations. Bidder i pays the marginal effect of his presence (on everyone else)." Iowa, April 2004 p. 4/36

6 The VCG (Direct) Mechanism A strategy-proof and efficient" rule must be a Groves mechanism (see Holmström, or Green and Laffont). The addition of individual rationality characterizes the Vickrey Clarke Groves mechanism: Allocate objects to maximize the sum of valuations. Bidder i pays the marginal effect of his presence (on everyone else)." Question: What ascending auction (if any) implements VCG payments? Iowa, April 2004 p. 4/36

7 Motivation Reasons to use Ascending Auctions Winners may reveal less information. Perception of fairness. (More difficult for auctioneer to manipulate.) Bidders may save costs of computation/communication of their valuations. See Cramton (EER 1998). Iowa, April 2004 p. 5/36

8 Model Heterogeneous objects, private values. Set of Goods G. Set of Bidders N. Valuation functions v i (S) for S G. (monotonic, integer, private values) Iowa, April 2004 p. 6/36

9 VCG payments: definition Coalition value: V (M) max µ Γ v i (µ i ). V (N) is achieved by some efficient allocation µ. M Iowa, April 2004 p. 7/36

10 VCG payments: definition Coalition value: V (M) max µ Γ v i (µ i ). V (N) is achieved by some efficient allocation µ. M Then [VCG payment] i V (N \ i) j i v j (µ j) Iowa, April 2004 p. 7/36

11 VCG payments: definition Coalition value: V (M) max µ Γ v i (µ i ). V (N) is achieved by some efficient allocation µ. M Then [VCG payment] i V (N \ i) j i v j (µ j) Hence [VCG payoff] i = v i (µ ) [VCG payment] i = V (N) V (N \ i) = i s marginal value Iowa, April 2004 p. 7/36

12 Ascending Auction: Special Cases Single object: English auction. (Japanese clock/button auction.) Iowa, April 2004 p. 8/36

13 Ascending Auction: Special Cases Single object: English auction. (Japanese clock/button auction.) Assignment problem: Bidders consume one object: v i (S) = max g S v i ({g}). Demange, Gale, and Sotomayor, Raise prices within minimally overdemanded set. Yields VCG payoffs. Iowa, April 2004 p. 8/36

14 Special Cases Homogeneous Goods: Objects are indistinguishable: H = H v i (H) =v i (H ) Ausubel s clinching" auction. Assume decreasing marginal values. Bidders reduce demand as price rises. When a bidder clinches" a unit, he buys immediately. Yields VCG. Iowa, April 2004 p. 9/36

15 Special Cases Homogeneous Goods: Objects are indistinguishable: H = H v i (H) =v i (H ) Ausubel s clinching" auction. Assume decreasing marginal values. Bidders reduce demand as price rises. When a bidder clinches" a unit, he buys immediately. Yields VCG. Note: Different bidders may pay different amounts for the same consumption VCG is not envy-free. Hence, we cannot use anonymous prices in the general case. Iowa, April 2004 p. 9/36

16 Other related literature Kelso Crawford (1984) and Gul Stacchetti (1999, 2000) provide an ascending auction that results in Walrasian prices under Gross Substitutes." (Uses additive prices.) G&S Impossibility result: there exists no ascending VCG auction that uses additive bundle prices. Iowa, April 2004 p. 10/36

17 Other related literature Kelso Crawford (1984) and Gul Stacchetti (1999, 2000) provide an ascending auction that results in Walrasian prices under Gross Substitutes." (Uses additive prices.) G&S Impossibility result: there exists no ascending VCG auction that uses additive bundle prices. Ausubel (2002) achieves VCG by using n +1versions of a Walrasian mechanism, such as Gul and Stacchetti. Iowa, April 2004 p. 10/36

18 Other related literature Kelso Crawford (1984) and Gul Stacchetti (1999, 2000) provide an ascending auction that results in Walrasian prices under Gross Substitutes." (Uses additive prices.) G&S Impossibility result: there exists no ascending VCG auction that uses additive bundle prices. Ausubel (2002) achieves VCG by using n +1versions of a Walrasian mechanism, such as Gul and Stacchetti. Ausubel Milgrom (2002) Package auction with proxies (revelation mechanism). Bids ascend; do (implicit) prices? Iowa, April 2004 p. 10/36

19 A Linear Programming Connection The Package Assignment Model," Bikhchandani and Ostroy JET, Linear Programming and Vickrey Auctions," Bikhchandani, de Vries, Schummer, and Vohra (2002). The connection: VCG payoff = bidder s marginal value (to V (N)). LP dual variable = marginal effect of changing a constraint. Iowa, April 2004 p. 11/36

20 A Linear Programming Connection The Package Assignment Model," Bikhchandani and Ostroy JET, Linear Programming and Vickrey Auctions," Bikhchandani, de Vries, Schummer, and Vohra (2002). The connection: VCG payoff = bidder s marginal value (to V (N)). LP dual variable = marginal effect of changing a constraint. Possibility: For some linear program, does (VCG payoff = dual variable)? Iowa, April 2004 p. 11/36

21 Consequences If (VCG payoffs = dual variables)... Computational Consequences Sealed-bid VCG auctions can be computed with a single linear program (as opposed to computing each V (N \ i)). See Bikhchandani,de Vries,Schummer,Vohra Iowa, April 2004 p. 12/36

22 Consequences If (VCG payoffs = dual variables)... Computational Consequences Sealed-bid VCG auctions can be computed with a single linear program (as opposed to computing each V (N \ i)). See Bikhchandani,de Vries,Schummer,Vohra Implementation Consequences (this paper) Interpret LP-solving algorithms as (ascending?) auctions. Iowa, April 2004 p. 12/36

23 Primal-Dual & Ascending Auctions Primal-Dual Algorithm: tells us how to converge to optimal dual variables. Primal-Dual Auction 1. Guess a solution for the Dual. 2. Try to find a Primal solution that satisfies CS. (Defines RP.) 3. If none exists, DRP tells you how to change your guess. Iowa, April 2004 p. 13/36

24 Primal-Dual & Ascending Auctions Primal-Dual Algorithm: tells us how to converge to optimal dual variables. Primal-Dual Auction 1. Guess a solution for the Dual. 2. Try to find a Primal solution that satisfies CS. (Defines RP.) 3. If none exists, DRP tells you how to change your guess. 1. Set prices low. 2. Try to find an allocation that satisfies bidders demand. 3. If none exists, adjust prices. Iowa, April 2004 p. 13/36

25 Not all LP s work. max j N s.t. S g S G v i (S)y j (S) S G j N y j (S) 1 g G y j (S) 1 j N 0 y j (S) 1 S G, j N Iowa, April 2004 p. 14/36

26 Not all LP s work. max j N s.t. S g S G v i (S)y j (S) S G j N y j (S) 1 g G y j (S) 1 j N 0 y j (S) 1 S G, j N Problem: fractional solutions (e.g. 3 chopsticks/3 bidders). Iowa, April 2004 p. 14/36

27 Primal (P) This one (sometimes) works. max v j (S)y j (S) j N S G s.t. y j (S) = δ µ j N, S G (p j (S)) S G µ:µ j =S y j (S) 1 j N (π j ) δ µ =1 (π s ) µ Γ 0 y j (S) S G, j N 0 δ µ µ Γ Iowa, April 2004 p. 15/36

28 Dual (D) min j N π j + π s s.t. π j + p j (S) v j (S) p j ( ) v j ( ) j N, S G j N π s j N p j (µ j ) 0 µ Γ p j (S) 0 π j 0 π s 0 j N, S G j N Iowa, April 2004 p. 16/36

29 Applying the PD algorithm approach If primal/dual variables are optimal, then they satisfy Complementary Slackness. Verbally, the CS conditions are that... A positive-surplus bidder must get something. Only demanded bundles can be assigned. Only seller-revenue-maximizing assignments can be used. Let N + = positive-surplus bidders. Γ (D) =revenue-maximizing assignments that assign only demanded bundles. Iowa, April 2004 p. 17/36

30 RP: add CS conditions to (P) (RP) y j (S) = µ Γ (D):µ j =S δ µ j N, S D j { } y j (S) = 1 j N + S D j y j (S) 1 j N \ N + S D j δ µ =1 µ Γ (D) 0 y j (S) j N, S D j { } 0 δ µ µ Γ (D) Note: two differences from (P); no new dual variables. Iowa, April 2004 p. 18/36

31 Dual of Restricted Primal (DRP) If (RP) is infeasible, adjust p j (S) s in proportion to ρ j (S) s... (DRP) λ s + λ j < 0 j N λ j + ρ j (S) 0 j N, S D j ρ j ( ) 0 j N + λ s ρ j (µ j ) 0 j N µ Γ (D) ρ j (S) 0 j N, S D j λ s 0 λ j 0 j N + λ j 0 j / N + Iowa, April 2004 p. 19/36

32 Do prices ascend? Find a solution to (DRP) such that ρ j (S) 0 for all j, S. Does it exist? Not if prices are too high." Iowa, April 2004 p. 20/36

33 Do prices ascend? Find a solution to (DRP) such that ρ j (S) 0 for all j, S. Does it exist? Not if prices are too high." Overdemand holds if the seller can maximize revenue at current prices by allocating only demanded bundles. Iowa, April 2004 p. 20/36

34 Do prices ascend? Find a solution to (DRP) such that ρ j (S) 0 for all j, S. Does it exist? Not if prices are too high." Overdemand holds if the seller can maximize revenue at current prices by allocating only demanded bundles. Say coalition K N + is undersupplied if they cannot be simultaneously satisfied at some revenue-maximizing assignment. K is minimally undersupplied if no K K is undersupplied. Iowa, April 2004 p. 20/36

35 Existence of non-negative price increase Theorem 1 If overdemand holds, then for any minimally undersupplied set K, there is a solution to (DRP) where ρ j (S) =1for j K, S D j, and ρ j (S) =0otherwise. We need overdemand to continue to hold after such a price adjustment, in order to use this. Iowa, April 2004 p. 21/36

36 Existence of non-negative price increase Theorem 1 If overdemand holds, then for any minimally undersupplied set K, there is a solution to (DRP) where ρ j (S) =1for j K, S D j, and ρ j (S) =0otherwise. We need overdemand to continue to hold after such a price adjustment, in order to use this. Fortunately... Theorem 2 Starting the algorithm at p j (S) 0, overdemand holds after each such price adjustment, until termination (when all bidders are satisfied). Iowa, April 2004 p. 21/36

37 The PD Auction 1. Set round-1 prices to zero: p 1 j (S) 0 for all j N, S G. 2. With respect to current (round-t) prices, ask the bidders to reveal their demand D t j. 3. No overdemand done. Otherwise Overdemand adjust prices: choose a minimally undersupplied set K, and set p t+1 j (S) =p t j (S)+1for each j K, S Dj t ; all other prices are unchanged. Return to Step 2. Iowa, April 2004 p. 22/36

38 The PD Auction 1. Set round-1 prices to zero: p 1 j (S) 0 for all j N, S G. 2. With respect to current (round-t) prices, ask the bidders to reveal their demand D t j. 3. No overdemand done. Otherwise Overdemand adjust prices: choose a minimally undersupplied set K, and set p t+1 j (S) =p t j (S)+1for each j K, S Dj t ; all other prices are unchanged. Return to Step 2. Corollary 2 A PD Auction weakly increases all prices p j (S) until termination and yields an efficient assignment. Iowa, April 2004 p. 22/36

39 Example Round Bidder 1 Bidder 2 Bidder 3 Seller Bundle a b c ab ac bc abc a b c ab ac bc abc a b c ab ac bc abc Sur- Value plus 1 Price Surplus Undersupply Any pair of bidders is minimally undersupplied; select {1,2}. Iowa, April 2004 p. 23/36

40 Example Round Bidder 1 Bidder 2 Bidder 3 Seller Bundle a b c ab ac bc abc a b c ab ac bc abc a b c ab ac bc abc Sur- Value plus 1 Price Surplus Undersupply Any pair of bidders is minimally undersupplied; select {1,2}. 2 Price Surplus Undersupply Bidder 3 himself is minimally undersupplied (as is {1,2}); select {3}. Iowa, April 2004 p. 24/36

41 Example Round Bidder 1 Bidder 2 Bidder 3 Seller Bundle a b c ab ac bc abc a b c ab ac bc abc a b c ab ac bc abc Sur- Value plus 1 Price Surplus Undersupply Any pair of bidders is minimally undersupplied; select {1,2}. 2 Price Surplus Undersupply Bidder 3 himself is minimally undersupplied (as is {1,2}); select {3}. 3 Price Surplus Undersupply Same as Round 1. Iowa, April 2004 p. 25/36

42 Example Round Bidder 1 Bidder 2 Bidder 3 Seller Bundle a b c ab ac bc abc a b c ab ac bc abc a b c ab ac bc abc Sur- Value plus 1 Price Surplus Undersupply Any pair of bidders is minimally undersupplied; select {1,2}. 2 Price Surplus Undersupply Bidder 3 himself is minimally undersupplied (as is {1,2}); select {3}. 3 Price Surplus Undersupply Same as Round 1. 4 Price Surplus Undersupply The unique minimally undersupplied set is {3} (since the assignment (bc, a, ) is in Γ (D)). Iowa, April 2004 p. 26/36

43 Example Round Bidder 1 Bidder 2 Bidder 3 Seller Bundle a b c ab ac bc abc a b c ab ac bc abc a b c ab ac bc abc Sur- Value plus 1 Price Surplus Undersupply Any pair of bidders is minimally undersupplied; select {1,2}. 2 Price Surplus Undersupply Bidder 3 himself is minimally undersupplied (as is {1,2}); select {3}. 3 Price Surplus Undersupply Same as Round 1. 4 Price Surplus Undersupply The unique minimally undersupplied set is {3} (since the assignment (bc, a, ) is in Γ (D)). 5 Price Surplus Undersupply Only {1,3} and {2,3} are minimally undersupplied; select {2,3}. Iowa, April 2004 p. 27/36

44 Example Round Bidder 1 Bidder 2 Bidder 3 Seller Bundle a b c ab ac bc abc a b c ab ac bc abc a b c ab ac bc abc Sur- Value plus 1 Price Surplus Undersupply Any pair of bidders is minimally undersupplied; select {1,2}. 2 Price Surplus Undersupply Bidder 3 himself is minimally undersupplied (as is {1,2}); select {3}. 3 Price Surplus Undersupply Same as Round 1. 4 Price Surplus Undersupply The unique minimally undersupplied set is {3} (since the assignment (bc, a, ) is in Γ (D)). 5 Price Surplus Undersupply Only {1,3} and {2,3} are minimally undersupplied; select {2,3}. 6 Price Surplus Undersupply The unique minimally undersupplied set is {1,3}. Iowa, April 2004 p. 28/36

45 Example Round Bidder 1 Bidder 2 Bidder 3 Seller Bundle a b c ab ac bc abc a b c ab ac bc abc a b c ab ac bc abc Sur- Value plus 1 Price Surplus Undersupply Any pair of bidders is minimally undersupplied; select {1,2}. 2 Price Surplus Undersupply Bidder 3 himself is minimally undersupplied (as is {1,2}); select {3}. 3 Price Surplus Undersupply Same as Round 1. 4 Price Surplus Undersupply The unique minimally undersupplied set is {3} (since the assignment (bc, a, ) is in Γ (D)). 5 Price Surplus Undersupply Only {1,3} and {2,3} are minimally undersupplied; select {2,3}. 6 Price Surplus Undersupply The unique minimally undersupplied set is {1,3}. 7 Price Surplus Undersupply The unique minimally undersupplied set is {1,2,3}. Iowa, April 2004 p. 29/36

46 Example Round Bidder 1 Bidder 2 Bidder 3 Seller Bundle a b c ab ac bc abc a b c ab ac bc abc a b c ab ac bc abc Sur- Value plus 1 Price Surplus Undersupply Any pair of bidders is minimally undersupplied; select {1,2}. 2 Price Surplus Undersupply Bidder 3 himself is minimally undersupplied (as is {1,2}); select {3}. 3 Price Surplus Undersupply Same as Round 1. 4 Price Surplus Undersupply The unique minimally undersupplied set is {3} (since the assignment (bc, a, ) is in Γ (D)). 5 Price Surplus Undersupply Only {1,3} and {2,3} are minimally undersupplied; select {2,3}. 6 Price Surplus Undersupply The unique minimally undersupplied set is {1,3}. 7 Price Surplus Undersupply The unique minimally undersupplied set is {1,2,3}. 8 Price Surplus Iowa, April 2004 p. 30/36

47 Obtaining VCG payoffs Agents are Substitutes if V (N) V (N \ M) j M[V (N) V (N \ j)] M N Theorem 3 (B&O) There exists a (D) solution yielding VCG payments iff Agents are Substitutes. Hence this is a necessary condition for a PD auction to yield such payments. Iowa, April 2004 p. 31/36

48 Obtaining VCG payoffs We have the following sufficiency. Theorem 4 If V () is submodular, then any PD Auction (i.e. regardless of choice of undersupplied sets) results in VCG payments. This submodularity condition is also a sufficient condition for Ausubel and Milgrom. Iowa, April 2004 p. 32/36

49 Obtaining VCG payoffs We have the following sufficiency. Theorem 4 If V () is submodular, then any PD Auction (i.e. regardless of choice of undersupplied sets) results in VCG payments. This submodularity condition is also a sufficient condition for Ausubel and Milgrom. If AS holds but submodularity does not: may or may not achieve VCG, depending on choice of undersupplied set. Iowa, April 2004 p. 32/36

50 Obtaining VCG payoffs Incentives: Corollary 2 (Under appropriately specified auction rules,) when submodularity holds, it is an ex-post equilibrium to bid truthfully in a PD auction. This is a revelation principle type of argument: truthful behavior yields VCG, so untruthful behavior must hurt. Iowa, April 2004 p. 33/36

51 Obtaining VCG payoffs Incentives: Corollary 2 (Under appropriately specified auction rules,) when submodularity holds, it is an ex-post equilibrium to bid truthfully in a PD auction. This is a revelation principle type of argument: truthful behavior yields VCG, so untruthful behavior must hurt. An observation (of little practical value?): Theorem 6 If the undersupplied set is chosen to exclude bidder j whenever possible, then the PD Auction yields bidder j s VCG payment. Iowa, April 2004 p. 33/36

52 Necessity of substitutability Consider Bundle Bidder a b ab α 2+α 8+α with α [ 1, 1]. (Note agents are substitutes" fails wrt {2,3}.) VCG: 2 pays 8 α for b; 3pays6fora. But α cannot be inferred until 3 demands", which requires p 3 (a) 8+α. Iowa, April 2004 p. 34/36

53 Necessity of substitutability More generally... Theorem 7 (Two objects.) If one bidder can have valuation for the two objects v i (ab) >v i (a)+v i (b), then on some domain of valuations, no ascending auction can implement VCG payments. Iowa, April 2004 p. 35/36

54 Summary Based on the observation that dual variables can represent VCG payments... Iowa, April 2004 p. 36/36

55 Summary Based on the observation that dual variables can represent VCG payments... This serves as motivation to interpret LP algorithms (like P-D) as auctions. Iowa, April 2004 p. 36/36

56 Summary Based on the observation that dual variables can represent VCG payments... This serves as motivation to interpret LP algorithms (like P-D) as auctions. Submodularity is sufficient (and substitutability is essentially necessary) for ascending VCG implementation. Iowa, April 2004 p. 36/36

57 Summary Based on the observation that dual variables can represent VCG payments... This serves as motivation to interpret LP algorithms (like P-D) as auctions. Submodularity is sufficient (and substitutability is essentially necessary) for ascending VCG implementation. DGS s minimally overdemanded sets of objects" generalize to minimally undersupplied sets of bidders." Iowa, April 2004 p. 36/36