1. 1 Auctions -2 Debasis Mishra QIP Short-Term Course on Electronic Commerce Indian Institute of Science, Bangalore February 17, 2006

2. QIP Course on E-Commerce : Auctions- 22 Outline Combinatorial Auctions Combinatorial Auctions The VCG Auction The VCG Auction Iterative VCG Auctions Iterative VCG Auctions –One-to-one assignment –General case: iBundle auction Winner Determination Problem Winner Determination Problem

3. February 17, 2006QIP Course on E-Commerce : Auctions- 23 Combinatorial Auctions Sale of multiple items simultaneously. Sale of multiple items simultaneously. Bidders can have non-additive values on items. Bidders can have non-additive values on items. Allows for bids on bundles of items. Allows for bids on bundles of items. An example: a seller wants to sell two items: An example: a seller wants to sell two items: –(a) a shopping complex in Goa –(b) a shopping complex in Mumbai. Two buyers with values: Two buyers with values: –5 for a, 6 for b, and 15 for a+b, –10 for a, 8 for b, and 10 for a+b.

4. February 17, 2006QIP Course on E-Commerce : Auctions- 24 Applications in Practice Spectrum wave auctions in different countries: US, European nations, and India too. Spectrum wave auctions in different countries: US, European nations, and India too. Transportation lane auctions: London bus routes, Home Depot in US. Transportation lane auctions: London bus routes, Home Depot in US. Airport time-slots by FAA (US). Airport time-slots by FAA (US). Sponsored search auctions by Google, Yahoo!, and Microsoft. Sponsored search auctions by Google, Yahoo!, and Microsoft.

5. February 17, 2006QIP Course on E-Commerce : Auctions- 25 Efficient Mechanism Design An economic and algorithmic view Input: “Bids” from buyers, Output: an allocation (assignment of items to buyers) and payments of buyers. Input: “Bids” from buyers, Output: an allocation (assignment of items to buyers) and payments of buyers. Allocation – efficient allocation: one that maximizes the total value of buyers or total payoff/welfare of the system. Allocation – efficient allocation: one that maximizes the total value of buyers or total payoff/welfare of the system. Payment – to make money AND to give an incentive to participate. Payment – to make money AND to give an incentive to participate.

6. February 17, 2006QIP Course on E-Commerce : Auctions- 26 Why are Incentives Important Bidders often indulge in strategizing in “badly” designed auctions – a fact from spectrum auctions in US and Europe: leads to low revenue and loss in efficiency. Bidders often indulge in strategizing in “badly” designed auctions – a fact from spectrum auctions in US and Europe: leads to low revenue and loss in efficiency. Strategizing is not easy in combinatorial auctions. Strategizing is not easy in combinatorial auctions. Incentive schemes makes strategies straightforward – no need to strategize, just bid in a straightforward manner. Leads to savings for bidders in terms of “auction preparation costs”. Incentive schemes makes strategies straightforward – no need to strategize, just bid in a straightforward manner. Leads to savings for bidders in terms of “auction preparation costs”.

7. February 17, 2006QIP Course on E-Commerce : Auctions- 27 The Model Set of items N={1,…,n}. Set of items N={1,…,n}. Set of buyers M={1,…,m}. Set of buyers M={1,…,m}. Bundles – any subset of items of N. Bundles – any subset of items of N. Buyers have values on bundles: v(i,S) Buyers have values on bundles: v(i,S) –A1- S is a subset of T means v(i,S) <= v(i,T) –A2- v(i,φ)=0 –A3 - if i pays amount p and gets bundle S, his payoff is v(i,S) - p

8. February 17, 2006QIP Course on E-Commerce : Auctions- 28 The Efficient Allocation (1 of 3) An allocation is X: a partition of items in N to bundles and assignment of these bundles to buyers (buyers can get the empty bundle) – X i is bundle of i. An allocation is X: a partition of items in N to bundles and assignment of these bundles to buyers (buyers can get the empty bundle) – X i is bundle of i. Efficient allocation: one which maximizes Σ i v(i,X i ). Efficient allocation: one which maximizes Σ i v(i,X i ). An economy consists of buyers and their valuations – E(B) is economy with buyers in B (subset of M). An economy consists of buyers and their valuations – E(B) is economy with buyers in B (subset of M). E(M) is main economy and E(M-i) is a marginal economy. E(M) is main economy and E(M-i) is a marginal economy.

9. February 17, 2006QIP Course on E-Commerce : Auctions- 29 The Efficient Allocation (2 of 3) If X is an efficient allocation of E(B), it is a disjoint partition of A but assigns bundles to buyers in B only. If X is an efficient allocation of E(B), it is a disjoint partition of A but assigns bundles to buyers in B only. If X is an efficient allocation of E(B), denote V(B)= Σ i v(i,X i ). If X is an efficient allocation of E(B), denote V(B)= Σ i v(i,X i ). For an integer programming formulation: For an integer programming formulation: –y(i,S) = 1 if i is assigned bundle S and zero otherwise.

10. February 17, 2006QIP Course on E-Commerce : Auctions- 210 The Efficient Allocation (3 of 3) Formulation (IP) for main economy Formulation (IP) for main economy V(M) = max ∑ i,S v(i,S)y(i,S) s.t. ∑ S y(i,S) = 1 for all i in M, ∑ i ∑ S: j in S y(i,S) = 1 for all j in N, y(i,S) in {0,1} for all i, S.

11. February 17, 2006QIP Course on E-Commerce : Auctions- 211 The VCG Mechanism (1 of 2) The Vickrey-Clarke-Groves (VCG) mechanism is a sealed-bid auction. The Vickrey-Clarke-Groves (VCG) mechanism is a sealed-bid auction. Allocation – Efficient allocation of main economy. Allocation – Efficient allocation of main economy. Payment - such that payoff of buyer i is V(M)- V(M-i): marginal contribution of i Payment - such that payoff of buyer i is V(M)- V(M-i): marginal contribution of i –Payment is: v(i,X i )-[V(M)-V(M-i)], where X is efficient allocation of main economy. Bidding one’s true value as bid is a dominant strategy. Bidding one’s true value as bid is a dominant strategy. –In many ways, it is a unique mechanism which is efficient and has a dominant strategy.

12. February 17, 2006QIP Course on E-Commerce : Auctions- 212 The VCG Mechanism (2 of 2) Requires solving (IP) (m+1) times in the worst case. Requires solving (IP) (m+1) times in the worst case. –(IP) is difficult to solve – NP Hard (Rothkopf et al., 1998, Management Science). Requires every buyer to submit an exponential-sized valuation function. Requires every buyer to submit an exponential-sized valuation function. Example: 3 buyers, 2 items Example: 3 buyers, 2 items V(1,2)=8+8=16; V(1)=12,V(2)=14. V(1,2)=8+8=16; V(1)=12,V(2)=14. –p 1 =8-[16-14]=6; p 2 =8-[16-12]=4 aba+b 18912 26814

13. February 17, 2006QIP Course on E-Commerce : Auctions- 213 An Easy Instance: The Assignment Problem (1 of 2) Every buyer is interested in at most one item. Every buyer is interested in at most one item. –Values of buyers can be written as: v(i,j) value of buyer i on item j. (IP) reduces to following (LPA) (IP) reduces to following (LPA) V(M) = max ∑ i,j v(i,j)y(i,j) s.t. ∑ j y(i,j) <= 1 for all i in M, ∑ i y(i,j) <= 1 for all j in N, y(i,j) >= 0 for all i, j.

14. February 17, 2006QIP Course on E-Commerce : Auctions- 214 An Easy Instance: The Assignment Problem (2 of 2) Dual of (LPA) is (DPA) Dual of (LPA) is (DPA) V(M) = min ∑ i q i + ∑ j p j s.t. p j + q i >= v(i,j) for all i, j, p j >= 0 for all i, q j >= 0 for all j. Leonard (1983) showed that there exists an optimal solution of (DPA) such that p j =VCG payoff of buyer who is assigned item j in optimal solution of (LPA). Leonard (1983) showed that there exists an optimal solution of (DPA) such that p j =VCG payoff of buyer who is assigned item j in optimal solution of (LPA). –This solution is obtained by maximizing ∑ i q i over all possible solutions of (DPA).

15. February 17, 2006QIP Course on E-Commerce : Auctions- 215 Iterative Auctions Iterative auctions – decentralized implementation of sealed-bid auctions. Iterative auctions – decentralized implementation of sealed-bid auctions. Better preference elicitation – buyers can work on bounds on valuations and no need to submit the entire valuation function. Better preference elicitation – buyers can work on bounds on valuations and no need to submit the entire valuation function. More transparent – shown to generate more revenue and efficiency (Cramton 1998, Eur. Econ. Rev.). More transparent – shown to generate more revenue and efficiency (Cramton 1998, Eur. Econ. Rev.). Popular in practice – English auction more popular than the Vickrey auction. Popular in practice – English auction more popular than the Vickrey auction.

16. February 17, 2006QIP Course on E-Commerce : Auctions- 216 Linear Price Iterative Auction For the assignment problem, p j denotes price on item j. For the assignment problem, p j denotes price on item j. Price vector p – prices on items. Price vector p – prices on items. Demand set of buyer i at p: D(i,p)={j in N:v(i,j)- p j >= v(i,k) – p k for all k in N} – payoff maximizing items. Demand set of buyer i at p: D(i,p)={j in N:v(i,j)- p j >= v(i,k) – p k for all k in N} – payoff maximizing items. An iterative auction (Demange et al., 1986): An iterative auction (Demange et al., 1986): –Start from zero price vector (or a low price). Initially no buyer is assigned. –A buyer bids when he is not assigned and his maximum payoff is more than zero. A buyer bids (truthfully) by increasing the price of an item in his demand set at the current price by ε. –The auction stops when there is no bidding. As ε approaches zero, this auction implements the VCG outcome and truthful bidding is a (ex post) Nash equilibrium for buyers. As ε approaches zero, this auction implements the VCG outcome and truthful bidding is a (ex post) Nash equilibrium for buyers.

17. February 17, 2006QIP Course on E-Commerce : Auctions- 217 Example Three buyers, two items: Three buyers, two items: v(1,1)=3, v(1,2)=4; v(2,1)=2, v(2,2)=5; v(3,1)=4,v(3,2)=4. At price (0,0) buyer 1 bids on item 2 to make it (0,1). Now, buyer 3 bids on item 1 to make it (1,1). Now, buyer 2 bids on item 2 to make it (1,2) … will converge approximately to (3,4). At price (0,0) buyer 1 bids on item 2 to make it (0,1). Now, buyer 3 bids on item 1 to make it (1,1). Now, buyer 2 bids on item 2 to make it (1,2) … will converge approximately to (3,4).

18. February 17, 2006QIP Course on E-Commerce : Auctions- 218 Complex Price Auctions For general combinatorial auction settings, iterative auctions require complex prices. For general combinatorial auction settings, iterative auctions require complex prices. –Non-linear (every bundle has a price) and non-anonymous (personalized prices for every buyer): p(i,S). –The underlying theory for such complex prices can be found in Bikhchandani and Ostroy (2002, J. of Econ. Theory). Sometimes, such complex prices can be represented in a simple way – when items are homogeneous and marginal values on units/items are non-increasing Ausubel (2004) designs an iterative auction that maintains a single price but implicitly maintains non- linear and non-anonymous prices (Bikhchandani and Ostroy, 2005, Games and Econ. Behavior, Forthcoming) Sometimes, such complex prices can be represented in a simple way – when items are homogeneous and marginal values on units/items are non-increasing Ausubel (2004) designs an iterative auction that maintains a single price but implicitly maintains non- linear and non-anonymous prices (Bikhchandani and Ostroy, 2005, Games and Econ. Behavior, Forthcoming)

19. February 17, 2006QIP Course on E-Commerce : Auctions- 219 iBundle Auction (1 of 4) First appeared in Parkes (1999, EC’99). First appeared in Parkes (1999, EC’99). –Maintains non-linear and non-anonymous prices. Given such a price vector p, demand set of buyer i is D(i,p) = {all bundles S: v(i,S)-p(i,S) >= v(i,T) – p(i,T) for all bundles T} Given such a price vector p, demand set of buyer i is D(i,p) = {all bundles S: v(i,S)-p(i,S) >= v(i,T) – p(i,T) for all bundles T} –payoff maximizing bundles at price p. Supply set of seller at price p as L(p) = {all allocations X: ∑ i p(i,X i ) >= ∑ i p(i,Y i ) for all allocations Y} Supply set of seller at price p as L(p) = {all allocations X: ∑ i p(i,X i ) >= ∑ i p(i,Y i ) for all allocations Y} –revenue maximizing allocations. Define L*(p)={all allocations X: X in L(p) and X i in D(i,p) or Xi = φ}: buyer compatible allocations in supply set. Define L*(p)={all allocations X: X in L(p) and X i in D(i,p) or Xi = φ}: buyer compatible allocations in supply set.

20. February 17, 2006QIP Course on E-Commerce : Auctions- 220 iBundle Auction (2 of 4) Start from zero prices (or low prices). Start from zero prices (or low prices). At every iteration with price p: At every iteration with price p: –Collect demand sets of buyers at p. –Find an allocation X (provisional allocation) from L*(p)  The auction ensures that L*(p) is not empty. –Define losers in X as: O(X,p)={i: X i notin D(i,p)}. –If O(X,p) is empty go to last step. Else for every i in O(X,p) and S in D(i,p) set p(i,S):=p(i,S) + 1 (this bid increment is for convenience) and repeat. The final allocation is final provisional allocation X and payment of buyer i is p(i, X i ). The final allocation is final provisional allocation X and payment of buyer i is p(i, X i ).

21. February 17, 2006QIP Course on E-Commerce : Auctions- 221 iBundle Auction (3 of 4) An example: An example: –v(1,a)= 4, v(1,b)=3, v(1,a+b)=6; –v(2,a)=2, v(2,b) = 5, v(2,a+b)=8. At price (0,0,0;0,0,0): At price (0,0,0;0,0,0): –D(1,p)=D(2,p)= {a+b}. L*(p)={1 gets a+b, 2 gets φ}. Next price (0,0,0;0,0,1) where D( ) are unchanged. Next price (0,0,0;0,0,1) where D( ) are unchanged. –L*(p)={2 gets a+b, 1 gets φ}. Next price (0,0,1;0,0,1). This goes on … Price reaches (0,0,2;0,0,2): Price reaches (0,0,2;0,0,2): –D(1,p)={a,a+b}, D(2,p)={a+b}, L*(p)={1 gets a+b}. Next price (0,0,2;0,0,3}: Next price (0,0,2;0,0,3}: –D(1,p)={a,a+b}, D(2,p)={b,a+b}, L*(p)={2 gets a+b} –Next price (1,0,3;0,0,3} … Finally, price reaches (3,2,5;0,2,5): Finally, price reaches (3,2,5;0,2,5): –D(1,p)={a,b,a+b}, D(2,p)={b,a+b}, L*(p)={1 gets a, 2 gets b}. –Auction ends with 1 paying 3 and 2 paying 2. –Note that these are VCG payments.

22. February 17, 2006QIP Course on E-Commerce : Auctions- 222 iBundle Auction (4 of 4) The fact that final payments in the iBundle auction is VCG payment is no coincidence in the example. The fact that final payments in the iBundle auction is VCG payment is no coincidence in the example. Ausubel and Milgrom (2002, Frontiers of Theo. Econ.) show that under a submodularity condition on V( ), satisfied in the example, final payments in iBundle will always be VCG payments. Ausubel and Milgrom (2002, Frontiers of Theo. Econ.) show that under a submodularity condition on V( ), satisfied in the example, final payments in iBundle will always be VCG payments. For valuations that do not satisfy the submodular condition, we have to apply iBundle for marginal economies too and give discounts to buyers at the end (Mishra and Parkes, 2005, J. of Econ. Theory, Forthcoming). For valuations that do not satisfy the submodular condition, we have to apply iBundle for marginal economies too and give discounts to buyers at the end (Mishra and Parkes, 2005, J. of Econ. Theory, Forthcoming). –These discounts are marginal contribution of a buyer to the revenue of the seller.

23. February 17, 2006QIP Course on E-Commerce : Auctions- 223 Winner Determination Problem (1 of 2) Computing L*(p) is called the winner determination problem (WDP). Computing L*(p) is called the winner determination problem (WDP). max ∑ i ∑ S in D(i,p) or φ p(i,S)y(i,S) s.t. ∑ S in D(i,p) or φ y(i,S) = 1 for all i in M, ∑ S: j in S ∑ i:S in D(i,p) y(i,S) = 1 for all j in N, y(i,S) in {0,1} for all i, S in D(i,p) or φ.

24. February 17, 2006QIP Course on E-Commerce : Auctions- 224 Winner Determination Problem (2 of 2) (WDP) is NP-Hard (WDP) is NP-Hard –Lot of research on quickly solving WDP (Rothkopf et al. 1998, Tuomas Sandholm’s papers). Observe that (WDP) is a smaller version of (IP). Observe that (WDP) is a smaller version of (IP). –In iterative auctions, we solve smaller instances of multiple (IP) instead of solving one huge instance of (IP).

25. February 17, 2006QIP Course on E-Commerce : Auctions- 225 Linear Programs and Iterative Auctions Close connection exists. Close connection exists. Almost all iterative auctions can be interpreted as a linear programming algorithm to solve an appropriate linear program (de Vries et al., 2005, J. of Econ. Theory, Forthcoming). Almost all iterative auctions can be interpreted as a linear programming algorithm to solve an appropriate linear program (de Vries et al., 2005, J. of Econ. Theory, Forthcoming). In particular, (almost) every auction in literature is either a primal-dual algorithm or a subgradient algorithm. In particular, (almost) every auction in literature is either a primal-dual algorithm or a subgradient algorithm.

26. February 17, 2006QIP Course on E-Commerce : Auctions- 226 Concluding Thoughts Similar to ascending auctions, possible to design descending auctions that implement the VCG outcome (Mishra and Parkes, 2004a, 2004b). Similar to ascending auctions, possible to design descending auctions that implement the VCG outcome (Mishra and Parkes, 2004a, 2004b). –Descending auctions have better preference elicitation properties than ascending ones. Combinatorial auction design is difficult due to complexity of the input. Combinatorial auction design is difficult due to complexity of the input. Carefully handling the WDP stage should help implement practical combinatorial auctions. Carefully handling the WDP stage should help implement practical combinatorial auctions. Besides incentives, stress should be given on simplicity (unfortunately, simplicity and incentives are not compatible) – simplicity in terms of prices, bidding languages (how to submit bids). Besides incentives, stress should be given on simplicity (unfortunately, simplicity and incentives are not compatible) – simplicity in terms of prices, bidding languages (how to submit bids).