The Simple Mathematics of Optimal Auctions Jason D. Hartline - PowerPoint PPT Presentation

The Simple Mathematics of Optimal Auctions Jason D. Hartline (joint with Maria-Florina Balcan, Nikhil Devanur, and Kunal Talwar) March 28, 2007

Economic Optimization Economic Optimization truthful fair prices Algorithmic Algorithmic Mechanism Pricing Design 1 O PTIMAL A UCTIONS – M ARCH 28, 2007

Overview = ⇒ 1. Review unlimited supply setting: (a) Algorithmic pricing. (b) Mechanism design via pricing. 2. Generalize to limited supply setting: (a) Algorithmic pricing. (b) Mechanism design via pricing. 3. Generality & conclusions. 2 O PTIMAL A UCTIONS – M ARCH 28, 2007

Example: Path Pricing Example: Edge pricing selling paths. v 2 v 5 v 3 v 4 v 1 v 6 Consumer 1 wants path from v 1 to v 2 for $5. Consumer 2 wants path from v 2 to v 3 for $3. . . . 3 O PTIMAL A UCTIONS – M ARCH 28, 2007

Example: Path Pricing Example: Edge pricing selling paths. v 2 v 5 $2 v 3 $2 v 4 $3 $2 $1 v 1 v 6 Consumer 1 wants path from v 1 to v 2 for $5. Consumer 2 wants path from v 2 to v 3 for $3. . . . 3 O PTIMAL A UCTIONS – M ARCH 28, 2007

Example: Path Pricing Example: Edge pricing selling paths. v 2 v 5 $2 v 3 $2 v 4 $3 $2 $1 v 1 v 6 Consumer 1 wants path from v 1 to v 2 for $5. (pays $4) Consumer 2 wants path from v 2 to v 3 for $3. . . . 3 O PTIMAL A UCTIONS – M ARCH 28, 2007

Example: Path Pricing Example: Edge pricing selling paths. v 2 v 5 $2 v 3 $2 v 4 $3 $2 $1 v 1 v 6 Consumer 1 wants path from v 1 to v 2 for $5. (pays $4) Consumer 2 wants path from v 2 to v 3 for $3. (not served) . . . 3 O PTIMAL A UCTIONS – M ARCH 28, 2007

Example: Path Pricing Example: Edge pricing selling paths. v 2 v 5 $2 v 3 $2 v 4 $3 $2 $1 v 1 v 6 Consumer 1 wants path from v 1 to v 2 for $5. (pays $4) Consumer 2 wants path from v 2 to v 3 for $3. (not served) . . . Goal: price edges to maximize objective. 3 O PTIMAL A UCTIONS – M ARCH 28, 2007

Unlimited Supply Algorithmic Pricing The Unlimited Supply Algorithmic Pricing problem: Given: • unlimited supply of stuff. • Set S of n consumers and their preferences for stuff. • class G of reasonable offers. Design: Algorithm to compute optimal offer from G . 4 O PTIMAL A UCTIONS – M ARCH 28, 2007

Unlimited Supply Algorithmic Pricing The Unlimited Supply Algorithmic Pricing problem: Given: • unlimited supply of stuff. • Set S of n consumers and their preferences for stuff. • class G of reasonable offers. Design: Algorithm to compute optimal offer from G . Notation: • p ( i, g ) = payoff from consumer i when offered g ∈ G . 4 O PTIMAL A UCTIONS – M ARCH 28, 2007

Unlimited Supply Algorithmic Pricing The Unlimited Supply Algorithmic Pricing problem: Given: • unlimited supply of stuff. • Set S of n consumers and their preferences for stuff. • class G of reasonable offers. Design: Algorithm to compute optimal offer from G . Notation: • p ( i, g ) = payoff from consumer i when offered g ∈ G . • p ( S, g ) = � i ∈ S p ( i, g ) . 4 O PTIMAL A UCTIONS – M ARCH 28, 2007

Unlimited Supply Algorithmic Pricing The Unlimited Supply Algorithmic Pricing problem: Given: • unlimited supply of stuff. • Set S of n consumers and their preferences for stuff. • class G of reasonable offers. Design: Algorithm to compute optimal offer from G . Notation: • p ( i, g ) = payoff from consumer i when offered g ∈ G . • p ( S, g ) = � i ∈ S p ( i, g ) . • opt G ( S ) = argmax g ∈G p ( S, g ) . 4 O PTIMAL A UCTIONS – M ARCH 28, 2007

Unlimited Supply Algorithmic Pricing The Unlimited Supply Algorithmic Pricing problem: Given: • unlimited supply of stuff. • Set S of n consumers and their preferences for stuff. • class G of reasonable offers. Design: Algorithm to compute optimal offer from G . Notation: • p ( i, g ) = payoff from consumer i when offered g ∈ G . • p ( S, g ) = � i ∈ S p ( i, g ) . • opt G ( S ) = argmax g ∈G p ( S, g ) . • OPT = OPT G ( S ) = max g ∈G p ( S, g ) .. 4 O PTIMAL A UCTIONS – M ARCH 28, 2007

Example: Digital Good Example: digital good • Single item for sale (unlimited supply). • Consumers have valuations for single copy of item, ( v 1 , . . . , v n ). • Consumers are indistinguishable. 5 O PTIMAL A UCTIONS – M ARCH 28, 2007

Example: Digital Good Example: digital good • Single item for sale (unlimited supply). • Consumers have valuations for single copy of item, ( v 1 , . . . , v n ). • Consumers are indistinguishable. • G = set of all prices, i.e., g q = “take-it-or-leave-it at price q ”. 5 O PTIMAL A UCTIONS – M ARCH 28, 2007

Example: Digital Good Example: digital good • Single item for sale (unlimited supply). • Consumers have valuations for single copy of item, ( v 1 , . . . , v n ). • Consumers are indistinguishable. • G = set of all prices, i.e., g q = “take-it-or-leave-it at price q ”. � q if q ≤ v i • p ( i, g q ) = . 0 o.w. 5 O PTIMAL A UCTIONS – M ARCH 28, 2007

Example: Digital Good Example: digital good • Single item for sale (unlimited supply). • Consumers have valuations for single copy of item, ( v 1 , . . . , v n ). • Consumers are indistinguishable. • G = set of all prices, i.e., g q = “take-it-or-leave-it at price q ”. � q if q ≤ v i • p ( i, g q ) = . 0 o.w. How can we compute opt G ? 5 O PTIMAL A UCTIONS – M ARCH 28, 2007

Example: Digital Good Example: digital good • Single item for sale (unlimited supply). • Consumers have valuations for single copy of item, ( v 1 , . . . , v n ). • Consumers are indistinguishable. • G = set of all prices, i.e., g q = “take-it-or-leave-it at price q ”. � q if q ≤ v i • p ( i, g q ) = . 0 o.w. How can we compute opt G ? 1. Sort valuations: v 1 ≥ . . . ≥ v n 2. Output v i to maximize i × v i . 5 O PTIMAL A UCTIONS – M ARCH 28, 2007

Literature Algorithmic Pricing in the Literature • unlimited supply (mostly). • many interesting special cases. • includes work of: Gagan Aggarwal, Maria-Florina Balcan, Avrim Blum, Patrick Briest, Shuchi Chawla, Eric Demaine, Tom´ as Feder, Uri Feige, Venkat Gurusuami, MohammadTaghi Hajiaghayi, Anna Karlin, David Kempe, Vladlin Koltun, Robert Kleinberg, Piotr Krysta, Clare Mathieu, Frank McSherry, Rajeev Motwani, and An Zhu. 6 O PTIMAL A UCTIONS – M ARCH 28, 2007

Literature Algorithmic Pricing in the Literature • unlimited supply (mostly). • many interesting special cases. • includes work of: Gagan Aggarwal, Maria-Florina Balcan, Avrim Blum, Patrick Briest, Shuchi Chawla, Eric Demaine, Tom´ as Feder, Uri Feige, Venkat Gurusuami, MohammadTaghi Hajiaghayi, Anna Karlin, David Kempe, Vladlin Koltun, Robert Kleinberg, Piotr Krysta, Clare Mathieu, Frank McSherry, Rajeev Motwani, and An Zhu. • hard (even to approximate). 6 O PTIMAL A UCTIONS – M ARCH 28, 2007

Overview 1. Review unlimited supply setting: (a) Algorithmic pricing. = ⇒ (b) Mechanism design via pricing. 2. Generalize to limited supply setting: (a) Algorithmic pricing. (b) Mechanism design via pricing. 3. Generality & conclusions. 7 O PTIMAL A UCTIONS – M ARCH 28, 2007

Auction Problem The Unlimited Supply Auction Problem : Given: • unlimited supply of stuff. • Set S of n bidders with preferences for stuff. • class G of reasonable offers. Design: Single round, sealed bid, truthful auction with profit near that of OPT G . Recall Notation: • g ( i ) = payoff from bidder i when offered g . • g ( S ) = � i ∈ S g ( i ) . • opt G ( S ) = argmax g ∈G g ( S ) . • OPT = OPT G ( S ) = max g ∈G g ( S ) . 8 O PTIMAL A UCTIONS – M ARCH 28, 2007

Random Sampling Auction Generalization of auction from [Goldberg, Hartline, Wright ’01]: Random Sampling Optimal Offer Auction, RSOO G 1. Randomly partition bidders into two sets: S 1 and S 2 . 2. compute g 1 (resp. g 2 ), optimal offer for S 1 (resp. S 2 ) 3. Offer g 1 to S 2 and g 2 to S 1 . S 9 O PTIMAL A UCTIONS – M ARCH 28, 2007

Random Sampling Auction Generalization of auction from [Goldberg, Hartline, Wright ’01]: Random Sampling Optimal Offer Auction, RSOO G 1. Randomly partition bidders into two sets: S 1 and S 2 . 2. compute g 1 (resp. g 2 ), optimal offer for S 1 (resp. S 2 ) 3. Offer g 1 to S 2 and g 2 to S 1 . S S 1 S 2 9 O PTIMAL A UCTIONS – M ARCH 28, 2007

Random Sampling Auction Generalization of auction from [Goldberg, Hartline, Wright ’01]: Random Sampling Optimal Offer Auction, RSOO G 1. Randomly partition bidders into two sets: S 1 and S 2 . 2. compute g 1 (resp. g 2 ), optimal offer for S 1 (resp. S 2 ) 3. Offer g 1 to S 2 and g 2 to S 1 . S S 1 g 1 = opt( S 1) g 2 = opt( S 2) S 2 9 O PTIMAL A UCTIONS – M ARCH 28, 2007

Random Sampling Auction Generalization of auction from [Goldberg, Hartline, Wright ’01]: Random Sampling Optimal Offer Auction, RSOO G 1. Randomly partition bidders into two sets: S 1 and S 2 . 2. compute g 1 (resp. g 2 ), optimal offer for S 1 (resp. S 2 ) 3. Offer g 1 to S 2 and g 2 to S 1 . S S 1 g 1 = opt( S 1) g 1 = opt( S 1) g 2 = opt( S 2) g 2 = opt( S 2) S 2 9 O PTIMAL A UCTIONS – M ARCH 28, 2007

Random Sampling Auction Generalization of auction from [Goldberg, Hartline, Wright ’01]: Random Sampling Optimal Offer Auction, RSOO G 1. Randomly partition bidders into two sets: S 1 and S 2 . 2. compute g 1 (resp. g 2 ), optimal offer for S 1 (resp. S 2 ) 3. Offer g 1 to S 2 and g 2 to S 1 . S S 1 g 1 = opt( S 1) g 1 = opt( S 1) g 2 = opt( S 2) g 2 = opt( S 2) S 2 Fact: RSOO G is truthful. 9 O PTIMAL A UCTIONS – M ARCH 28, 2007