Sequences of Take-It-or-Leave-it Offers: Near-Optimal Auctions Without Full Valuation Revelation Tuomas Sandholm and Andrew Gilpin Carnegie Mellon University Computer Science Department Agent-Mediated Electronic Marketplaces Group Workshop on Agent-Mediated Electronic Commerce (AMEC-V) July 15, 2003

Take-It-or-Leave-It Auctions AMEC-V 2 / 32 Outline Introduction Take-It-or-Leave-It Auction Optimizing the offers Economic performance Conclusions

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 3 / 32 Introduction Seller has a good she wishes to sell Group of n interested buyers –Buyer i has valuation v i drawn from PDF f i Q: How can the seller maximize revenue?

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 4 / 32 Introduction Seller has a good she wishes to sell Group of n interested buyers –Buyer i has valuation v i drawn from PDF f i Q: How can the seller maximize revenue? –A: Hold an auction!

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 5 / 32 English auction Buyers announce increasingly higher prices –Buyer “drops out” when price is too expensive Ends when no buyer wishes to go higher –(“Going once, going twice, sold!”) Despite popularity, English auctions (and other popular auctions) are sub-optimal –Example: two buyers, valuation uniform on [0,1] English: 0.33 Fixed price of 0.5: Myerson (maximum possible): –With asymmetric buyers, the revenue loss is worse

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 6 / 32 Optimal auctions Individual Rationality (IR) –A losing buyer pays nothing –A winning buyer i pays no more than v i Optimal auction for our setting is known –Roger B. Myerson. Optimal auction design. Mathematics of Operation Research, Among all IR mechanisms, the Myerson auction achieves optimal expected revenue

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 7 / 32 Myerson auction Buyer i reveals valuation v i Compute “virtual valuation” Ψ i for each buyer –Ψ i ( v i ) = v i - (1 - F i ( v i ))/f i ( v i ) Select buyer i * with max virtual valuation Allocate good to buyer i * only if Ψ i * > 0 –Winning buyer makes smallest winning payment

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 8 / 32 Myerson auction (cont.) Despite optimality, there are drawbacks: –Full valuation revelation required –“Rules of the game” difficult to understand –Submitting true valuations is unintuitive Myerson auctions are not used in practice Goal: Design an auction that: 1.Does not require full valuation revelation 2.Has easily explainable rules 3.Yields close to optimal expected revenue

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 9 / 32 Outline Introduction Take-It-or-Leave-It Auction Equilibrium Analysis Optimizing the Offers Economic Performance Conclusions

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 10 / 32 Take-It-or-Leave-It Auction (TLA) An instance of a TLA is: At the j th step buyer b j receives an offer of a j Buyer b j can “take-it” or “leave-it” Entire sequence of offers is revealed to all Single-offer vs. multiple-offer

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 11 / 32 Equilibrium analysis When facing an offer, what do you do? If it is your last offer, answer truthfully –Prop: Truth is a dominant strategy in single-offer TLA If not, the best thing to do is to “gamble” –Compute probability you receive another offer –Buyers update beliefs about other buyers’ valuations –Other buyers are gambling as well Threshold strategy –Deterministic plan for a bidder in a TLA

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 12 / 32 Example Two buyers –Uniformly distributed on [0,1] Four offers –All offers are announced to both buyers –The first offer is to buyer 1

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 13 / 32 Example (cont.) Should buyer 1 accept first offer of 0.625? –If v 1 < 0.625, then of course not. –If v 1 > 0.625, then maybe. It may be better for buyer 1 to reject, even though she stands to profit from accepting! When is buyer 1 indifferent between accepting and rejecting? –When v 1 – a 1 = F 2 ( t 2 ) ( v 1 – a 3 )

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 14 / 32 Equilibrium analysis We have the following system t 1 – a 1 = F 2 ( t 2 ) ( t 1 – a 3 ) t 2 – a 2 = F 1 ( a 3 )/ F 1 ( t 1 ) ( t 2 – a 4 ) Buyer 1’s revenue if she accepts first offer

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 15 / 32 Equilibrium analysis We have the following system t 1 – a 1 = F 2 ( t 2 ) ( t 1 – a 3 ) t 2 – a 2 = F 1 ( a 3 )/ F 1 ( t 1 ) ( t 2 – a 4 ) Probability buyer 2 rejects offer 2

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 16 / 32 Equilibrium analysis We have the following system t 1 – a 1 = F 2 ( t 2 ) ( t 1 – a 3 ) t 2 – a 2 = F 1 ( a 3 )/ F 1 ( t 1 ) ( t 2 – a 4 ) Buyer 1’s revenue if she accepts third offer

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 17 / 32 Equilibrium analysis We have the following system t 1 – a 1 = F 2 ( t 2 ) ( t 1 – a 3 ) t 2 – a 2 = F 1 ( a 3 )/ F 1 ( t 1 ) ( t 2 – a 4 ) Buyer 1’s expected revenue if she rejects first offer

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 18 / 32 Equilibrium analysis We have the following system t 1 – a 1 = F 2 ( t 2 ) ( t 1 – a 3 ) t 2 – a 2 = F 1 ( a 3 )/ F 1 ( t 1 ) ( t 2 – a 4 ) Updating: probability that buyer 1 rejects offer 3, given that she has already rejected offer 1

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 19 / 32 Equilibrium analysis We have the following system A solution to this system yields the optimal threshold strategies –Theorem: In Perfect Bayesian Equilibrium (PBE), all buyers play according to their thresholds t 1 – a 1 = F 2 ( t 2 ) ( t 1 – a 3 ) t 2 – a 2 = F 1 ( a 3 )/ F 1 ( t 1 ) ( t 2 – a 4 )

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 20 / 32 Outline Introduction Take-It-or-Leave-It Auction Optimizing the Offers Economic Performance Conclusions

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 21 / 32 Optimizing the offers: single-offer Symmetric setting: –Order of buyers does not matter –Simply compute the offers in reverse order

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 22 / 32 Optimizing the offers: single-offer Symmetric setting: –Order of buyers does not matter –Simply compute the offers in reverse order Rev = 0 For i from #Buyers down to 1 a i = argmax a (1 – F ( a )) a + F ( a ) Rev Rev = (1 – F ( a i )) a i + F ( a ) Rev

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 23 / 32 Optimizing the offers: single-offer Asymmetric setting –For specific distributions (e.g., uniform, exponential), optimization is easy –Basic idea Sort buyers by some property Then use previous algorithm to compute offer levels –No known efficient algorithm yet for general valuation distributions

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 24 / 32 Optimizing the offers: multiple-offer Optimization is much more complicated For certain distributions, efficient algorithms exist –E.g. 2 buyers, uniform and symmetric distributions, there is an O (#Offers) algorithm An efficient general algorithm is not known –We solve the problem as a non-linear optimization using general solvers such as Matlab

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 25 / 32 Optimizing the offers: complexity SymmetricAsymmetric Single-offer O(n)O(n) O ( n log n ) for many distributions Multiple-offer O ( k ) for many distributions No general efficient algorithm yet n Number of buyers k Number of offers

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 26 / 32 Optimal TLAs For a given setting, there exists an optimal TLA such that: –Prop: No buyer receives consecutive offers –Prop: Each buyer individually receives decreasing offers But offers may not decrease over time

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 27 / 32 Outline Introduction Take-It-or-Leave-It Auction Optimizing the Offers Economic Performance Conclusions

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 28 / 32 Economic performance Theorem: The revenue loss in an optimal k- offer TLA with 2 symmetric buyers is O (1/ k 2 ) –Proof based on result in: Liad Blumrosen and Noam Nisan. Auctions with severely bounded communication. In FOCS, Expect similar result for general distributions –The analysis becomes increasingly complex

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 29 / 32 Economic performance: example Two buyers, uniform on [0,1]

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 30 / 32 Outline Introduction Take-It-or-Leave-It Auction Optimizing the Offers Economic Performance Conclusions

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 31 / 32 Conclusions TLAs reduce valuation revelation TLAs are intuitive to play –Playing threshold strategies is optimal Close-to-optimal revenue generation Optimal TLA markets can be designed quickly in many settings

July 15, 2003Take-It-or-Leave-It Auctions AMEC-V 32 / 32 Future work Algorithms for general asymmetric preferences and multiple offers Multiple units of the item, and multiple distinguishable items Comparison of information revelation with commonly used auctions