1. 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

2. July 8, 2004Take-It-or-Leave-It Auctions Games 2004 2 / 32 Outline Introduction Take-It-or-Leave-It Auction Optimizing the offers Economic performance Conclusions

3. July 8, 2004Take-It-or-Leave-It Auctions Games 2004 3 / 32 Introduction Setting: –Seller has a good she wishes to sell –Buyer i = 1..n has private valuation v i drawn independently from common-knowledge PDF f i –Risk-neutral, quasi-linear preferences Seller wishes to maximize expected utility –One possibility: run an auction

4. July 8, 2004Take-It-or-Leave-It Auctions Games 2004 4 / 32 English auction Seller announces increasingly higher prices –At any point in time, there is a set of current bidders Buyers exit when the price exceeds their valuation Auction terminates when no buyer wishes to go higher –(“Going once, going twice, sold!”) Good is sold to the remaining bidder at the final price Despite popularity, English auctions (and other popular auctions) are sub-optimal –Example: two buyers, valuations uniform on [0,1] English: 0.33 Fixed price of 0.5: 0.375 Myerson (maximum possible): 0.4167 –With asymmetric buyers, the revenue loss may be worse

5. July 8, 2004Take-It-or-Leave-It Auctions Games 2004 5 / 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, 1981. Among all IR mechanisms, the Myerson auction achieves optimal expected revenue

6. July 8, 2004Take-It-or-Leave-It Auctions Games 2004 6 / 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

7. July 8, 2004Take-It-or-Leave-It Auctions Games 2004 7 / 32 Myerson auction (cont.) Despite optimality, there are drawbacks: –Requires full valuation –Rules of the game (especially virtual valuation) difficult to explain/understand –Submitting true valuations is unintuitive Myerson auctions are not used in practice Goal: Design an auction that: 1.Limits valuation revelation 2.Is easy to explain 3.Yields close to optimal expected revenue

8. July 8, 2004Take-It-or-Leave-It Auctions Games 2004 8 / 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 up front Single-offer vs. multiple-offer

9. July 8, 2004Take-It-or-Leave-It Auctions Games 2004 9 / 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 by taking into account the possibility that one will receive a better offer later –Buyers compute probability they receive another offer –Buyers update beliefs about other buyers’ valuations –Buyers counter-speculate other buyers beliefs as well Buyers can compute a threshold strategy –Deterministic plan for a bidder in a TLA

10. July 8, 2004Take-It-or-Leave-It Auctions Games 2004 10 / 32 Example Two buyers –Uniformly distributed on [0,1] Four offers –All offers are announced to both buyers –The first offer is 0.625 to buyer 1

11. July 8, 2004Take-It-or-Leave-It Auctions Games 2004 11 / 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 )

12. July 8, 2004Take-It-or-Leave-It Auctions Games 2004 12 / 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

13. July 8, 2004Take-It-or-Leave-It Auctions Games 2004 13 / 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

14. July 8, 2004Take-It-or-Leave-It Auctions Games 2004 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 third offer

15. July 8, 2004Take-It-or-Leave-It Auctions Games 2004 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 ) Buyer 1’s expected revenue if she rejects first offer

16. July 8, 2004Take-It-or-Leave-It Auctions Games 2004 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 ) Updating: probability that buyer 1 rejects offer 3, given that she has already rejected offer 1

17. July 8, 2004Take-It-or-Leave-It Auctions Games 2004 17 / 32 Equilibrium analysis We have the following system A solution to this system yields the optimal threshold strategies –Theorem: In perfect Bayesian equilibrium, 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 )

18. July 8, 2004Take-It-or-Leave-It Auctions Games 2004 18 / 32 Optimizing the offers: single-offer Symmetric setting: –Order of buyers does not matter –Algorithm: compute the offers in last-first order Rev = 0 For i from n down to 1 a i = argmax a (1 – F ( a )) a + F ( a ) Rev Rev = (1 – F ( a i )) a i + F ( a ) Rev

19. July 8, 2004Take-It-or-Leave-It Auctions Games 2004 19 / 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 for general valuation distributions

20. July 8, 2004Take-It-or-Leave-It Auctions Games 2004 20 / 32 Optimizing the offers: multiple-offer Optimization is much more complicated –Objective and constraints are non-linear, non-convex For certain distributions, efficient algorithms exist –E.g. 2 buyers, uniform and symmetric distributions, we have an O (#offers) algorithm An efficient general algorithm is not known –We solve the general problem as a non-linear optimization using solvers such as Matlab May yield locally optimal solutions

21. July 8, 2004Take-It-or-Leave-It Auctions Games 2004 21 / 32 Optimizing the offers: summary SymmetricAsymmetric Single-offerO(n) O(n log n) for many distributions Multiple-offer O(k) for many distributions No general efficient algorithm

22. July 8, 2004Take-It-or-Leave-It Auctions Games 2004 22 / 32 Characteristics of 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

23. July 8, 2004Take-It-or-Leave-It Auctions Games 2004 23 / 32 Worst-case revenue loss Theorem: The revenue loss in an optimal k- offer TLA with 2 symmetric buyers is O (1/ k 2 ) –Follows from result in: Liad Blumrosen and Noam Nisan. Auctions with severely bounded communication. In FOCS, 2002. –Given optimal thresholds, we can compute the offer levels for the optimal TLA In general, no known worst-case bounds, but expected revenue loss can be computed

24. July 8, 2004Take-It-or-Leave-It Auctions Games 2004 24 / 32 Economic performance: example Two buyers, uniform on [0,1]

25. July 8, 2004Take-It-or-Leave-It Auctions Games 2004 25 / 32 Conclusions TLAs reduce valuation revelation TLAs are intuitive to play –Playing truthfully is optimal in 1-offer TLAs –Playing threshold strategies is optimal generally Close-to-optimal revenue Optimal TLAs can be designed quickly in many settings

26. July 8, 2004Take-It-or-Leave-It Auctions Games 2004 26 / 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

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