Post on 20-Dec-2015
Sequences of Take-It-or-Leave-it Offers:Near-Optimal Auctions Without Full Valuation Revelation
Tuomas Sandholm and Andrew Gilpin
Carnegie Mellon UniversityComputer Science Department
Agent-Mediated Electronic Marketplaces Group
Workshop on Agent-Mediated Electronic Commerce (AMEC-V)July 15, 2003
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Outline
• Introduction
• Take-It-or-Leave-It Auction
• Optimizing the offers
• Economic performance
• Conclusions
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Introduction• Seller has a good she wishes to sell
• Group of n interested buyers– Buyer i has valuation vi drawn from PDF fi
• Q: How can the seller maximize revenue?
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Introduction• Seller has a good she wishes to sell
• Group of n interested buyers– Buyer i has valuation vi drawn from PDF fi
• Q: How can the seller maximize revenue?
– A: Hold an auction!
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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: 0.375• Myerson (maximum possible): 0.4167
– With asymmetric buyers, the revenue loss is worse
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Optimal auctions
• Individual Rationality (IR)– A losing buyer pays nothing– A winning buyer i pays no more than vi
• 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
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Myerson auction
• Buyer i reveals valuation vi
• Compute “virtual valuation” Ψi for each buyer
– Ψi(vi) = vi - (1 - Fi (vi ))/fi (vi)
• Select buyer i* with max virtual valuation
• Allocate good to buyer i* only if Ψi* > 0– Winning buyer makes smallest winning payment
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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 revelation2. Has easily explainable rules3. Yields close to optimal expected revenue
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Outline
• Introduction
• Take-It-or-Leave-It Auction
• Equilibrium Analysis
• Optimizing the Offers
• Economic Performance
• Conclusions
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Take-It-or-Leave-It Auction (TLA)
• An instance of a TLA is:
• At the jth step buyer bj receives an offer of aj
• Buyer bj can “take-it” or “leave-it”
• Entire sequence of offers is revealed to all• Single-offer vs. multiple-offer
kk abababA ,,...,,,, 2211
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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
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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
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Example (cont.)
• Should buyer 1 accept first offer of 0.625?– If v1 < 0.625, then of course not.
– If v1 > 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 v1 – a1 = F2(t2) (v1 – a3)
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Equilibrium analysis
• We have the following system
t1 – a1 = F2(t2) (t1 – a3)
t2 – a2 = F1(a3)/F1(t1) (t2 – a4)
Buyer 1’s revenue if she accepts first offer
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Equilibrium analysis
• We have the following system
t1 – a1 = F2(t2) (t1 – a3)
t2 – a2 = F1(a3)/F1(t1) (t2 – a4)
Probability buyer 2 rejects offer 2
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Equilibrium analysis
• We have the following system
t1 – a1 = F2(t2) (t1 – a3)
t2 – a2 = F1(a3)/F1(t1) (t2 – a4)
Buyer 1’s revenue if she accepts third offer
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Equilibrium analysis
• We have the following system
t1 – a1 = F2(t2) (t1 – a3)
t2 – a2 = F1(a3)/F1(t1) (t2 – a4)
Buyer 1’s expected revenue if she rejects first offer
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Equilibrium analysis
• We have the following system
t1 – a1 = F2(t2) (t1 – a3)
t2 – a2 = F1(a3)/F1(t1) (t2 – a4)
Updating: probability that buyer 1 rejects offer 3, given that she has already rejected offer 1
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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
t1 – a1 = F2(t2) (t1 – a3)
t2 – a2 = F1(a3)/F1(t1) (t2 – a4)
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Outline
• Introduction
• Take-It-or-Leave-It Auction
• Optimizing the Offers
• Economic Performance
• Conclusions
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Optimizing the offers: single-offer
• Symmetric setting:– Order of buyers does not matter– Simply compute the offers in reverse order
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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
ai = argmaxa (1 – F(a)) a + F(a) RevRev = (1 – F(ai)) ai + F(a) Rev
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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
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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
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Optimizing the offers: complexity
Symmetric Asymmetric
Single-offer 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
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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
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Outline
• Introduction
• Take-It-or-Leave-It Auction
• Optimizing the Offers
• Economic Performance
• Conclusions
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Economic performance
• Theorem: The revenue loss in an optimal k-offer TLA with 2 symmetric buyers is O(1/k2)– Proof based on result in:
• Liad Blumrosen and Noam Nisan. Auctions with severely bounded communication. In FOCS, 2002.
• Expect similar result for general distributions– The analysis becomes increasingly complex
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Economic performance: example
Two buyers, uniform on [0,1]
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Outline
• Introduction
• Take-It-or-Leave-It Auction
• Optimizing the Offers
• Economic Performance
• Conclusions
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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