Sequences of Take-It-or-Leave-it Offers: Near-Optimal Auctions Without Full Valuation Revelation...

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

July 15, 2003 Take-It-or-Leave-It Auctions AMEC-V

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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


• 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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• 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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t1 – a1 = F2(t2) (t1 – a3)

t2 – a2 = F1(a3)/F1(t1) (t2 – a4)

Probability buyer 2 rejects offer 2


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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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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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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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• 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




• 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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• 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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• 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




• 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




• 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


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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

Sequences of Take-It-or-Leave-it Offers: Near-Optimal Auctions Without Full Valuation Revelation...

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