What I Really Wanted To Know About Combinatorial Auctions

What I Really Wanted To Know

About Combinatorial

AuctionsArne AnderssonTrade ExtensionsUppsala University

Content

1. A glance at today’s topic

2. Research Background

3. Industrial Background

4. Auction Protocols

5. The Problem: combiatorial vs. simultaneous auctions

6. The Proof

7. Summary

Content






6. The Proof

7. Summary

Combinatorial Auction

Bid A Bid B Bid C Bid D Bid E

Item 1 100 102 x

Item 2 103 99 x x

Item 3 100 x x

Item 4 105 106 x x

CombPrice

200 205 305

Single Bids Combinatorial Bids


Maximize 100 A1 + 103 A2 + 100 A3 + 105 A4 + 102 B1 + 99 B2 + 106 B4 + 200 C + 205 D + 305 E

subject toA1 + B1 + C ≤ 1 (only one bid can win Commodity 1)A2 + B2 + D + E ≤ 1 (only one bid can win Commodity 2)A3 + D + E ≤ 1 (only one bid can win Commodity 3)A4 + B4 + C + E ≤ 1 (only one bid can win Commodity 4)


Item 1 100 102 x

Item 2 103 99 x x

Item 3 100 x x

Item 4 105 106 x x

CombPrice

200 205 305

...and here is why Computer Scientists

care about these auctions

What if....


Item 1 ? ? ?

Item 2 ? ? ? ?

Item 3 ? ? ?

Item 4 ? ? ? ?

...we do not allow combinatorial bids,but only single bids?(Simultaneous auction)

Will the auctioneer earn higher or lower revenue?

Content






6. The Proof

7. Summary

Research Background

• 1985-2000: Research in Algorithms and Data Structures

• 1995-2000: Applied project on Optimization and Resource Allocation in the Energy Sector• Resource allocation handled as Markets• Electronic Markets• Combinatorial Auctions

• 2000, co-Founded TradeExtensions• 2008, Finally left permanent university position,

my hart still belongs to research

Why do Research?

• In the beginning: It’s fun! (And it might help the career)

• After a while: you need to ”build your CV” to get a good job

• Finally: You can afford to be a bit relaxed on counting publications, and go for the important problems instead.

What is an important problem?

?

Decide yourself!

!

Content






6. The Proof

7. Summary

Industrial Background

• Trade Extensions founded 2000

• The world’s first on-line combinatorial auction 2001

• Today a world-leading provider of on-line bidding and optimization, handling millions of bid values with complex constraints, where combinatorial bids are just one special case

• Largest application area is Logistics

Content






6. The Proof

7. Summary

A Real World Problem

• A number of items for sale• A number of bidders• An Auctioneer• The Auctioneer’s Goal:

• (Maximize Efficiency)• Maximize Revenue

Which auction mechanism should the auctioneer use?

The Ideal Solution

”Let every bidder tell his true preferences and solve the resulting optimization problem”

Bidders

Auctioneer

Bids Allocations

The Real World

”Bidders will speculate, and the auctioneer has to be cool about it”

Bidders

Auctioneer

Bids Allocations

One idea (incentive-compatability)

”Use an auction mechanisms where thebidder’s best strategy is to bid truthfully”

Bidders

Auctioneer

Bids Allocations

Incentive-compatability is great, but....

• It is not a goal in itself, just a tool to reach good efficiency or high revenue

• Incentive-compatible protocols are often less uesful in practice

Instead, we should emphasise

• Simple and practical protocols

• Example: First-price auctions

Which protocol is ”best”?

• An incentive-compatible protocol with low revenue?

• A simple protocol where the Nash equilibrium is known to give high revenue, but the optimal strategies are unknown?

• The second one is more likely to be used in practice

Content






6. The Proof

7. Summary



Item 1 100 102 x

Item 2 103 99 x x

Item 3 100 x x

Item 4 105 106 x x

CombPrice

200 205 305

Single Bids Combinatorial Bids


Maximize 100 A1 + 103 A2 + 100 A3 + 105 A4 + 102 B1 + 99 B2 + 106 B4 + 200 C + 205 D + 305 E

subject toA1 + B1 + C ≤ 1 (only one bid can win Commodity 1)A2 + B2 + D + E ≤ 1 (only one bid can win Commodity 2)A3 + D + E ≤ 1 (only one bid can win Commodity 3)A4 + B4 + C + E ≤ 1 (only one bid can win Commodity 4)


Item 1 100 102 x

Item 2 103 99 x x

Item 3 100 x x

Item 4 105 106 x x

CombPrice

200 205 305

...and here is why Computer Scientists

care about these auctions

What if....


Item 1 ? ? ?

Item 2 ? ? ? ?

Item 3 ? ? ?

Item 4 ? ? ? ?

...we do not allow combinatorial bids,but only single bids?(Simultaneous auction)

Will the auctioneer earn higher or lower revenue?

Intuition

• Combinatorial Auction• Bid High: No risk of winning just a few items, so I can

afford to bid above my single-bid valuation• Bid Low: If I win, my combination is part of a puzzle

with many other winning combinations. If they bid high I can still bid low and our puzzle wil win anyway. (threshold)

• Simultaneous Auction• Bid High: If I bid high enough, I will beat all others and

win my entire combination.• Bid Low: Potential risk of winning just a few items,

dangerous to bid above single-bid valuation (exposure)

Previous knowledge

• For 2 items and 3 bidders, the simultaneous auction gives higher revenue (Krishna & Rosentahl)

A Nobel Price Problem

• There exists no theoretical evidence for the belief that combinatorial auctions provide higher revenue

• Could we provide any such evidence?

The Plan

1. Provide theoretical evidence that combinatorial auctins give higher revenue

2. Humbly accept the Nobel Price

Content






6. The Proof

7. Summary

The Proof

1. Formal problem

2. Upper bound on simultaneous auctions

3. Lower bounds on combinatorial auctions

4. Comparison

The Proof

1. Formal problem



4. Comparison

Formal Problem

• M items

• N Single Bidders per item

• N Synnergy Bidders, each interested in k items, getting a synnergy α iff all are won.

• Bidders have random valuations, the combinations are randomly selected.

Bidder A Bidder B Bidder C Bidder D Bidder E Bidder F

Item 1 0.78 0.98

Item 2 0.56 0.98

Item 3 0.77 0.98

Item 4 0.55 0.56

Item 5 0.64 0.77

Item 6

Item 7 0.56 0.98

Item 8 0.56

Item 9 0.77

Item 10 0.77

Synnergy 1 1 1

Total Value

0.78 0.55 0.64 6.24 7.08 7.92

Our initial view on the problem

• Traditional game-theoretic approaches can not be used

• No hope in deriving the actual equilibrium strategies

• Try to find some bounds on what is possible to achieve with the two auction protocols.

The Proof

1. Formal problem



4. Comparison

Upper bound on revenue in simultaneous autcions

Main idea: Prove that exposure is a real problem

Lemma:

• We can assume the bidders to be ordered, highest, 2nd, ...

Bidder A

Bidder B

Bidder C

Bidder D

Bidder E

Proof: Adversary argument

Observation:

• You realize synnergy iff you do not collide with any higher bidder

Bidder A

Bidder B

Bidder C

Bidder D

Bidder E

Combinatorial argument: You only realize synnergy if you do not collide with a higher bidder

Given two synnergy bidders, the probability that they do not collide is

The probability that the jth highest bidder gets his synnergy is

Summing over all bidders, theexpected total realizd synnergy is

Adding a maximum valuation of 1 per item,an upper bound on total utility is

Theorem: Upper bound on revenue for simultaneous auctions

The Proof

1. Formal problem



4. Comparison

Lower Bound on combinatorial auction

Main idea: Prove that free riding / threshold problem does not have a major effect

Idea: if a bidder with high valuation bids low, there will be someone else that can benefit from this by raising his bid.

Proving lower bounds on strategies,the general idea:

• Strategies are monotone• Therefore, a bidder X with low valuation has low

probability of winning (since there will be many bidders above him)

• Therefore, X has low expected revenue• Let W be the expected value of a winning bid• If W is low enough, X can bid above W and get a higher

expected revenue than theoretically possible.• We have a contradiction. • So, W can not be too low.

An Example

• 10 items• Combination size k=4 (so only two combinatorial bids can

win)• Millions of bidders• A bidder X with valuation 0.95 per item has very low

chance of winning, since there will probably be two non-colliding bidders with higher valuation than 0.95.

• Suppose the expected value of the lowest winning combinatorial bid is 1.9 per item. Then, since the probability that X does not collide with the other winning bid is quite high, X can get a good expected revenue by bidding 1.91.

• We have a contradiction

Asympotic Result

Lemma: In the combinatorial auction, as the number of bidders approaches infinity, the lowest winning bidapproaches the maximum value k(1+α)

Proof: By contradiction: If the winning bids are lower, therewill be bidders that can get impossibly high revenue by bidding higher

Theorem: In the first-price combinatorial auction as the number of bidders approach infininty, the expected revenueapproaches

A Paremeterized Lower Bound onCombinatorial Auctions

The Proof

1. Formal problem



4. Comparison

A First Comparison

Corollary: As the number of bidders approach infininty, the expectedrevenue of the first-price combinatorial auction is higher than that ofthe simultaneous auction, give M ≥ 2k and K > 2.

A second comparison, finding specific examples

Content






6. The Proof

7. Summary

Summary

Finally, after a couple of years, I know the answer: There is a theoretical support for combinatorial auctions. It does not coverall thinkable cases, but it covers by far more than previous theoretical studies.

What’s next?

Documents

What I Really Wanted To Know About Combinatorial Auctions