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					                      What I Really
                     Wanted To Know
                         About
Arne Andersson        Combinatorial
                        Auctions
Trade Extensions
Uppsala 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

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
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
              Comb                     200      205    305
              Price



                 Single Bids          Combinatorial Bids
Combinatorial Auction
                                      Bid A    Bid B Bid C Bid D Bid E
 ...and here is why         Item 1       100       102       x
Computer Scientists
  care about these          Item 2       103        99              x     x
      auctions              Item 3       100                        x     x
                            Item 4       105       106       x            x
                            Comb                          200     205   305
                            Price
  Maximize
  100 A1 + 103 A2 + 100 A3 + 105 A4
  + 102 B1 + 99 B2 + 106 B4 + 200 C + 205 D + 305 E
  subject to
  A1 + 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)
What if....
                                   Bid A   Bid B Bid C Bid D Bid E
...we do not allow
combinatorial bids,       Item 1       ?      ?       ?
but only single bids?
                          Item 2       ?      ?            ?     ?
(Simultaneous auction)
                          Item 3       ?                   ?     ?
                          Item 4       ?      ?       ?          ?




       Will the auctioneer earn higher or lower revenue?
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
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

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
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

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
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
       Bids                   Allocations




                 Auctioneer
The Real World


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



                  Bidders
       Bids                   Allocations




                 Auctioneer
One idea (incentive-compatability)
”Use an auction mechanisms where the
bidder’s best strategy is to bid truthfully”




                    Bidders
       Bids                     Allocations




                   Auctioneer
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

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
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
              Comb                     200      205    305
              Price



                 Single Bids          Combinatorial Bids
Combinatorial Auction
                                      Bid A    Bid B Bid C Bid D Bid E
 ...and here is why         Item 1       100       102       x
Computer Scientists
  care about these          Item 2       103        99              x     x
      auctions              Item 3       100                        x     x
                            Item 4       105       106       x            x
                            Comb                          200     205   305
                            Price
  Maximize
  100 A1 + 103 A2 + 100 A3 + 105 A4
  + 102 B1 + 99 B2 + 106 B4 + 200 C + 205 D + 305 E
  subject to
  A1 + 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)
What if....
                                   Bid A   Bid B Bid C Bid D Bid E
...we do not allow
combinatorial bids,       Item 1       ?      ?       ?
but only single bids?
                          Item 2       ?      ?            ?     ?
(Simultaneous auction)
                          Item 3       ?                   ?     ?
                          Item 4       ?      ?       ?          ?




       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

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
The Proof

1.   Formal problem
2.   Upper bound on simultaneous auctions
3.   Lower bounds on combinatorial auctions
4.   Comparison
The Proof

1.   Formal problem
2.   Upper bound on simultaneous auctions
3.   Lower bounds on combinatorial auctions
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         0.78     0.55      0.64     6.24     7.08     7.92
Value
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
2. Upper bound on simultaneous
   auctions
3. Lower bounds on combinatorial auctions
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, the
expected 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
2. Upper bound on simultaneous auctions
3. Lower bounds on combinatorial
   auctions
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 bid
approaches the maximum value k(1+α)

Proof: By contradiction: If the winning bids are lower, there
will 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 revenue
approaches
A Paremeterized Lower Bound on
Combinatorial Auctions
The Proof

1.   Formal problem
2.   Upper bound on simultaneous auctions
3.   Lower bounds on combinatorial auctions
4.   Comparison
A First Comparison
Corollary: As the number of bidders approach infininty, the expected
revenue of the first-price combinatorial auction is higher than that of
the simultaneous auction, give M ≥ 2k and K > 2.
A second comparison, finding
specific examples
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
Summary

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

What’s next?

				
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