Practical Public Sector Combinatorial Auctions

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Practical Public Sector Combinatorial Auctions

• S. Raghavan University of Maryland
• (joint work with Robert Day, University of
  Connecticut)
• Full paper
• Fair Payments for Efficient Allocations in
  Public Sector Combinatorial Auctions, Management
  Science, Vol 53, No 9, September 2007,
  pp1389-1406.

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What is a Combinatorial Auction?

• Any auction for multiple items in which bidders
  may bid on combinations of items, rather than
  placing bids on items individually.
• Advantages
• Complements dont have to get stuck with less
  than what you want
• Substitutes dont have to get stuck with more
  than what you want
• Disadvantages
• Potential computational difficulty for
  determination of winners, payments, and
  strategies

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C.A. Applications

• Landing slots at airports to control congestion
• Award of Spectrum to Telecom Cos
• Shipping Lanes (reverse auction)
• Industrial Procurement (reverse auction)
• CombineNet world leader in industrial CAs

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Auction Goals

• Efficiency the items being auctioned go to those
  who value them most.
• Price Discovery an iterative process allows
  competitors to learn about supply-demand
  pressures to determine market value of items

5
The General Winner Determination Problem
Maximize wd(J) ? ? bj(S) xj(S)
subject to
j?J S in I

• ? ? xj(S) ? 1 , for each good i
• ? xj(S) ? 1 , for each bidder j

j?J S i ? S
S in I
Where xj(S) 1 if bidder j receives set S
0 otherwise bj(S) bidder js bid on
set S
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Complexity of the General Winner Determination
Problem (WD)

• NP-hard for arbitrary bids
• Many special cases are solvable in
  polynomial-time (Rothkopf, Pekec, Harstad)
• Given advances in computing power, and
  optimization methodologies, most practical WD
  problems can be solved to optimality in practice.
• Additionally, hybrid variations of combinatorial
  auctions have been proposed where initially
  separate parallel auctions are conducted (clock
  phase) for the items, followed by one final round
  of combinatorial auction (proxy phase).

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Classic auction theory

• English auction 1 item, price rises until only
  one willing buyer remains
• Provably optimal strategy stay in until price
  reaches your true value
• Winner pays one bid increment above second
  highest bidders value
• Sealed-bid auction
• Each bidder submits value for item
• First-price variation pay-what-you bid
• Second-price variation winner pays second
  highest bid

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Properties of the Second-price Sealed-bid Auction
for 1 item

• Individual Rationality (IR) Bidders each expect
  a non-negative payoff
• Efficiency the highest bid wins
• Dominant Strategy Incentive Compatibility
  Misreporting value never gives an advantage
• The Core property no coalition can form a
  mutually beneficial renegotiation among
  themselves (notion of core akin to a stable
  outcome)

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The Beautiful GeneralizationThe
Vickrey-Clarke-Groves (VCG) Mechanism

• Focus in mechanism design on Incentive
  Compatibility
• There is a unique mechanism that satisfies
• Individual Rationality
• Efficiency
• Dominant Strategy Incentive Compatibility
• for a general set of items with arbitrary
  preferences.
• Each winning bidder j gets a discount equal to
• wd(J) wd(J \ j )

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VCG example (substitutes)

• b1(A) 4, b1(B) 3, b1(AB) 6
• b2(A) 3, b2(B) 4, b2(AB) 5
• Efficient solution bidder 1 gets A, bidder 2
  gets B
• Discount to bidder 1 wd(1,2) 8, wd(2) 5,
  discount 8 5 3, payment 1.
• Discount to bidder 2 wd(1,2) 8, wd(1) 6,
  discount 8 6 2, payment 2.
• Interpretation Each bidder pays the min. amount
  necessary to take her good away from the other

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The problem with VCG!The Quintessential Example

• b1(A) 2, b2(B) 2, b3(AB) 2
• VCG outcome Bidder 1 and 2 pay zero
• The seller would be better off bargaining with
  bidder 3 for non-zero payment (both would prefer
  it)
• Thus, this outcome is not in the Core
• Non-monotonicity over bids
• More bids can mean lower revenue for the seller!
• Such (non-core) payments are not acceptable in a
  public sector setting.

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Impossibility result in the combinatorial auction
setting

• Suppose we want a sealed-bid Combinatorial
  Auction that has all the nice properties of the
  second-price auction for one item
• Impossibility Result
• No mechanism can simultaneously satisfy
• Individual Rationality
• Efficiency
• Dominant Strategy Incentive Compatibility
• the Core property
• for a general set of items with arbitrary
  preferences

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The Practical GeneralizationCore-Selecting
Mechanisms

• Prevailing attitude in Mechanism Design
  literature Incentive Compatibility must be
  upheld (a constraint.)
• Since VCG is not practically viable we must drop
  DS Incentive Compatibility as a hard constraint
• (IR and Efficiency must stay)
• The perspective of core-selecting mechanisms
  Incentive compatibility is an objective
• Maintain IR, efficiency, and the core property
  (with respect to submitted bids) as constraints
• minimize the incentives to misreport

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The Core

• An Allocation / Payment outcome is blocked if
  there is some coalition of bidders that can
  provide more revenue to the seller in an
  alternative outcome that is weakly preferred to
  the initial outcome by every member of the
  coalition.
• An unblocked outcome is in the core.
• A Core-Selecting Mechanism computes payments in
  the core with respect to submitted bids.

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5 bidder example with bids on A,B

• b1A 28
• b2B 20
• b3AB 32
• b4A 14
• b5B 12

Winners
VCG prices
p1 14
p2 12
16
The Core
b4A 14
b1A 28
b3AB 32
Bidder 2Payment
Efficient outcome
b2B 20
20
The Core
12
b5B 12
Bidder 1Payment
14
32
28
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VCG prices How much can each winners bid be
reduced holding others fixed?
b4A 14
b1A 28
Bidder 2Payment
b3AB 32
b2B 20
20
The Core
12
b5B 12
VCG prices
Problem Bidder 3 can offer seller more (32 gt 26)!
Bidder 1Payment
14
32
28
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Bidder-optimal core prices Jointly reduce
winning bids as much as possible
b4A 14
b1A 28
Bidder 2Payment
b3AB 32
b2B 20
20
The Core
Bidder-optimal core
12
b5B 12
VCG prices
Problem bidder-optimal core prices are not
unique!
Bidder 1Payment
14
32
28
19
Core point closest to VCG prices
b4A 14
b1A 28
Bidder 2Payment
b3AB 32
b2B 20
20
Uniquecore prices
15
12
b5B 12
VCG prices
Minimize incentive to distort bid!
14
32
28
17
Bidder 1Payment
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So why core (stable) pricing?

• Truthful bidding nearly optimal
• Simplifies bidding
• Improves efficiency
• Same as VCG if VCG in core (e.g., substitutes)
• Avoids VCG problems with complements
• Prices that are too low
• Revenue is monotonic in bids and bidders
• Minimizes incentive to distort bids

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Representing the Core

• Formulation
• pj wdC(p)
• Exponential number of coalitions so exponential
  number of constraints!
• Use constraint generation to determine minimum
  sum total payments over the core.

For all coalitions C in J
j ? W
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The Separation ProblemFinding the most
violated blocking coalition for a given payment
vector pt

• At pt , reduce each of the winning bidders bids
  by her current surplus
• That is let bj(S) bj(S) (bj(Sj) - pjt )
• Re-solve the Winner Determination Problem
• If the new Winner Determination value
• gt Total Payments
• Then a violated coalition has been found
• Add to core formulation and re-iterate

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Adjusting payments

• Minimize ? pj
• ? pj wd(pt) - ? pjt for each t t
• and for each j ? W
• pjVCG ? pj ? bj(Sj)

j ? W
j ? W \ Ct j ? W nCt
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Example of the Procedure
Winning Bids
Non-Winning Bids
b4 28
b5 26
b1 20
b6 10
b2 20
b3 20
b7 10
b8 10
VCG payments p1 10, p2 10, p3 10
Blocking Coalition p4 28, p3 10
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Example of the Procedure
Winning Bids
Non-Winning Bids
b4 28
b5 26
b1 10
b6 10
b2 10
b3 10
b7 10
b8 10
VCG payments p1 10, p2 10, p3 10
Blocking Coalition p4 28, p3 10
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Adjusting payments (1)

• Minimize ? pj
• p1 p2 38 10 28
• for each j ? W
• pjVCG ? pj ? bj(Sj)

j ? W
New payments p1 14, p2 14, p3 10
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Example of the Procedure
Winning Bids
Non-Winning Bids
b4 28
b5 26
b1 14
b6 10
b2 14
b3 10
b7 10
b8 10
New payments p1 14, p2 14, p3 10
Blocking Coalition p2 14, p5 26
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Adjusting payments (2)

• Minimize ? pj
• p1 p2 28
• p1 p3 26
• for each j ? W
• pjVCG ? pj ? bj(Sj)

j ? W
New payments p1 16, p2 12, p3 10
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Winning Bids
Non-Winning Bids
b4 28
b5 26
b1 16
b6 10
b2 12
b3 10
b7 10
b8 10
New payments p1 16, p2 12, p3 10
No Blocking Coalition exists These payments are
final
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Conclusions

• Core mechanisms provide a practical alternative
  to VCG when VCG does not work well
• Separation problem provides complexity result
  finding a core point is NP-hard iff Win. Det. is
  NP-hard
• Analogous to Second Price Mechanism if bids were
  replaced by payments, they would just be enough
  to be winning
• When VCG does work well (is in the core) the
  outcomes are the same (Ausubel and Milgrom)
• Government combinatorial auctions (FAA (USA) and
  OfCom (UK)) using the quadratic rule described
  here are ongoing