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.
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: don’t have to get stuck with less
than what you want
– Substitutes: don’t have to get stuck with more than
what you want
• Disadvantages:
– Potential computational difficulty for determination
of winners, payments, and strategies
C.A. Applications
• Landing slots at airports to control
congestion
• Award of Spectrum to Telecom Co’s
• Shipping Lanes
(reverse auction)
• Industrial Procurement (reverse auction)
– CombineNet: world leader in industrial CAs
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
The General Winner Determination Problem:
Maximize: wd(J) =
  bj(S) xj(S)
jєJ S in I
subject to:
  xj(S)  1 , for each good i
jєJ
S|iєS
 xj(S)  1 ,
for each bidder j
S in I
Where
xj(S) = 1 if bidder j receives set S
= 0 otherwise
bj(S) = bidder j’s bid on set S
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).
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 bidder’s 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
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)
The Beautiful Generalization:
The 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 } )
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
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.
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
The Practical Generalization:
Core-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
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.
5 bidder example with bids on {A,B}
• b1{A} = 28
Winners
• b2{B} = 20
• b3{AB} = 32 VCG prices:
• b4{A} = 14 p1= 14
• b5{B} = 12 p2= 12
The Core
Bidder 2
Payment
b4{A} = 14
b3{AB} = 32
b1{A} = 28
Efficient outcome
b2{B} = 20
20
The
Core
12
b5{B} = 12
14
28
32
Bidder 1
Payment
VCG prices: How much can each winner’s
bid be reduced holding others fixed?
Bidder 2
Payment
b4{A} = 14
b1{A} = 28
b3{AB} = 32
b2{B} = 20
20
The
Core
12
VCG
prices
b5{B} = 12
Problem: Bidder 3
can offer seller more
(32 > 26)!
14
28
32
Bidder 1
Payment
Bidder-optimal core prices: Jointly reduce
winning bids as much as possible
Bidder 2
Payment
b4{A} = 14
b1{A} = 28
b3{AB} = 32
b2{B} = 20
20
The
Core
12
VCG
prices
Problem: bidderoptimal core prices are
not unique!
14
b5{B} = 12
28
32
Bidder 1
Payment
Core point closest to VCG prices
Bidder 2
Payment
b4{A} = 14
b1{A} = 28
b3{AB} = 32
b2{B} = 20
20
Unique
core prices
15
12
VCG
prices
b5{B} = 12
Minimize
incentive to
distort bid!
14 17
28
32
Bidder 1
Payment
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
Representing the Core
• Formulation
 pj ≥ wdC(p)
For all coalitions C in J
jєW
Exponential number of coalitions so
exponential number of constraints!
Use constraint generation to determine
minimum sum total payments over the
core.
The Separation Problem:
Finding “the most violated blocking coalition”
for a given payment vector pt
• At pt , reduce each of the winning bidder’s
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
> Total Payments
• Then a violated coalition has been found
• Add to core formulation and re-iterate
Adjusting payments
Minimize  pj
jєW
 pj ≥ wd(pτ) -  pjτ for each τ ≤ t
j є W \ Cτ
j є W ∩Cτ
and for each j є W
pjVCG  pj  bj(Sj)
Example of the Procedure
Winning Bids
b1 = 20
Non-Winning Bids
b4 = 28
b5 = 26
b6 = 10
b2 = 20
b3 = 20
b7 = 10
b8 = 10
VCG payments
Blocking Coalition
p1 = 10, p2 = 10, p3 = 10
p4 = 28, p3 = 10
Example of the Procedure
Winning Bids
b’1 = 10
Non-Winning Bids
b4 = 28
b5 = 26
b6 = 10
b’2 = 10 b’3 = 10
b7 = 10
b8 = 10
VCG payments
Blocking Coalition
p1 = 10, p2 = 10, p3 = 10
p4 = 28, p3 = 10
Adjusting payments (1)
Minimize  pj
jєW
p1 + p2 ≥ 38 – 10 = 28
for each j є W
pjVCG  pj  bj(Sj)
New payments
p1 = 14, p2 = 14, p3 = 10
Example of the Procedure
Winning Bids
b’1 = 14
Non-Winning Bids
b4 = 28
b5 = 26
b6 = 10
b’2 = 14 b’3 = 10
b7 = 10
b8 = 10
New payments
Blocking Coalition
p1 = 14, p2 = 14, p3 = 10
p2 = 14, p5 = 26
Adjusting payments (2)
Minimize  pj
jєW
p1 + p2 ≥ 28
p1 + p3 ≥ 26
for each j є W
pjVCG  pj  bj(Sj)
New payments
p1 = 16, p2 = 12, p3 = 10
Winning Bids
b’1 = 16
Non-Winning Bids
b4 = 28
b5 = 26
b6 = 10
b’2 = 12 b’3 = 10
b7 = 10
b8 = 10
New payments
No Blocking Coalition exists:
p1 = 16, p2 = 12, p3 = 10
These payments are final
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