“Fair Payments for Efficient
Allocations in Public Sector
Combinatorial Auctions”
• To appear in Management Science
• Plus some other unpublished research
Robert Day
S. Raghavan
Paul Milgrom
University of CT
University of MD
Stanford
What to take away from this talk:
• Combinatorial auctions are an exciting new area with
many applications and research opportunities
• An understanding of how O.R. concepts enable
better economic outcomes
• An understanding of the core in economics
• Core auctions provide the most practical payment
schemes for combinatorial auctions in general
• Combinatorial Auction Test Suite (CATS) data provide
a set of benchmarks for testing new auction
algorithms
• Read Combinatorial Auctions, Cramton Shoham,
Steinberg eds.
Combinatorial auctions
• Multiple different items are sold
simultaneously
• Bidders can bid on combinations of items
• When goods are complements, bidders can
be sure not to get a partial set
• When goods are substitutes, can be sure not
to pay too much
• Forward and reverse, iterative and sealed-bid
variations exist
Industrial Auction Applications
• CombineNet is the world-leader in
hosting “expressive commerce” events
• Reverse auction applications include
procurement events for a variety of
resources including shipping lanes
Government Auction Applications
• FCC sells spectrum licenses and has
considered package bidding.
• In the UK, OfCom is close to adopting a
combinatorial spectrum license auction using
the techniques described here. (I coded it)
• FAA: combinatorial landing slot auctions have
been proposed to control congestion, but less
likely to happen soon.
A Practical Auction format
• In the clock-proxy auction (due to Ausubel,
Cramton, and Milgrom) linear prices go up until
there is no excess demand
• Activity rules usher bidding along
• A final sealed-bid auction is needed to correct
for the limitations of linear prices and allow for
efficiency
• Here we focus on the final sealed-bid round
Notation
I
J
bj (S)
Sj
W
pj
zC
z(p)
=
=
=
=
=
=
=
=
set of items being auctioned
set of bidders
bid by j on some set S in I
set won by j in an efficient sol’n
the winners in the efficient sol’n
payment made by j
win-determ value over C in J
w-d value after discounting each
bid by surplus at pay vector p
O.R. Perspective, a Set Packing Problem Variation:
General Winner Determination Problem (XOR)
Maximize
  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
Vickrey-Clarke-Groves
payment mechanism
• Each bidder gets a discount equal to:
zJ – zJ \ j
• Provably dominant-strategy incentive-compatible
(truthful)
• Vickrey won the Nobel prize for this line of work
• Wrought with problems, however, including:
Vulnerable to shill-bidding and collusion
Low (sometimes zero) revenues
“Unfair!”
• Not used in practice
Example: Bids on {A,B,C}
•
•
•
•
•
•
b1{AB} = 18
b2{C} = 12
b3{A} = 3
b4{B} = 3
b5{C} = 3
b6{ABC} = 12
Winners
Bidder 2
Pay-as-bid
Payment
(18,12)
= p2
The Core
6
VCG
3
(6,3)
p1+p2 >= 12
6
9
Bidder 1
Payment = p1
What is the Core?
• From Wikipedia: “The core is the set of feasible
allocations in an economy that cannot be
improved upon by a subset of the set of the
economy's consumers (a coalition).”
• Example:
N>1 miners find many large gold bars.
It takes two to carry a bar home.
If N is even each gets ½ bar (in the core.)
If N is odd the core is empty. (NTU result)
The Core in Auctions
• 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 Mechanism computes payments in
the core with respect to submitted bids.
Representing the core
(naïve approach)
Define the core with coalitional offerings qC , where
qC is the most money the coalition C will offer to
pay the seller for a reallocation in their favor:
 p j ≥ qC
for each subset C of J
jєW
pjVCG  pj  bj(Sj)
Defining the Core:
Problems and Solutions
• A winning bidder’s contribution to a blocking coalition
varies with his payment, i.e., qc ≠ zc
Cancel out contributions of coalition
members who are also winners
• There are an exponential number of blocking
coalitions to consider, each requiring solution of an
NP-hard problem
Generate constraints only as they are
violated, i.e. only consider coalitions that
block potential solutions.
(Main Contribution of the M.S. paper.)
Representing the Core
• MS Paper formulation
 pj ≥ z(p
jєW\C
t) –
 pj
t
For all coalitions C in J
jєW∩C
• Equivalent (static) formulation
 pj ≥ zC –  bj (Sj)
jєW\C
jєW∩C
For all coalitions C in J
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
Simplest objective
we consider
 pj ≥ z(pτ) -  pjτ
j є W \ Cτ
for each τ ≤ t
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
Other Properties and supporting results:
• For any core mechanism, the Nash equilibria in semisincere strategies correspond exactly to the BPO
Core payments
• Therefore, we can expect efficient core outcomes
when using a core mechanism
• If coordination is sufficiently expensive, then truthtelling by all is a Nash equilibrium
For a payment-minimizing core mechanism:
• A form of profitable collusion to reduce total
payments is eliminated
• The sum of all individual incentives for unilateral
deviation from truth-telling is minimized
• Run time compares favorably with other techniques
for computing core payments
• See MS paper for details
Conclusions on MS material
• We developed a method that is simple to
describe for computing core payments
• The general algorithm works in any
environment where WD is solved explicitly,
allowing it to be applied for any “bid
language” environment.
• We have heuristically minimized the number
of NP-hard WDs to solve, making this a fast
method
• Drastically faster than existing algorithms
Newer results
• A shill-proof mechanism must be a coremechanism
• Using a symmetric strictly convex objective w/
super-additive derivative applied to the core,
shill-bidding is dominated
• Certain Quadratic objectives provide a practical
example
• Auctioneer can adjust for publicly known pricing
information, entice bidding with multipliers, and
uniquely decompose payments according to
KKT conditions.
Open avenues
• Combinatorial auctions with stochastic
demand have barely been explored; nothing
exists in combinatorial auctions core theory
• Experimental work with bidding languages
possible
• Elicitation and bidding language work has
begun, but still interesting
• Endogenous bidding in combinatorial
auctions unexplored -> my new technique for
bid weights has no guiding theory-> weights
must be set exogenously