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Fair Payments for Efficient Allocations in
Public Sector Combinatorial Auctions

• To appear in Management Science
• Plus some other unpublished research
• Robert Day University of CT
• S. Raghavan University of MD
• Paul Milgrom Stanford

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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.

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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

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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

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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.

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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

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Notation

• I set of items being auctioned
• J set of bidders
• bj (S) bid by j on some set S in I
• Sj set won by j in an efficient soln
• W the winners in the efficient soln
• pj payment made by j
• zC win-determ value over C in J
• z(p) w-d value after discounting each bid
  by surplus at pay vector p

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O.R. Perspective, a Set Packing Problem
VariationGeneral Winner Determination Problem
(XOR)
Maximize ? ? 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
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Vickrey-Clarke-Grovespayment 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

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Example Bids on A,B,C

• b1AB 18
• b2C 12
• b3A 3
• b4B 3
• b5C 3
• b6ABC 12

Winners
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Bidder 2 Payment p2
Pay-as-bid (18,12)
The Core
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VCG (6,3)
3
p1p2 gt 12
Bidder 1 Payment p1
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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
• Ngt1 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)

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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.

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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
• ? pj qC for each subset C of J
• pjVCG ? pj ? bj(Sj)

j ? W
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Defining the Core Problems and Solutions

• A winning bidders contribution to a blocking
  coalition varies with his payment, i.e., qc ? zc
• There are an exponential number of blocking
  coalitions to consider, each requiring solution
  of an NP-hard problem

Cancel out contributions of coalition members who
are also winners Generate constraints only as
they are violated, i.e. only consider coalitions
that block potential solutions. (Main
Contribution of the M.S. paper.)
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Representing the Core

• MS Paper formulation
• ? pj z(p t) ? pj t
• Equivalent (static) formulation
• ? pj zC ? bj (Sj)

For all coalitions C in J
j ? W \ C
j ? W n C
For all coalitions C in J
j ? W \ C
j ? W n C
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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 z(pt) - ? pjt for each t t
• and for each j ? W
• pjVCG ? pj ? bj(Sj)

• Simplest objective we consider

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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Other Properties and supporting results

• For any core mechanism, the Nash equilibria in
  semi-sincere 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
  truth-telling 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

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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

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Newer results

• A shill-proof mechanism must be a core-mechanism
• 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.

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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 -gt my new technique for bid weights
  has no guiding theory-gt weights must be set
  exogenously