“Fair Payments for Efficient Allocations in Public Sector Combinatorial Auctions” To appear in Management Science Plus some other unpublished research

“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• Plus some other unpublished research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

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

II = = set of items being auctionedset of items being auctioned

JJ = = set of biddersset of bidders

bbjj (S) (S) == bid by j on some set S in Ibid by j on some set S in I

SSjj == set won by j in an efficient sol’nset won by j in an efficient sol’n

WW = = the winners in the efficient sol’nthe winners in the efficient sol’n

ppjj == payment made by jpayment made by j

zzCC == win-determ value over C in Jwin-determ value over C in J

z(p)z(p) == w-d value after discounting each w-d value after discounting each bid by surplus at pay vector pbid by surplus at pay vector p

O.R. Perspective, a Set Packing Problem Variation:O.R. Perspective, a Set Packing Problem Variation:General Winner Determination Problem (XOR)General Winner Determination Problem (XOR)

xxjj(S)(S) 1 , for each good 1 , for each good i i

xxjj(S)(S) 1 , for each bidder 1 , for each bidder jj

Where Where xxjj(S)(S) = 1 if bidder = 1 if bidder jj receives set receives set SS

= 0 otherwise= 0 otherwise

jєJ S | i є S

S in I

MaximizeMaximize bbjj(S) x(S) xjj(S)(S)

subject to:subject to:

jєJ S in I

Vickrey-Clarke-Grovespayment mechanism

• Each bidder gets a discount equal to:

zzJJ – z – zJ \ jJ \ 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 collusionVulnerable to shill-bidding and collusionLow (sometimes zero) revenuesLow (sometimes zero) revenues““Unfair!”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

Payment

= p2

Bidder 1

Payment = p1

3

p1+p2 >= 12

6

Pay-as-bid

(18,12)

9

6

VCG

(6,3)

The Core

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.

Define the core with coalitional offerings Define the core with coalitional offerings qqCC , where , where

qqCC is the most money the coalition is the most money the coalition CC will offer to will offer to

pay the seller for a reallocation in their favor:pay the seller for a reallocation in their favor:

ppjj ≥ ≥ qqCC for each subset for each subset CC of of JJ

ppjjVCG VCG ppjj bbjj(S(Sjj))

Representing the core(naïve approach)

j є W

Defining the Core:Defining the Core: Problems and Solutions Problems and Solutions

• A winning bidder’s contribution to a blocking coalition A winning bidder’s contribution to a blocking coalition varies with his payment, i.e., varies with his payment, i.e., qqcc ≠ ≠ zzcc

• There are an exponential number of blocking There are an exponential number of blocking coalitions to consider, each requiring solution of an coalitions to consider, each requiring solution of an NP-hard problemNP-hard problem

Cancel out contributions of coalition Cancel out contributions of coalition members who are also winnersmembers who are also winners

Generate constraints only as they are Generate constraints only as they are violated, i.e. only consider coalitions that violated, i.e. only consider coalitions that

block potential solutions.block potential solutions.

(Main Contribution of the M.S. paper.)(Main Contribution of the M.S. paper.)

Representing the Core

• MS Paper formulation

pj ≥ z(p t) – pj t

• Equivalent (static) formulation

pj ≥ zC – bj (Sj)j є W \ C j є W ∩ C

j є W \ C j є W ∩ C

For all coalitions C in J

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 paymentsAdjusting payments

Minimize Minimize ppjj

ppjj ≥ ≥ z(z(ppττ) - ) - ppjjττ for each for each ττ ≤≤ tt

and and for each for each j є j є

WW


j є W \ Cτ j є W ∩Cτ

j є W

Simplest objective we consider

b1 = 20

b2 = 20

b4 = 28 b5 = 26

b6 = 10

b3 = 20

Winning Bids

b7 = 10 b8 = 10

Non-Winning Bids

VCG paymentsVCG payments

pp11 = 10, p = 10, p22 = 10, p = 10, p33 = 10 = 10

Blocking CoalitionBlocking Coalition

pp44 = 28, p = 28, p33 = 10 = 10

Example of the ProcedureExample of the Procedure

b’1 = 10

b’2 = 10

b4 = 28 b5 = 26

b6 = 10

b’3 = 10

Winning Bids

b7 = 10 b8 = 10

Non-Winning Bids

VCG paymentsVCG payments

pp11 = 10, p = 10, p22 = 10, p = 10, p33 = 10 = 10


pp44 = 28, p = 28, p33 = 10 = 10


Adjusting payments (1)Adjusting payments (1)


pp11 + + pp22 ≥ 38 – 10 = 28 ≥ 38 – 10 = 28

for each for each j є Wj є W


j є W

New paymentsNew payments

pp11 = 14, p = 14, p22 = 14, p = 14, p33 = 10 = 10

b’1 = 14

b’2 = 14

b4 = 28 b5 = 26

b6 = 10

b’3 = 10

Winning Bids

b7 = 10 b8 = 10

Non-Winning Bids


pp11 = 14, p = 14, p22 = 14, p = 14, p33 = 10 = 10


pp22 = 14, p = 14, p55 = 26 = 26


Adjusting payments (2)Adjusting payments (2)


pp11 + + pp22 ≥ 28 ≥ 28

pp11 + + pp33 ≥ 26 ≥ 26

for each for each j є Wj є W


j є W


pp11 = 16, p = 16, p22 = 12, p = 12, p33 = 10 = 10

b’1 = 16

b’2 = 12

b4 = 28 b5 = 26

b6 = 10

b’3 = 10

Winning Bids

b7 = 10 b8 = 10

Non-Winning Bids


pp11 = 16, p = 16, p22 = 12, p = 12, p33 = 10 = 10

No Blocking CoalitionNo Blocking Coalition exists:exists:

These payments are finalThese payments are final

Other Properties and supporting results:

• For any core mechanism, the Nash equilibria in semi-For any core mechanism, the Nash equilibria in semi-sincere strategies correspond exactly to the BPO sincere strategies correspond exactly to the BPO Core paymentsCore payments

• Therefore, we can expect efficient core outcomes Therefore, we can expect efficient core outcomes when using a core mechanismwhen using a core mechanism

• If coordination is sufficiently expensive, then truth-If coordination is sufficiently expensive, then truth-telling by all is a Nash equilibriumtelling by all is a Nash equilibrium

For a payment-minimizing core mechanism:For a payment-minimizing core mechanism:• A form of profitable collusion to reduce total A form of profitable collusion to reduce total

payments is eliminatedpayments is eliminated• The sum of all individual incentives for unilateral The sum of all individual incentives for unilateral

deviation from truth-telling is minimizeddeviation from truth-telling is minimized• Run time compares favorably with other techniques Run time compares favorably with other techniques

for computing core paymentsfor computing core payments

• See MS paper for detailsSee MS paper for details

Conclusions on MS materialConclusions on MS material

• We developed a method that is simple to We developed a method that is simple to describe for computing core paymentsdescribe for computing core payments

• The general algorithm works in any environment The general algorithm works in any environment where WD is solved explicitly, allowing it to be where WD is solved explicitly, allowing it to be applied for any “bid language” environment.applied for any “bid language” environment.

• We have heuristically minimized the number of We have heuristically minimized the number of NP-hard WDs to solve, making this a fast NP-hard WDs to solve, making this a fast methodmethod

• Drastically faster than existing algorithmsDrastically faster than existing algorithms

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.

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

Documents

“Fair Payments for Efficient Allocations in Public Sector Combinatorial Auctions” To appear in Management Science Plus some other unpublished research