Combinatorial Auctions By: Shai Roitman e-mail: [email protected].

Combinatorial Auctions

By: Shai Roitman e-mail: [email protected]

Auctions

• One to many mechanism

• Efficient Allocation of the items.

• Seller Auctions

• Buyer Auctions - Reversed Auctions

Known Auction Types

• Open Cry Auctions– English– Dutch

• Sealed Bid Auctions– First Price – Second Price

The equivalence of auctions

• True Valuations– English – Sealed Bid Second Price

• Winners Curse– Dutch – Sealed Bid First Price

Sealed Bid Auctions advantages

• Communication efficient

• The value of the bid can be kept private.

Items Value

• Private Value - An Item has a value to the bidder regardless of the value to the other bidders– Example: Consumer goods

• Public Value - The item has value in the context of other bidder estimations– Example: Stocks

Strategies for the Auctionsunder private value assumptions

• English Auction– Small increments until maximum price(true

value) reached.

• Second price Sealed Auction– Submit the evaluated value as the bid

• First price Sealed Auction & Dutch Auction– Need to evaluate others evaluation (may use

some distribution on the values of the other bidders) and use this evaluation for setting the bid.

– Winners Curse

• Complex analysis

Strategies for Auctions - continued

Multi Item Auctions - Multi Stage Auction

• Scenario– A set of items has to be sold

• Naive Solution– Hold auctions for each item or set of items one

at a time

Multi Item Auctions - Problems

• How to choose the order of the items to be sold?

• How to bundle several dependant items?

• If the items have dependencies multi stage auctions can lead to inefficient allocation

Combinatorial auction

• Items may be grouped as bundles.

• => Takes into considerations the dependencies between the items.

• => Greater economic efficiency

The Utility function

• Private - Public Value

• Super Additive - Supplemental items

• Sub Additive - Complementary items

• Monotonic - The more the better

• Convex - Diversity

Uses for combinatorial auctions

• FCC Radio spectrum

• Logistics

• Scheduling

• Any purchase of dependant multiple items.

Logistic explicit use case of combinatorial auctions

• Logistics.com - OptiBid(TM)– Trucking companies bid on bundles of lanes– Logistics.com - More than $5 billion in

transportation contracts been bid to date (January 2000) (Ford, Wal-Mart, K-Mart).

Incentive Issues - An example

• 3 bidders {1,2,3}• 2 items {x,y}• Bidder 1 values

– {x,y}=100 {x}={y}=0

• Bidder 2 values– {x,y}=0 {x}={y}=75

• Bidder 3 values– {x,y}=0 {x}={y}=40

Incentive Issues - An example - continued

• If bid truthfully - x->2 , y->3 (Revenue 115)• If Bidder 2 and Bidder 3 belief that the others

truthfully bid their values– Bidder 2 can shade his value of {x} and {y} to 65

and still get the same x->2 y->3 (Revenue 105)– Bidder 3 can shade his value of {x} and {y} to 30

and still get the same x->2 y->3 (Revenue 105)

• If Bidder 2 & Bidder 3 shade their value (65 & 30) then they will lose as {x,y}->1

• => Lost of economic efficiency

Incentive Issues - An example - continued

Threshold Problem

• a collections of bidders whose combined valuation for distinct portions of a subset of items exceed the bid submitted on that subset by some other bidder.

• Difficulty in coordination of their bids to outbid the single large bidder on that subset

Auction Scheme assumptions

• Independent private values for bidders

• values draw from a commonly known distribution

• risk neutral

Auction Design- An optimal mechanism

• Truth Revelation - revelation principle

• No Bidder is made worse off by participating

• Seller Maximum Expected Revenue

Efficiency

• If the allocation of objects to bidders chosen by the seller solves the following equations than the auction is efficient

NjMSjSy

NjjSy

MijSy

ts

jSySv

AuctionEfficient

Otherwise

jtoallocatedisSBundlejSy

objectsofBundlesMSS

objectsdistinctofSetM

BiddersofSetN

Let

MS

Si Nj

Nj MS

j

,1,0),(

1),(

1),(

..

),()(max

0

1),(

.

:

General CAP Formalization

NjMSjSy

NjjSy

MijSy

ts

jSySb

problemAuctionialCombinatorCAP

Otherwise


SbSb

SbundleforannouncedhasNjagentthatbidTheSb

objectsofBundlesMSS


BiddersofSetN

Let

MS

Si Nj

Nj MS

j

i

Nj

i

,1,0),(

1),(

1),(

..

),()(max

)(

0

1),(

)(max)(

.)(

.

:

Vickrey Clarke Groves (VCG) - part 1

*

,1,0),(

1),(

1),(

..

),()(max

.2

.1

yallocationoptimalthisCall

NjMSjSy

NjjSy

MijSy

ts

jSySvV

solvesthatallocationthechoosessellerThe

vreportsjAgent

MS

Si Nj

Nj MS

j

j


k

MS

Si kNj

kNj MS

jk

ysolutionthisCall

kNjMSjSy

kNjjSy

MijSy

ts

jSySvV

Nkeachfor

makemustbiddereachthatpaymentthecomputeTo

\,1,0),(

\1),(

1),(

..

),()(max

:.3

\

\

0)(

),()()(

.4

*

kp

kSySvVVkp

toequalismakeskbidderthatpaymentThe

MS

kk


Nk

k VVV

venueSellerRe

• If no agent has a significant effect on the average V is close to V^(-k) thus the revenue is close to the maximum revenue defined in the General CAP.

Problems in the VCG mechanism

• Solving the CAP problem is hard (NP-Hard)

• Using Approximate solutions => Not incentive compatible

• Payments in VCG are sensitive to the choice of the solution

General CAP Formalization

NjMSjSy

NjjSy

MijSy

ts

jSySb


Otherwise


SbSb


objectsofBundlesMSS


BiddersofSetN

Let

MS

Si Nj

Nj MS

j

i

Nj

i

,1,0),(

1),(

1),(

..

),()(max

)(

0

1),(

)(max)(

.)(

.

:

Multiple object in the CAP Formulation

NjMSjSy

NjjSy

itemstheinisiobjecttimesofnumberlMiljSy

ts

jSySb


Otherwise


SbSb


objectsofBundlesMSS


BiddersofSetN

Let

MS

Si Nj

Nj MS

j

i

Nj

i

,1,0),(

1),(

),(

..

),()(max

)(

0

1),(

)(max)(

.)(

.

:

The CAP (Combinatorial Auction Problem)

• Bidders must submit bid for every subset

• Transmitting the bid sets in a succinct manner

Restriction of conditions => solvable solution - an example

• Restriction– All bidders complement each other– all bidders are symmetric

• Solution– Auction all the items as one item in an optimal

single item auction

Cybernomics experiments

• Performed tests for additive values and valuations with synergies of small , medium or high intensity

• Results– Combinatorial multi round auctions always

superior in efficiency but lower in revenue– Slower convergence (finishing the auctions)

The CAP - continued

• partial solutions– Restriction on the way the bids are transmitted

• OR / OR* Trees

• Single mind restriction

– Sending an Oracle

• Problem of deciding the collection of bids to accept

The SPP Problem

• Given a set of M elements

• collection V of subsets with weights

• Find the largest weight collection of subsets that are pairwise disjoint.

The SPP Formalization

Vjx

Mixa

ts

xc

oblemSPPThe

otherwise

MielementcontainsVinsetjtheifa

otherwise

selectediscweightwithVinsetjtheifx

j

Vjjij

Vjjj

th

ij

jth

j

1,0

1

..

max

Pr

0

1

0

1

SPP Related Problem - Set Partitioning Problem (SPA)

Vjx

Mixa

ts

xc

oblemSPAThe

otherwise

MielementcontainsVinsetjthefa

otherwise


j

Vjjij

Vjjj

th

ij

jth

j

1,0

1

..

max

Pr

0

1

0

1

SPP Related Problem - Set Covering Problem

Vjx

Mixa

ts

xc

oblemSCPThe

otherwise

MielementcontainsVinsetjthefa

otherwise


j

Vjjij

Vjjj

th

ij

jth

j

1,0

1

..

min

Pr

0

1

0

1

What is the complexity of SPP?

• SPP Is a NP-Hard / Complete problem

• SPP Problem is exponential in |V| (V the number of subsets of M)!

• No Hope??

Effective solution to the CAP Problems

• Requirements– Number of distinct bids is not large– Underlying SPP problem can be solved

reasonable quick.

SPP Approximation

• There is no Polynomial algorithm that can deliver a worst case ration larger than n^(E-1) for any E>0

• There is a worst case ratio of O(n/(log n)^2) algorithm (Polynomial algorithm)

Other Approaches

• Decentralized Methods– Setting up a fictitious market determining an

allocation and prices– Choosing an allocation and bidders are required

to send improvements

Conclusions

• Combinatorial Auctions can lead to higher economic efficiency

• Practical Combinatorial Auctions are hard to implement with compliance to the truth revelation principle

Combinatorial Auctions By: Shai Roitman e-mail: [email protected].

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Transcript of Combinatorial Auctions By: Shai Roitman e-mail: [email protected].