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Combinatorial Auctions: A Survey. Sven de Vries & Rakesh Vohra (2000). Contents. Introduction CAP Decentralized Methods. Introduction(1). Complimentarities between different assets Bidders have preferences not just for particular items but for sets of bundels of items Traveling to LA - PowerPoint PPT Presentation
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Combinatorial Auction s: A Survey Sven de Vries & Rakesh Vohra (200 0)
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Page 1: Combinatorial Auctions:  A Survey

Combinatorial Auctions: A Survey

Sven de Vries & Rakesh Vohra (2000)

Page 2: Combinatorial Auctions:  A Survey

Contents

1. Introduction

2. CAP

3. Decentralized Methods

Page 3: Combinatorial Auctions:  A Survey

Introduction(1)

• Complimentarities between different assets– Bidders have preferences not just for particular items but for

sets of bundels of items– Traveling to LA

• (restaurants and hotels for the intermediate cities, car)or (airline ticket, taxi)

• Auctions where bidders submit bids on combinations : recently been aroused– Jackson(1976),Caplice(1996),Rothkopf(1998),Fujishima(199

9),Sandholm(1999)– Increases in computing power

Page 4: Combinatorial Auctions:  A Survey

Introduction(2)

• Tools– ‘SBIDS’ by SAITECH-INC– ‘OptiBid’ by Logistics.com

• Combinatorial Auction Problem (CAP)– Selecting the winning set of bids.– Can be formulated as an Integer Program

Page 5: Combinatorial Auctions:  A Survey

1. Introduction

2. CAP

3. Decentralized Methods

Page 6: Combinatorial Auctions:  A Survey

CAP

1. CAP2. SPP3. Solvable Instances of SPP4. Exact Methods5. Approximate Methods

Page 7: Combinatorial Auctions:  A Survey

CAP(1)

Difficulty Resolution

– Each bidder must submit a bid for every subset of objects he is interested in– How to transmit this bidding function in a succinct way to the auctioneer

–To restrict the kinds of combinations that bid

ders may bid on

– How to decide which collection of bids to accept - Solving CAP

CAP(Combinatorial Auction Problem)

-Selecting the winning set of bids-

Page 8: Combinatorial Auctions:  A Survey

CAP(2)

Difficulty Resolution

– Each bidder must submit a bid for every subset of objects he is interested in– How to transmit this bidding function in a succinct way to the auctioneer

–To restrict the kinds of combinations that bid

ders may bid on

– How to decide which collection of bids to accept - Solving CAP

CAP(Combinatorial Auction Problem)

-Selecting the winning set of bids-

Page 9: Combinatorial Auctions:  A Survey

CAP(3)

• Notations– N : the set of bidders– M : the set of m distinct objects– S : subset of M– bj(S) : the bid that agent j in N has announced he is willing to

pay for S– ( ) max ( )j

j Nb S b S

Page 10: Combinatorial Auctions:  A Survey

CAP(4)

• CAP formula :

max ( ) ( )

. . ( ) 1

( ) 0,1

S M

i S

b S x S

s t x S i M

x S S M

Page 11: Combinatorial Auctions:  A Survey

CAP(4)

• CAP formula :

– x(S) = 1 : the highest bid on the set S is to be accepted 0 : no bid on the set S are accepted

max ( ) ( )

. . ( ) 1

( ) 0,1

S M

i S

b S x S

s t x S i M

x S S M

Page 12: Combinatorial Auctions:  A Survey

CAP(4)

• CAP formula :

– : no object in M is assigned to more than one bidder

max ( ) ( )

. . ( ) 1

( ) 0,1

S M

i S

b S x S

s t x S i M

x S S M

( ) 1i S

x S i M

Page 13: Combinatorial Auctions:  A Survey

CAP(4)

• CAP formula :

• Call this formulation CAP1

max ( ) ( )

. . ( ) 1

( ) 0,1

S M

i S

b S x S

s t x S i M

x S S M

Page 14: Combinatorial Auctions:  A Survey

CAP(5)

• Superadditive : – for all j∈N and A,B⊂M

such that

• CAP1 correctly models CAP when the bid functions bj are all superadditive– The goods complement each other.

• When goods are substitutes, CAP1 is incorrect.– Why ?

• Superadditive formula doesn’t hold for some j,A,B.• An optimal solution to CAP1 may assign A,B to bidder j and inco

rrectly record a revenue of bj(A)+bj(B) rather than

( ) ( ) ( )j j jb A b B b A B

( )jb A B

A B

Page 15: Combinatorial Auctions:  A Survey

CAP(6)

• How to obviate this difficulty ?– Through the introduction of dummy good g

• bj(A) => bj(A∪{g}) bj(B) => bj(B∪{g}) bj(A∪B) remains the same

M => M∪{g}• If A is assigned to j, then B cannot be assigned to j.

– Through the formula CAP2

Page 16: Combinatorial Auctions:  A Survey

CAP(7)

• CAP2 formulation

max ( ) ( , )

. . ( , ) 1

( , ) 1

( , ) 0,1 ,

j

j N S M

i S j N

S M

b S y S j

s t y S j i M

y S j j N

y S j S M j N

• CAP1 formulation

max ( ) ( )

. . ( ) 1

( ) 0,1

S M

i S

b S x S

s t x S i M

x S S M

Page 17: Combinatorial Auctions:  A Survey

CAP(8)

• CAP2 formulation

max ( ) ( , )

. . ( , ) 1

( , ) 1

( , ) 0,1 ,

j

j N S M

i S j N

S M

b S y S j

s t y S j i M

y S j j N

y S j S M j N

No bidder receives more than one subset

Page 18: Combinatorial Auctions:  A Survey

CAP(9)

• CAP2 formulation

max ( ) ( , )

. . ( , ) 1

( , ) 1

( , ) 0,1 ,

j

j N S M

i S j N

S M

b S y S j

s t y S j i M

y S j j N

y S j S M j N

Overlapping sets of goods are never assigned

Page 19: Combinatorial Auctions:  A Survey

CAP(10)

• Assumption of CAP1,CAP2 – There is at most one copy of each object.

• Extending the formulation– The case when there are multiple copies of the same object

and each bidder wants at most one copy of each object :• The right hand sides of the contraints in CAP1, CAP2 take on val

ues larger than 1.– The case when there are multiple copies and the bidder may

want more than one copy of the same object :• Multi-unit combinatorial auctions (Leyton-Brown 2000)

Page 20: Combinatorial Auctions:  A Survey

CAP

1. CAP2. SPP3. Solvable Instances of SPP4. Exact Methods5. Approximate Methods

Page 21: Combinatorial Auctions:  A Survey

SPP(1)

• Set Packing Problem– Given a ground set M of elements and a collection V of subs

ets with non-negative weights, find the largest weight collection of subsets that are pairwise disjoint.

Page 22: Combinatorial Auctions:  A Survey

SPP(2)

• Set Packing Problem– Given a ground set M of elements and a collection V of subs

ets with non-negative weights, find the largest weight collection of subsets that are pairwise disjoint.

• Notation– x(j) = 1 if the j-th set in V with weight c(j) is selected

0 otherwise– a(i,j) = 1 if the j-th set in V contains element i∈M

0 otherwise

Page 23: Combinatorial Auctions:  A Survey

SPP(3)

• Notation– x(j) = 1 if the j-th set in V with weight c(j) is selected

0 otherwise– a(i,j) = 1 if the j-th set in V contains element i∈M

0 otherwise

• SPP Formulation

max ( ) ( )

. . ( , ) ( ) 1

( ) 0,1

j V

j V

c j x j

s t a i j x j i M

x j j V

Page 24: Combinatorial Auctions:  A Survey

SPP(3)

• Notation– x(j) = 1 if the j-th set in V with weight c(j) is selected

0 otherwise– a(i,j) = 1 if the j-th set in V contains element i∈M

0 otherwise

• SPP Formulation

max ( ) ( )

. . ( , ) ( ) 1

( ) 0,1

j V

j V

c j x j

s t a i j x j i M

x j j V

• CAP Formulation

max ( ) ( )

. . ( ) 1

( ) 0,1

S M

i S

b S x S

s t x S i M

x S S M

Page 25: Combinatorial Auctions:  A Survey

SPP(3)

• Notation– x(j) = 1 if the j-th set in V with weight c(j) is selected

0 otherwise– a(i,j) = 1 if the j-th set in V contains element i∈M

0 otherwise

• SPP Formulation

max ( ) ( )

. . ( , ) ( ) 1

( ) 0,1

j V

j V

c j x j

s t a i j x j i M

x j j V

• CAP Formulation

max ( ) ( )

. . ( ) 1

( ) 0,1

S M

i S

b S x S

s t x S i M

x S S M

Page 26: Combinatorial Auctions:  A Survey

SPP(4)

Other related Prolems

Set Partitioning Problem(SPA)

Set Covering Problem(SCP)

max ( ) ( )

. . ( , ) ( ) 1

( ) 0,1

j V

j V

c j x j

s t a i j x j i M

x j j V

max ( ) ( )

. . ( , ) ( ) 1

( ) 0,1

j V

j V

c j x j

s t a i j x j i M

x j j V

Page 27: Combinatorial Auctions:  A Survey

SPP(5)

Set Partitioning Problem(SPA)

– Bidders are sellers (rather than buyers).

– Trucking companies bidding for the opportunity to ship goods from a particular warehouse to retail outlet.

max ( ) ( )

. . ( , ) ( ) 1

( ) 0,1

j V

j V

c j x j

s t a i j x j i M

x j j V

Page 28: Combinatorial Auctions:  A Survey

SPP(6)

Set Covering Problem(SCP)

– Auction problems in procurement rather than selling terms.

– Scheduling of crews for railways.

max ( ) ( )

. . ( , ) ( ) 1

( ) 0,1

j V

j V

c j x j

s t a i j x j i M

x j j V

Page 29: Combinatorial Auctions:  A Survey

Complexity of SPP

• No polynomial time algorithm for SPP is known.• Any algorithm for the CAP that uses directly the bids f

or the sets, must scan the bids and the number of such bids could be exponential in |M|.– |M| : the number of variables

=> |V| : the number of solutions to check = 2|M|

• SPP : NP-hard (NP-complete)• Effective solution procedures for CAP

– The number of distinct bids is not large– Be structured in computationally useful ways.

Page 30: Combinatorial Auctions:  A Survey

CAP

1. CAP2. SPP3. Solvable Instances of SPP4. Exact Methods5. Approximate Methods

Page 31: Combinatorial Auctions:  A Survey

Solvable Instances of SPP

1. Total Unimodularity

2. Balanced Matrices

3. Perfect Matrices

4. Graph Theoretic Methods

5. Using Preferences

Page 32: Combinatorial Auctions:  A Survey

Solvable Instances of SPP

• Usual way in which instances SPP can be solved by a polynomial algorithm– When the extreme points of the polyhedron

are all integral, i.e. 0-1.– In these cases, we can simply drop the integrality requireme

nt from the SPP and solve it as a linear program

• A polyhedron with all integral extreme points is called integral.

( ) { : ( , ) ( ) 1 ; ( ) 0 }j V

P A x a i j x j i M x j j V

Page 33: Combinatorial Auctions:  A Survey

Total Unimodularity(TU) (1)

• A matrix is TU if the determinant of every square submatrix is 0,1 or –1.

• A : TU At : TU• If A={a(i,j)}i∈M,j∈V is TU, then all extreme point of the po

lyhedron P(A) are integral.• There is a polynomial time algorithm to decide whethe

r a matrix is TU.

Page 34: Combinatorial Auctions:  A Survey

Total Unimodularity(TU) (2)

• Theorem 2.1) Let B be a matrix each of whose entries is 0,1 or -1. Suppose each subset S of columns of B can be divided into two sets L and R such that

then B is TU. The converse is also true.

• Theorem 2.2) All 0-1 matrices with the consecutive ones property are TU.– A 0-1 matrix has the consecutive ones property if the non-zer

o entries in each column occur consecutively.

0,1ij ijj S L j S R

b b i

Page 35: Combinatorial Auctions:  A Survey

Total Unimodularity(TU) (3)

• For example, – Objects to be auctioned : parcels of land along a shore line

• Shore line is important : it imposes a linear order on the parcels– Make a restriction to bid only contiguous parcels

• The most interesting combinations would be contiguous, in the bidders eyes.

– Two computational consequences.• Number of distinct bids would be limited by a polynomial in the

number of objects.• The constraint matrix A of the CAP would have the consecutive o

nes property in the columns.

Page 36: Combinatorial Auctions:  A Survey

Balanced Matrices(1)

• A 0-1 matrix B is balanced if it has no square submatrix of odd order with exactly two 1’s in each row and column.

• Theorem 2.3) Let B be a balanced 0-1 matrix. Then the following linear program :

has an integral optimal solution whenever the c(j)’s are integral.

max ( ) ( ) : ( ) 1 , ( ) 0ijj j

c j x j b x j i x j j

Page 37: Combinatorial Auctions:  A Survey

Balanced Matrices(2)

• For example,– Consider a tree T with a distance function d.

• v : vertex of T• N(v,r) : set of all vertices in T that are within distance r of v.

– The vertices represent parcels of land connected by a read network with no cycles.

– Bidders bid for subsets of parcels which is to be of the form N(v,r).

– Row of the constraint matrix : for each vertexColumn : for each set of the form N(v,r)

– This constraint matrix is balanced.

Page 38: Combinatorial Auctions:  A Survey

Perfect Matrices

• If the contraints matrix A can be identified with the vertex-clique adjacency matrix of what is known as a perfect graph, then SPP can be solved in polynomial time.

• A simple graph G is perfect if, for every induced subgraph H of G, the number of vertices in a maximum clique is – , the chromatic number of H, is the minumum k for whi

ch H is k-colorable.( )H

( )H

Page 39: Combinatorial Auctions:  A Survey

Graph Theoretic Methods

• There are situations where P(A) is not integral yet the SPP can be solved in polynomial time because the contraint matrix A admits a graph theoretic interpretation in terms of an easy problem.– When each column of the matrix A contains at most two 1’s.

=> maximum weight matching problem (can be solved in polynomial time)

• At most two 1’s per row of A => NP-hard– When A has the circular ones property.

• A 0-1 has the circular ones property if the non-zero entries in each column (row) are consecutive

• First and last entries in each column (row) are treated consecutive

• Note the resemblance to the consecutive ones property

Page 40: Combinatorial Auctions:  A Survey

Graph Theoretic Methods

• There are situations where P(A) is not integral yet the SPP can be solved in polynomial time because the contraint matrix A admits a graph theoretic interpretation in terms of an easy problem.– When each column of the matrix A contains at most two 1’s.

=> maximum weight matching problem (can be solved in polynomial time)

• At most two 1’s per row of A => NP-hard– When A has the circular ones property.

=> A can be identified with the vertex-clique adjacency matrix of a circular arc graph.=> maximum weight independent set problem for a circular arc graph. (can be solved in poly time)

Page 41: Combinatorial Auctions:  A Survey

Using Preferences(1)

• Restrictions in the preference orderings of the bidders– Suppose that bidders come in two types

• Type one : bj(.) = g1(.)• Type two : bj(.) = g2(.)

where g1 and g2 are non-decreasing integer valued supermodular functionsThe dual of CAP2 is :

1 1

2 2

min

. . ( ) ,

( ) ,

, 0 ,

i ji M j N

i ji S

i ji S

i j

p q

s t p q g S S M j N

p q g S S M j N

p q i M j N

Page 42: Combinatorial Auctions:  A Survey

Using Preferences(1)

• Restrictions in the preference orderings of the bidders– Suppose that bidders come in two types

• Type one : bj(.) = g1(.)• Type two : bj(.) = g2(.)

where g1 and g2 are non-decreasing integer valued supermodular functionsThe dual of CAP2 is :

1 1

2 2

min

. . ( ) ,

( ) ,

, 0 ,

i ji M j N

i ji S

i ji S

i j

p q

s t p q g S S M j N

p q g S S M j N

p q i M j N

This Problem is an instance of the polymatroid intersection problem.(polynomially solvable)

Page 43: Combinatorial Auctions:  A Survey

Using Preferences(1)

• Restrictions in the preference orderings of the bidders– Suppose that bidders come in two types

• Type one : bj(.) = g1(.)• Type two : bj(.) = g2(.)

where g1 and g2 are non-decreasing integer valued supermodular functions

– Using the method to solve problems with three or more types of bidders is not possible. • It is known in those cases that the dual problem above admits f

ractional extreme points.• The problem of finding an in integer optimal solution for the inte

rsection of three or more polymatroids is NP-hard.

Page 44: Combinatorial Auctions:  A Survey

Using Preferences(2)

• Restrictions in the preference orderings of the bidders– When each of the bj(.) have the gross substitutes property, C

AP2 reduces to a sequence of matroid partition problems, each of which can be solved in polynomial time.

Page 45: Combinatorial Auctions:  A Survey

CAP

1. CAP2. SPP3. Solvable Instances of SPP4. Exact Methods5. Approximate Methods

Page 46: Combinatorial Auctions:  A Survey

Exact Methods(1)

• The upper bound on the optimal solution value is obtained by solving a relaxation of the optimization problem.– Replace the given problem by one with a larger feasible regio

n that is more easily solved.• Lagrangean relaxation

– Will be discussed later• Linear programming relaxation

– Only the integrality constraints are relaxed

Page 47: Combinatorial Auctions:  A Survey

Exact Methods(2)

• Exact methods– Branch and bound– Cutting planes– Hybrid called branch and cut

Page 48: Combinatorial Auctions:  A Survey

Exact Methods(2)• Exact methods

– Branch and bound1. At each stage, after solving the LP, a fractional variable x j is se

lected and two subproblems are set up, one where x j=1 and the other where xj=0. (Branch)

2. Solve the LP relaxation of the two subproblems. 3. From each subproblem with a nonintegral solution we branch a

gain to generate two subproblems and so on.4. By comparing the LP bound across nodes in different branches

of the tree, one can prune some branches in advance. (Bound)– Cutting planes– Hybrid called branch and cut

Page 49: Combinatorial Auctions:  A Survey

Exact Methods(3)

• Exact methods– Branch and bound– Cutting planes

• Find linear inequalities (cuts) that are violated by a solution of a given relaxation but are satisfied by all feasible zero-one solution.

• If one adds enough cuts, one is left with integral extreme points.

– Hybrid called branch and cut

Page 50: Combinatorial Auctions:  A Survey

Exact Methods(4)

• Exact methods– Branch and bound– Cutting planes– Hybrid called branch and cut

• Works like branch and bound, but tightens the bounds in every node of the tree by adding cuts.

• Since even small instances of the CAP1 may involve a huge number of columns (bids), this method needs to be augmented with another method known as column generation.(It works by generating a column when needed rather than all at once.)

Page 51: Combinatorial Auctions:  A Survey

Exact Methods(5)

• How successful exact approaches are :– Being able to find an optimal solution to an instance of SPA

with 1,053,137 variables and 145 constraints in under 25 minutes.

• Major impetus behind the desire to solve large instances of SPA(SPC) quickly has been the airline industry.– Assinging crews to routes can be formulated as an SPA.

• The rows of the SPA correspond to flight legs.• The columns correstpond to a sequence of flight legs that would

be assigned to a crew.

Page 52: Combinatorial Auctions:  A Survey

CAP

1. CAP2. SPP3. Solvable Instances of SPP4. Exact Methods5. Approximate Methods

Page 53: Combinatorial Auctions:  A Survey

Approximate Methods• Probably every heuristic approach for solving general i

nteger programming problems has been applied to the SPP.– Greedy, Interchange/steepest ascent approach, genetic algor

ithms, probabilistic search, simulated annealing, neural networks

• Give up on finding the optimal solution. – Rather one seeks a feasible solution fast and hopes that it is

near optimal.– How close to optimal is the solution ?

• Worst-case analysis• Probabilistic analysis• Empirical testing

Page 54: Combinatorial Auctions:  A Survey

1. Introduction

2. CAP

3. Decentralized Methods

Page 55: Combinatorial Auctions:  A Survey

Decentralized Methods

1. Duality in Integer Programming

2. Lagrangean Relaxation

Page 56: Combinatorial Auctions:  A Survey

Decentralized Methods

• One way of reducing some of the computational burden in solving the CAP.– Auctioneer : sets prices for the objects

Agents : announce which sets of objects they will purchase ar the posted prices

– If two or more agents compete for the same object, the auctioneer adjusts the price vector.

– Bidders : save from specifying their bids for every possible combinationauctioneer : saves from having to process each bid function

Page 57: Combinatorial Auctions:  A Survey

Duality in Integer Programming(1)

• Decentralized approach– Auctioneer chooses a feasible solution.– Bidders are asked to submit improvements.– Auctioneer agrees to share a portion of the revenue gain with

the bidder.

• Above method can be viewed as instances of dual based procedures for solving an integer program.

Page 58: Combinatorial Auctions:  A Survey

Duality in Integer Programming(2)

• The (superadditive) dual to SPP – the problem of finding a superadditive, non-decreasing functi

on such that

– If the primal integer program has the integrality property, there is an optimal integer solution to its LP relaxation, and the dual function F will be linear,i.e.,

1: mF R R

min (1)

. . ( )

(0) 0

jj

F

s t F a c j V

F

( ) i ii

F u y u

Page 59: Combinatorial Auctions:  A Survey

Duality in Integer Programming(3)

• The (superadditive) dual to SPP – If the primal integer program has the integrality property, ther

e is an optimal integer solution to its LP relaxation, and the dual function F will be linear,i.e.,The dual becomes :

( ) i ii

F u y u

min

. .

0

ii

ij i ji

i

y

s t a y c j V

y i M

Page 60: Combinatorial Auctions:  A Survey

Duality in Integer Programming(3)

• The (superadditive) dual to SPP – If the primal integer program has the integrality property, ther

e is an optimal integer solution to its LP relaxation, and the dual function F will be linear,i.e.,The dual becomes :

( ) i ii

F u y u

Superadditive dual reduces to the dual of the linear programming relaxation of SPP.yi : can be interpreted as the price of the object i.

min

. .

0

ii

ij i ji

i

y

s t a y c j V

y i M

Page 61: Combinatorial Auctions:  A Survey

Duality in Integer Programming(4)• Solving the superadditive dual problem is as hard as

solving the original primal problem.• It is possible to reformulate the superadditive dual pr

oblem as a linear program.– The number of variables is exponential in the size of the origi

nal problem.– For small or specially structured problems, this can provide s

ome insight.• In general, one relies on the solution to the LP dual a

nd uses its optimal value to guide the search for an optimal solution to the original primal integer program.=> Lagrangean Relaxation

Page 62: Combinatorial Auctions:  A Survey

Lagrangean Relaxation(1)

• Relax some of the constraints of the original problem by moving them into the objective function with a penalty term.– Infeasible solutions : allowed but penalized in proportion to t

he amount of infeasibility.

Page 63: Combinatorial Auctions:  A Survey

Lagrangean Relaxation(2)

• Recall the SPP:

• Notation– ZLP : optimal objective function value to LP relaxation of SPP.

(Note that Z ≤ ZLP)–

s.t.

max ( ) ( )

. . ( , ) ( ) 1

( ) 0,1

j V

j V

Z c j x j

s t a i j x j i M

x j j V

( ) max ( ) ( ) (1 ( , ) ( ))ij V i M j V

Z c j x j a i j x j

0 ( ) 1x j j

Page 64: Combinatorial Auctions:  A Survey

Lagrangean Relaxation(3)

• Theorem3.2)

– Computing Z(∧) is easy.• Simply set x(j)=1 if

0 otherwisesince

( ( ) ( , )) 0ii M

c j a i j

( ) ( ) (1 ( , ) ( )) ( ( ) ( , )) ( )i i ij V i M j V j V i M i M

c j x j a i j x j c j a i j x j

0min( ( ))LPZ Z

Page 65: Combinatorial Auctions:  A Survey

Lagrangean Relaxation(3)

• Theorem3.2)

– Computing Z(∧) is easy.– Using subgradient algorithm, finding ∧ which minimizes Z(∧)

can be done.– Therefore, ZLP can be found in a fast procedure.

• Lagrangean relaxation is not guaranteed to find the optimal solution to the underlying problem.– It finds an optimal solution to a relaxation of it.– The resulting solution may not be too infeasible, so could be

fudged into a feasible solution without a great reduction in objective function value.

0min( ( ))LPZ Z

Page 66: Combinatorial Auctions:  A Survey

Lagrangean Relaxation(4)

• Market Interpretation– Auctioneer chooses a price vector ∧ for the objects.– Bidders submit bids.– If the highest bid c(j) for the jth bundle exceeds

this bundle is tentatively assigned to that bidder.

• SAA (simultaneous ascending auction)– Bidders bid simultaneously in rounds.– Bids must be increased by a specified minimum from one rou

nd to the next.– Bidders adjust prices which is different from the way of Lagra

ngean Relaxation.– Exposure problem occurs.

( , )ii M

a i j

Page 67: Combinatorial Auctions:  A Survey

Lagrangean Relaxation(5)

• Exposure Problem– Bidders pay too much for individual items or bidders with pref

erences for certain bundles drop out early to limit losses.– For example,

• A bidder A values the bundle of goods i and j at $100 but each at $0.

• In SAA, A has to submit high bids on i and j to secure them.• Suppose that it loses the bidding on i.• A is left standing with a high bid j which A valued at $0.

– Any auction scheme that relies on prices for individual items will face this problem.

Page 68: Combinatorial Auctions:  A Survey

Lagrangean Relaxation(6)

• AUSM (Adaptive User Selection Mechanism)– Asynchronous in that bids on subsets can be submitted at an

y time.– Difficult to connect to the Lagrangean ideas.

• Iterative auction scheme– Hybrid of the SAA and AUSM– Easier to connect to the Lagrangean framework.– Bidders submit bids on packages rather than on individual ite

ms.

Page 69: Combinatorial Auctions:  A Survey

The End

Even if the researcher does not find

what was initially expected,

the pursuit of a personally important topic is still rewarding and

generally produces continuing researches.


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