dimacs workshop on computational issues in auction design, oct. 7, 2004 shmuel oren multi-item...

DIMACS Workshop on Computational Issues in Auction Design, Oct. 7, 2004 Shmuel Oren

Multi-Item Auctions with Credit Limits

Shmuel Orenhttp://www.ieor.berkeley.edu/~oren

Shehzad WadalawalaU.C. Berkeley

October 7, 2004

2Multi-Item Auctions with Credit Limits Shmuel Oren

SIGNIFICANT CONSTRAINTSIN ERCOT SUMMER 2001


720 MW

1200 MW

3750 M

W

2200 M

W580 MW310 M

W

WEST2001LOAD 3,700 MWGEN 5,300 MW

NORTH2001LOAD 20,700 MW GEN 22,000 MW

SOUTH2001LOAD 33,000 MWGEN 45,200 MW

Trading Pattern in ERCOT Summer 2001


The ERCOT Zonal Congestion Management Model

• Three zones and four “Commercially Significant Constraints (CSC)”

• Zonal spot prices and shadow prices on CSCs determined by a zonal economic dispatch algorithm (Min generation cost s.t CSCs)

• Bilateral transactions between zones charged zonal price differences for congestion

• Congestion charges can be hedged by buying Transmission Congestion Rights (TCRs) that constitute financial entitlements to the real time shadow prices on CSCs


Hedging Congestion Charges with TCRs• Full hedging of the congestion charge for 1MW sent

from A to B requires a portfolio of TCRs in proportion to the Power Transfer Distribution Factors (PTDF)

A

B

C

1MW

1MW

2/3

1/3

1/3


Bid Format in the TCR Auction• Bidders submit price and quantity pairs for vectors of

flow distribution Portfolio Bids on CSCs

Bid S-T G-P STP-DowBid

(per. MW)Quantity

(MW)

A1 0.2 0.3 0.5 $10.00 300

A2 1.0 0.0 0.0 $5.00 185

B 0.2 0.5 0.3 $11.25 250

C1 0.6 0.3 0.1 $7.50 240

C2 1.0 0.0 0.0 $1.00 100

D1 0.2 0.4 0.4 $9.50 320

D2 0.0 1.0 0.0 $3.00 140

D3 0.0 0.0 1.0 $2.50 170The letter identifies the bidder while the number identifies bid


Resource Constraints

Available TCRs TCR Limit

S-T 447 S-T 186

G-P 339 G-P 141

STP-Dow 419 STP-Dow 174


LP Formulation of Clearing Algorithm

, ( )

, ( )

( )

max ( designates bid j of bidder )

.

( ) \

, ( ) ( )

( )

ij iji j J i

ij iji j J i

ij ij

ij ij ij J i

b x ij i

st

A x K transmission capacity Shadow price vector

x C i j J i bid quantity

x A L i ownership bound on bidder i

0 , ( )

Clearing unit price for award is

0

or for some resource

ij

Tij ij

Tij ij ij

Tij ij ij ij ijk ijk ik

j

x i j J i

x A

x A b

x C A b x A L k


LP Solution

Total TCRs Awarded (rounded

down)

Bid Award S-T G-P STP-DowPaid total

Paid per MW Bid

A1 300 60 90 150 $2,265.00 $7.55 $10.00

A2 126 126 0 0 $126.00 $1.00 $5.00

B 236.7 47 118 71 $2,655.25 $11.22 $11.25

C1 240 144 72 24 $1,692.00 $7.05 $7.50

C2 40.3 40 0 0 $40.00 $0.99 $1.00

D1 146.7 29 58 58 $1,363.00 $9.29 $9.50

D2 0 0 0 0 $0.00 $3.00

D3 115.3 0 0 115 $258.75 $2.24 $2.50

Total Awarded 446 338 418 $8,400.00

Clearing Price $1.00 $20.75 $2.25


Credit Limits

• Awards to any bidder may be constrained by credit limits on total cost of awards

• Bidders may want to self-impose limits on spending in the auction

• Self-imposed credit limits often serve as a proxi for contingent constraints• EXAMPLE XOR constraints that would require an MIP

clearing engine


Criteria for Settlement Rules

• Allocate objects efficiently• Objects given to those bidders who value them most• No withholding to support prices

• Incentive Compatibility• Induce truthful revelation of values and constraints

• Market Clearing• Accepted bids have greater valuation than prices and

rejected bids have lower valuation than prices or insufficient funds


Bid Based Enforcement of Credit Limit

1. Impose Credit limit on submitted bids (prescreening)

2. Introduce new constraint for each bidder to LP formulation

( )

( )

ij ij ij J i

b C M

where J i is the set of bids

submitted by bidder i

( )

( )

ij ij ij J i

b x M

where J i is the set of bids

submitted by bidder i

•Justification: Any bid could set the market clearing price


Consequences of Bid Based Approach

• Over-enforces budget constraints• High bidders will see their allocations limited due to their

budget constraint even when clearing price is much lower than their submitted bid (violates market clearing condition)

• Provides incentive to shade bids towards the anticipated clearing price. • Since bidding a high value can sometimes decrease the

probability of being allocated an object, bidders will start to shade bids down and flatten their demand curves


EXAMPLE with Bid Based Approach

Bidder A B

Bid Price

$2 $1

Bid Quantity

100 100

Budget $150 N/A

LP Results with 100 units

Item clears at $1Bidder A receives 75 unitsBidder B receives 25 units

At a price of $1, Bidder A can argue that he should be allocated all 100 units

If he had bid in the range (1, 1.5], he would have received all 100 units


Exhausting Budget Approach

• If a person’s budget is violated then she would maximize her surplus by exhausting her entire budget (under a price taking assumption)

• Method• Solve LP excluding budget constraints• Find budgets that are exceeded• Adjust prices to meet budget constraints with minimal

distortions to allocations and clear the market


EXAMPLE of Price Adjustment

• One object example• A: $120 budget, $2 bid, 100 unit maximum• B: No budget constraint, $1.50 bid, 25 unit maximum• C: No budget constraint, $1 bid, 150 unit maximum• 100 units available• LP with over enforcement, A 60, B 25, C 15, P = $1 • LP no budget constraint, A 100, B 0, C 0, P = $2 (A is over budget)• LP with adjustment A 80 B 20 C 0 P =$1.5

• $1.50 clears market AND exhausts bidder A’s budget• Market clearing conditions satisfied, Efficient allocation• Prices depend on budgets (incentive for A to shade budget)


Non-existence of market clearing with marginal value based uniform pricing

Bidder A B

Bid Price $2 $1

Bid Quantity 100 100

Budget $150 N/A

•At P=$2, A cannot afford all the units and B is not willing to pay for the left over

•At P=$1, A can afford all the units so marginal value is $2

•Market clearing price that will clear the market efficiently is not unique and not incentive compatible P (1, 1.5]


MPEC Formulation

, ( )

, ( )

( )

max

.

( )

0 ( ) , ( )

( )

ij iji j J i

ij iji j J i

ij ij

Tij ij i

j J i

b x

st

A x K transmission capacity

x C bid quantity i j J i

p A x M budget i

This is a parametric LP contingent on price vector p

(For simplicity we omit ownership constraints)


Equilibrium conditions for vector p

( )

( ) ( ) , ( )

exhaust budget low or marginal bid

0 ( ) , ( )

high or marginal bid

T Tij ij ij ij i ij ij

j J i

Tij ij ij

x C p A x M or p A b i j J i

x p A b i j J i


Discrete Object CaseVickrey Model

• Notation

• Winner Determination Problem

• *If any valuations are subadditive, dummy objects will need to be added to exclude Simultaneous awards of separate objects with joint subadditive valuation (deVries and Vohra 2003)

| |2

{0,1}

M

is

is

M set of objects

S

v valuation of bundle s by bidder i

x

jx

st

xvV

i sjsis

i sisis

1

.

max

:


VCG Mechanism• Winner determination without bidder k

• Vickrey payment

• Outcome efficient and Incentive compatible

jx

st

xvV

ki sjsis

ki sisisk

1

.

max

:

)( ks

ksks VVxv


VCG auction with self imposed budgetsBidder Valuation of

AValuation of

BValuation of

ABBudget

1 100 100 200 120

2 75 0 75 999

3 0 65 65 999

How would they bid to prevent budget violation?Bidder 1 would reasonably do one of the following: 1. Bid equally for each object2. Bid aggressively for one object and conservatively on the

other


Budget issue (cont)

Bidder Valuation of A Valuation of B Valuation of AB

Budget

1 60 60 120 120

2 75 0 75 999

3 0 65 65 999

If Bidder 1 allocates resources equally, and is risk averse (under no circumstances will he violate his budget)

Applying the VCG mechanism, the following allocation and prices would result: Bidder 2 receives object A and pays $60Bidder 3 receives object B and pays $60Total value awarded is $75 + $65 = $140


Budget issue (cont)

• An allocation with Bidder 1 receiving either of the objects would be better from a welfare point of view

• If he had bid more aggressively on one of the two objects, he would have taken one, but he might have guessed incorrectly.

• Similarly, a situation where Bidder 1 would have been better off bidding equally than aggressively could be created


Incorporating budget constraints into auction design

• Allows bidders to submit a budget constraint explicitly.

• Develop award determination algorithm and pricing so as to support market clearing conditions:

When bidders are not allocated an object either their bid was too low or they have insufficient funds to secure the item


Formulation for discrete case

max (maximize award value)

(cannot allocate more than a

Altern

v

ative objecti

ailable)

(budget constraint)

1 0 (c

ve: m

an only be awarded

x )

if

a (

ij

ij iji j

ij ji

j ij ij

ij j ij

j iji j

b x

st

x c

p

b p x

x M i

x p b

bid high enough)

0 ( 0) ( )

(if object not awarded either bid is at or below price or

remaining funds at or below price*)

*object may not be awarded if there is a tie in remaini

ij j ij i k ik jk

x p b or M p x p

ng funds


Placement Bidding System at the U of Chicago School of Business

(Graves, Sankaran and Schrage, 1993)

• Students get 1000 points per season to bid on interview slots and use them over several interview rounds

• Under current system total bids placed by a student in a round cannot exceed his/hers remaining budget (worst case enforcement)

• Auction cleared so as to maximize award value


Numerical results for discrete formulation with price based enforcement

(Linus Schrage – personal communication)

(the results assume that each student has a budget of 350 points allocated to each round)

Data Bidders Bids Objects No budget

Price based

Bid based

Oct/21/02 193 725 40 61596 61393 45717

Nov/04/02 293 1078 57 91696 90794 77575

Nov/11/02 197 382 11 28014 27919 27790


Loss of Incentive Compatibility

•With truthful bidding the unconstrained VCG will award A and B to agent 1 for $140 but that violates his budget constraint.

• With truthful bidding the budget constrained formulation with surplus maximization will award A to agent 1 at $75 and B to agent 3 at $45.

•If agent 2 bids $85 while agents 1 and 3 bid truthfully then the procedure will award B to agent 1 at $65 and A to agent 2 at $55 (agent 2 surplus increases from 0 to 20 while overall award value decreases from 180 to 175))

•Agent 2 has an incentive to increase its bid .

Bidder V(A) V(B) Budget

1 115 100 120

2 75 0 100

3 0 65 100


Summary

• Budget introduces new gaming behavior depending on settlement rule• For bid based enforcement, bidders will shade bids• For actual price based enforcement, bidders may submit

lower budgets• In discrete case, bidders may benefit from bidding beyond

valuations to exhaust competitor budgets

• Multi-round auction with activity rules and bid based enforcement of budget may provide a way for bidders to “tune” their bids to reduce the over enforcement effect.

dimacs workshop on computational issues in auction design, oct. 7, 2004 shmuel oren multi-item...

Documents