MultiItem Auctions with Credit Limits Shmuel Oren http

Multi-Item Auctions with Credit Limits Shmuel Oren http: //www. ieor. berkeley. edu/~oren Shehzad Wadalawala U. C. Berkeley October 7, 2004 DIMACS Workshop on Computational Issues in Auction Design, Oct. 7, 2004 Shmuel Oren

SIGNIFICANT CONSTRAINTS IN ERCOT SUMMER 2001 Multi-Item Auctions with Credit Limits 2 Shmuel Oren

Trading Pattern in ERCOT Summer 2001 1200 MW 58 0 3750 MW 720 MW 2200 MW WEST 2001 LOAD 3, 700 MW GEN 5, 300 MW NORTH 2001 LOAD 20, 700 MW GEN 22, 000 MW W M 31 0 W M SOUTH 2001 LOAD 33, 000 MW GEN 45, 200 MW Multi-Item Auctions with Credit Limits 3 Shmuel Oren

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 Multi-Item Auctions with Credit Limits 4 Shmuel Oren

Hedging Congestion Charges with TCRs • Full hedging of the congestion charge for 1 MW sent from A to B requires a portfolio of TCRs in proportion to the Power Transfer Distribution Factors (PTDF) 1 MW A 2/3 B 1 MW 1/3 C Multi-Item Auctions with Credit Limits 5 Shmuel Oren

Bid Format in the TCR Auction • Bidders submit price and quantity pairs for vectors of flow distribution Portfolio Bids on CSCs Bid (per. MW) Quantity (MW) Bid S-T G-P STP-Dow A 1 0. 2 0. 3 0. 5 $10. 00 300 A 2 1. 0 0. 0 $5. 00 185 B 0. 2 0. 5 0. 3 $11. 25 250 C 1 0. 6 0. 3 0. 1 $7. 50 240 C 2 1. 0 0. 0 $1. 00 100 D 1 0. 2 0. 4 $9. 50 320 D 2 0. 0 1. 0 0. 0 $3. 00 140 D 3 0. 0 1. 0 $2. 50 170 The letter identifies the bidder while the number identifies bid Multi-Item Auctions with Credit Limits 6 Shmuel Oren

Resource Constraints Available TCRs TCR Limit S-T 447 S-T 186 G-P 339 G-P 141 STPDow 419 STP 174 Dow Multi-Item Auctions with Credit Limits 7 Shmuel Oren

LP Formulation of Clearing Algorithm Multi-Item Auctions with Credit Limits 8 Shmuel Oren

LP Solution Total TCRs Awarded (rounded down) Bid Award S-T G-P STP-Dow Paid total A 1 300 60 90 150 $2, 265. 00 $7. 55 $10. 00 A 2 126 0 0 $126. 00 $1. 00 $5. 00 B 236. 7 47 118 71 $2, 655. 25 $11. 22 $11. 25 C 1 240 144 72 24 $1, 692. 00 $7. 05 $7. 50 C 2 40. 3 40 0 0 $40. 00 $0. 99 $1. 00 D 1 146. 7 29 58 58 $1, 363. 00 $9. 29 $9. 50 D 2 0 0 $0. 00 D 3 115. 3 0 0 115 $258. 75 Total Awarded 446 338 418 $8, 400. 00 Clearing Price $1. 00 $20. 75 $2. 25 Multi-Item Auctions with Credit Limits 9 Paid per MW Bid $3. 00 $2. 24 $2. 50 Shmuel Oren

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 Multi-Item Auctions with Credit Limits 10 Shmuel Oren

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 Multi-Item Auctions with Credit Limits 11 Shmuel Oren

Bid Based Enforcement of Credit Limit • Justification: Any bid could set the market clearing price 1. Impose Credit limit on submitted bids (prescreening) 2. Introduce new constraint for each bidder to LP formulation Multi-Item Auctions with Credit Limits 12 Shmuel Oren

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 Multi-Item Auctions with Credit Limits 13 Shmuel Oren

EXAMPLE with Bid Based Approach Bidder A Bid $2 Price Bid 100 Quantity Budget B $1 LP Results with 100 units 100 Item clears at $1 Bidder A receives 75 units Bidder B receives 25 units $150 N/A 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 Multi-Item Auctions with Credit Limits 14 Shmuel Oren

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 Multi-Item Auctions with Credit Limits 15 Shmuel Oren

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) Multi-Item Auctions with Credit Limits 16 Shmuel Oren

Non-existence of market clearing with marginal value based uniform pricing Bidder Bid Price Bid Quantity Budget A $2 100 $150 B $1 100 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 Multi-Item Auctions with Credit Limits 17 Shmuel Oren

MPEC Formulation This is a parametric LP contingent on price vector p (For simplicity we omit ownership constraints) Multi-Item Auctions with Credit Limits 18 Shmuel Oren

Equilibrium conditions for vector p Multi-Item Auctions with Credit Limits 19 Shmuel Oren

Discrete Object Case Vickrey 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 (de. Vries and Vohra 2003) Multi-Item Auctions with Credit Limits 20 Shmuel Oren

VCG Mechanism • Winner determination without bidder k • Vickrey payment • Outcome efficient and Incentive compatible Multi-Item Auctions with Credit Limits 21 Shmuel Oren

VCG auction with self imposed budgets Bidder Valuation of A B AB Budget 1 100 200 120 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 object 2. Bid aggressively for one object and conservatively on the other Multi-Item Auctions with Credit Limits 22 Shmuel Oren

Budget issue (cont) If Bidder 1 allocates resources equally, and is risk averse (under no circumstances will he violate his budget) Bidder Valuation of A Valuation of B Valuation of AB Budget 1 60 60 120 2 75 0 75 999 3 0 65 65 999 Applying the VCG mechanism, the following allocation and prices would result: Bidder 2 receives object A and pays $60 Bidder 3 receives object B and pays $60 Total value awarded is $75 + $65 = $140 Multi-Item Auctions with Credit Limits 23 Shmuel Oren

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 Multi-Item Auctions with Credit Limits 24 Shmuel Oren

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 Multi-Item Auctions with Credit Limits 25 Shmuel Oren

Formulation for discrete case Multi-Item Auctions with Credit Limits 26 Shmuel Oren

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 Multi-Item Auctions with Credit Limits 27 Shmuel Oren

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 Multi-Item Auctions with Credit Limits 28 Shmuel Oren

Loss of Incentive Compatibility Bidder V(A) V(B) Budget 1 115 100 120 2 75 0 100 3 0 65 100 • 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. Multi-Item Auctions with Credit Limits 29 Shmuel Oren

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. Multi-Item Auctions with Credit Limits 30 Shmuel Oren