MultiItem Auctions with Credit Limits

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Multi-Item Auctions with Credit Limits
Shmuel Oren http//www.ieor.berkeley.edu/oren Sh
ehzad Wadalawala U.C. Berkeley October 7, 2004
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(No Transcript)
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Trading Pattern in ERCOT Summer 2001
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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

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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)

1/3
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Bid Format in the TCR Auction

• Bidders submit price and quantity pairs for
  vectors of flow distribution

The letter identifies the bidder while the number
identifies bid
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Resource Constraints
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LP Formulation of Clearing Algorithm
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LP Solution
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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

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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

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Bid Based Enforcement of Credit Limit

• Justification Any bid could set the market
  clearing price

• Impose Credit limit on submitted bids
  (prescreening)
• Introduce new constraint for each bidder to LP
  formulation

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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

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EXAMPLE with Bid Based Approach
LP Results with 100 units Item clears at
1 Bidder A receives 75 units Bidder 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
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Exhausting Budget Approach

• If a persons 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

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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 As
  budget
• Market clearing conditions satisfied, Efficient
  allocation
• Prices depend on budgets (incentive for A to
  shade budget)

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

• At P2, A cannot afford all the units and B is
  not willing to pay for the left over
• At P1, 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

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MPEC Formulation
This is a parametric LP contingent on price
vector p (For simplicity we omit ownership
constraints)
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Equilibrium conditions for vector p
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Discrete Object CaseVickrey Model

• Notation
• Winner Determination Problem

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VCG Mechanism

• Winner determination without bidder k
• Vickrey payment
• Outcome efficient and Incentive compatible

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VCG auction with self imposed budgets

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

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Budget issue (cont)
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
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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

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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

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Formulation for discrete case
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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

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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)

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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 .

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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.