What format for multiunit auctions: Uniform, discriminatory or Vickrey

1
What format for multi-unit auctions Uniform,
discriminatory or Vickrey?

• Hailu Atakelty (ARE/UWA)
• Thoyer Sophie (ENSAM/Lameta)
• WEHIA 2005 presentation
• University of Essex, Colchester, UK
• 14 June 2005

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Outline

• Multi-unit auction formats
• Structural properties of equilibrium strategies
• the incompleteness of the theory
• The agent-based modelling approach and
  reinforcement learning
• Comparing three standard auction formats
• for different competition (rationing) levels and
  for different bidder population structures
• Evaluating relative performance

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Multi-unit Auctions

• Examples of use
• Electricity market in United Kingdom (Wolfram,
  1998)
• Allocation of foreign currency (Tenorio, 1993)
• Sale of Treasury bonds ( Binmore and
  Swierzbinski, 2000)
• Allocation of railway lines? (Caillaud, 2002)
• Advantage Avoid the lumpy bid problem

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The auction setting

• Sealed-bid procurement auction
• bidders are allowed to make multiple bids
• auctioneer has a target (demand level) and
  chooses the clearing price and buys units
  accordingly.
• payments to bidders depend on auction format
• Private independent values
• Risk-neutral bidders

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Payments under alternative formats for multi-unit
auctions
Discriminatory pricing (pay-as-bid) each winning
bidder is paid based on its own bids (the
production or opportunity cost implied by their
individual bids) Uniform pricing winning
bidders are paid the clearing price for all units
sold. Generalized Vickrey (Clinched or Ausubel
auction in an open-cry format) The payment is
equal to the price that would have been paid if
the unit had to be sourced from the other bidders.
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Numerical Example
2 bidders with 4 units to sale each. Total demand
is 4 units
Uniform Price 4 R1 448 R2
448 Discriminatory R1 134 R2
246 Generalized Vickrey (clinched) R1 9514
R2 7613
Bidder 1 Bidder 2
1 2 3 4 6 5 7
9
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Payment rules
Supply schedule by bidder i
Qi(b)
Cut-off price
Residual demand facing bidder i
Di(b) DT - ? Q-i(b)
Payment under discriminatory
Payment under generalized Vickrey
Payment under uniform
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What auction design?

• Very few theoretical results available because
  multiplicity of equilibria
• generalized Vickrey is truthful (weakly
  dominant)
• overbidding with other formats.
• only structural properties of equilibrium
  strategies are known
• Controversy on the best payment scheme ( Binmore
  and Swierzbinski 2000)
• Few empirical data analysis (Wolfram, 1998)
• Simplified experiments (Alemgeest et al, 1998
  Kagel and
• Levine, 2001 etc.)
• Need for rapid simulations development of
  agent-based
• models (Bower and Bunn, 2001, Binmore and
  Swierzbinski 2000)

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The model

• Model in which the government agent uses
  sealed-bid multi-unit auctions with three payment
  formats uniform, discriminatory and generalized
  Vickrey
• Agents learn to bid through repeated action. Each
  agent i has a true cost function
• Pi ai bi Q
• After each auction, it exploits previous bids or
  experiments with new bids
• PiL ai L(t) bi L(t) Q

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Reinforcement learning algorithm(Roth Erev
1995 Erev Roth 1998)

• Asserts that the propensity to use an action or a
  strategy is positively related to the results
  obtained from it (exploiting known strategies)
• And agents also experiment with strategies
  similar to those that they have tried and
  benefited from
• Recent experience has more impact than past
  experience
• Learning algorithm suitable for the auction
  problem
• Individual learning
• No need to evaluate payoffs of foregone
  strategies

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Propensity of player i to choose strategy (a,b)
R(1-?) if a c and b d Ecd (a, b, R)
R. (?/n) if (a, b) is neighbouring strategy of
(c,d) 0 otherwise

• Qiab (t1) is the propensity updating strategy
• is a recency parameter
• R is reinforcment from previous choice of
  strategy (c,d)
• ? is experimentation

Neighbours
Strategy
b choice

• Law of effect
• Experimentation
• Recency

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Simulation convergence

• Agents learning towards a mixed strategy
  choice is probabilistic
• Convergence defined in terms of choice
  probability convergence
• - the highest probability of choice must be gt 0.5
• highest probability must be at least three times
  as big as the second highest
• These are stringent criteria but were met easily
  for all demand levels in the case of
  discriminatory auctions and at lower demand
  levels for the other formats.

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Experimental set up of simulation

• Simulation experiments with four different
  bidder populations (POPs)
• flat marginal cost curves (a 0.5, b 0)
• POP1 - homogeneous ( 8 bidders with size of ms
  0.5)
• POP2 - heterogeneous (4 with ms0.25, 4 with
  ms0.75)
• slopping marginal cost curves (a 0.5, b 0.5)
• POP3 - homogeneous ( 8 with ms 0.5)
• POP4 - highly heterogeneous population (4 types
  of bidders)
• flatter MC curves and small capacity (b0.25,
  ms0.25) 2 bidders
• flatter MC curves and large capacity (b0.25,
  ms0.75) 2 bidders
• steeper MC curves and small capacity (b0.75,
  ms0.25) 2 bidders
• steeper MC curves and large capacity (b0.75,
  ms0.75) 2 bidders

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Experimental set up

• Different levels of rationing (competition)
  simulated by taking demand levels of 0.5, 1.0,
  1.5 and 2.0
• degree of rationing ranging from 12.5 to 50 of
  the aggregate supply capacity of 2.0

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Bidding strategies Summary of results

• For all populations and rationing levels, Vickrey
  leads to high frequencies of sincere learnt
  bidding
• Confirming theoretical predictions (and coherence
  of results from our learning algorithm)
• Discriminatory Overbidding takes the
  theoretically predicted form (high flat bidding)
  only when competition is weak (high rationing
  levels). Supply inflation (true entry price but
  steeper slope) is more predominant.
• Explanation risk of being completely priced out
  is lowered than with high flat bidding.
• Uniform two types of bidding strategies
  observed
• Supply inflation (theoretically expected) when
  competition is strong or when the bidder has a
  competitive disadvantage
• Truthful bidding increases with decline in
  competition especially among small bidders
  (free-riding on price risks taken by others)

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NE checks of learnt bidding

• NE status check based testing if any bidder could
  have improved over the learnt bid if others kept
  their bids unchanged
• High NE pass ratios (82 to 95) in Vickrey for
  populations 1 and 2. Lower for other populations.
• NE pass rates for discriminatory and uniform are
  much lower and decline with decline in
  competition.
• Prices coordinated through learning susceptible
  to unilateral deviation

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Budgetary performance summaries

• Performance difference more in terms of budgetary
  outcomes rather than allocation efficiency
• Vickrey least expensive under most settings for
  procuring agency
• Advantages clearer when the MC is flat especially
  at higher demand levels
• For rising MC, the performance of Vickrey and
  uniform are similar, especially at lower demand
  levels. Uniforms performance deteriorates with
  increase in demand for heterogeneous populations.
• Discriminatory generally most expensive but with
  qualifications (D better than U)
• Low demand levels and homogeneous populations
• High demand levels and heterogeneous populations

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Conclusions

• Bidding behaviour cannot be completely
  characterized by auction format
• Competition levels and population structure
  important
• Bidding in uniform auctions is especially
  sensitive to context
• The picture is more complex than what the limited
  existing theoretical analysis indicates
• Format choice needs to be informed by context, if
  possible

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Further improvements

• This study used linear learnt bid curves.
  However, these linear bid responses can be an
  approximation, at best. These need to be
  generalized to nonlinear learnt bid curves. We
  have done that extension and will have newer
  results soon.