Competitive Bidding in a Certain Class of Auctions

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Competitive Bidding in a Certain Class of Auctions

• Mathias JohanssonUppsala University, Dirac
  ResearchSweden

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Problem background

• Fixed pricing is used today in mobile data
  networks (such as GSM, 3G)
• Could automatic mechanisms for dynamic pricing be
  used to obtain a desired service level?
• Why shouldnt prices reflect supply/demand?
• Radio channels fluctuate strongly and
  unpredictably

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Problem background

• Quality of service (QoS) requirements differ
  among users
• Typically defined in terms of throughput and
  delay requirements
• Streaming media requires tight service levels
• An automatic auctioning procedure could be used
  in order to obtain desired QoS

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The rules of the auction

• The resource can be used by only one user at a
  time
• For each user the resource carries a certain
  utility, the user-specific capacity of the
  resource
• Each user submits one sealed bid to the
  auctioneer, stating
• the price the user is willing to pay per unit
  resource, and
• the users capacity

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Auction set-up

• The winning total bid (bid per unit resource
  times the capacity) obtains the resource for a
  specific period of time
• This process is repeated many times
• Different bidders may have different capacities
• Bids and capacities of other users are hidden
• Future capacities are typically uncertain at the
  time of the bid

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Whats the best bid?

• Each user determines the probability for winning
  and a loss function reflecting the value of the
  resource.
• Clearly, some information of past auctions must
  be given to the users
• The average winning price-capacity product will
  be announced along with its sample variance over
  a given time
• This is announced after every lth auction

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User us probability for winning

• Let qu be the bid per unit resource of user u, cu
  the corresponding capacity
• If v is the user with the largest bid-capacity
  product among all users except u, the probability
  for winning is P(qvcv lt qucu cu,qu,I)
• If cu is uncertain, we must also marginalize over
  cu

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User us probability for winning

• The probability for user u to win is thuswhere
  y cvqv.
• But how do we assign P(ycu,qu, I)?

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Probability assignments

• P(cu I)
• We assume that cu can only take one of K possible
  values.
• Assuming that our only further information is a
  past history of how many times the K different
  levels have ocurred, Laplaces rule of succession
  applies

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Probability assignments

• P(y cu qu I)
• Only the mean winning bid-capacity product and
  its sample variance is known
• According to the maximum entropy principle, we
  assign a Gaussian distribution.
• Note! The announced information regards all
  users, but y should not include user u
• A correction is made by subtracting the
  contributions from us wins.

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Typical loss functions

• Constant throughput
• A user wants ?u resource units per time
  unitwhere xu (qu) is the obtained throughput
• Price-performance ratio
• A user may want to raise her bid if that results
  in a substantially better throughput (i.e.
  stockpiling when capacity is cheap)

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Price-performance criterion

• A possible formalization isi.e. a price
  increase of 1 unit is ok if the throughput
  increases by a factor of a. If the throughput is
  less than b, the resource is of no value.

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Expectations and computations

• The expected constant throughput loss is
• The expected price-performance loss is
  approximated by

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Examples

• 4 users
• Every 20th time unit, mean-variance information
  is broadcast and bids are updated
• 4 different possible capacities, c0, 74, 92,
  106
• Rate probabilities are updated by Laplaces rule
  (rates are generated as Gaussian numbers with avg
  80, std dev 20)
• All users have a maximum bid per unit 5.

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Example 1

• Constant rate loss for all users
• ?1 15, ?2 20, ?3 20, ?4 30,
• Resulting average throughput over 600 time units
  (30 price updates)
• x1 14, x2 21, x3 21, x4 33,
• More competitive setting
• ?1 15, ?2 20, ?3 25, ?4 30,
• x1 13, x2 19, x3 26, x4 31,
• and the average paid price per bit nearly doubles

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Example 2

• Price-performance loss for user 1 and constant
  rate loss for users 2-4
• ?2 10, ?3 20, ?4 20,
• Resulting average throughput
• x1 34, x2 11, x3 21, x4 21.

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Evolution of bids
Throughput per time unit
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Price-to-throughput ratio
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Comments

• Bidding can be used to satisfy QoS demands
• But what are the long-term customer reactions?
• Other types of bidding situations call for
  Bayesian treatment!
• Challenge (MaxEnt 2007?) What is the expected
  winning bid in the sale of an apartment given
  knowledge of existing bid history?
• Previous bids 500k, 550k, and 565k
• How would you, as a devoted Bayesian, bid?