Bidding Strategy and Auction Design

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Bidding Strategy and Auction Design

• Josh Ruffin, Dennis Langer, Kevin Hyland
  and Emmet Ferriter

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Auctions

• Auctions entail the transfer of a particular
  object from a seller to a buyer for a certain
  price. It is a useful way to sell commodities of
  undetermined quality. Auctions can be used for
  single items such as a work of art and for
  multiple units of a homogeneous item such as gold
  or Treasury securities. It is the purest of
  markets a seller wishes to obtain as much money
  as possible, and a buyer wants to pay as little
  as necessary.

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Why do we care?

• Auctions are a multi-billion dollar business
• The US Treasury uses auctions to determine
  mortgage rates
• The FCC uses competitive bidding to allocate
  licenses for broadcasting on the electromagnetic
  spectrum
• Stores such as Filenes Basement use a pricing
  strategy that reduces the price on items
  remaining on the racks for longer than a certain
  time

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Auction Formats

• First Price Auction The individual who submits
  the highest bid wins the auction, and pays that
  submitted highest bid can be applied to any type
  of auction
• Second Price Auctions The individual who
  submits the highest bid wins the auction, and
  pays the amount submitted by the second highest
  bidder can be applied to any type of auction
• Sealed Bid Auctions Bids are submitted
  privately and each bid is evaluated
  simultaneously the highest bidder becomes the
  winner can be applied to any type of auction
• Open-outcry Auctions Bids are submitted openly
  and publicly for rival bidders to evaluate the
  highest bidder becomes the winner can be applied
  to any type of auction
• English Auction The auctioneer announces a low
  price and invites ascending bids until no one is
  willing to go above the last bid made the last
  bidder wins
• Dutch Auction The auctioneer announces a high
  price and then announces successively lower bids
  the bidder who calls a halt to such announcements
  first wins the auction

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• Common-value Auctions The value of the object
  is the same for all bidders each bidders
  estimated valuation vary slightly also known as
  Objective-value
• Private-value Auctions Each bidder places
  his/her own and unique valuation to the good for
  example, valuations can be influenced by
  sentimental issues and/or imprecise monetary
  estimates also known as Subjective-value
• Winners Curse When the winner, and highest
  bidder, are forced to bid and pay a higher price
  for the good than its true value (to win)
• Risk-averse Bidders A bidder is more concerned
  about the losses caused by underbidding than the
  costs associated with bidding at or close to
  their true valuations risk-averse bidders want
  to win without ever overbidding

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Which format is the best?

• The answer depends upon many variables.
• 1.Sellers perspective - tries to reach the
  highest selling price
• - decrease incentives to cheat
• -affiancy (for perisible items)

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Winners Curse
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Vickreys Truth Serum

• In open outcry auctions, the winning bidder
  essentially pays the valuation of the second
  highest bidder, since the winner will not bid
  more than the minimum on the final bid.
• In sealed envelope auctions, the winner pays his
  bid, regardless of the distance between it and
  the second highest bid.
• As a result, a strategic bidder shades his bid
  lower than his true valuation in order to retain
  some profit.
• William Vickrey devised a method to ensure that
  bidders would bid their true valuation the
  highest bidder wins, but only pays the second
  highest bid.
• So, the second-price sealed-bid auction is called
  Vickreys Truth Serum.

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The Sellers Choice

• Before showing why the second-price auction
  works, we will consider the position of the
  seller.
• Its clear that, by selling the item for the
  second highest price, the seller is making less
  profit. In essence, he is buying information (the
  true valuation of each bidder).
• However, in the first-price auction, the seller
  is also making less profit when the bidders shade
  their bids.
• The seller must decide which form of auction will
  reduce his profit less.

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Vickreys Claim

• For any sealed-bid auction, the bidder has three
  strategies
• A) bid their true valuation
• B) bid under their valuation
• C) bid over their valuation
• For a first-price sealed-bid auction, the best
  strategy is for the bidder to bid under his
  valuation.
• Vickrey says that in the second-price auction,
  the best strategy for every bidder is to bid
  their true valuation.

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Why Vickreys Claim is True

• Consider a second-price sealed-bid private-value
  auction for some item, and let
• v your true valuation of the item
• b your bid
• r the highest bid besides yours
• All bids less than r are irrelevant, since they
  have no effect on whether you win or lose.
• Now, we must consider two cases
• a) where b lt v
• b) where b gt v

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Shading Up

• Suppose you bid higher than your true valuation
  (b gt v)
• Then if
• I) (r lt v) i.e., the next highest bid is less
  than your valuation, so you win the item and turn
  a profit. However, if you had bid (b v), you
  wouldve still won, and wouldve turned the same
  profit.

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Shading Up

• Else if
• II) (v lt r lt b) i.e., the next highest bid is
  between your valuation and your bid, then you win
  the item, but you must purchase it for more than
  your valuation. So, you shouldve bid (b v)
  although you wouldve lost the item, you wouldnt
  have sustained a loss.

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Shading Up

• Else if
• III) (b lt r) i.e., you do not have the highest
  bid. If you had bid (b v), you still wouldve
  lost.

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1st Summary

• In cases I and III, bidding (b gt v) has the same
  result as bidding (b v).
• In case II, bidding (b gt v) is worse than bidding
  (b v).
• So, there is no reason to bid (b gt v) instead of
  (b v) since 1/3 of the time, the result is
  worse, and 2/3 of the time, the result is equal.

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Shading Down

• Suppose you bid lower than your true valuation (b
  lt v)
• Then if
• I) (r lt b) i.e., the next highest bid is less
  than your bid, so you win the item and turn the
  profit. However, if you had bid (b v), you
  wouldve still won, and wouldve turned the same
  profit.

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Shading Down

• Else if
• II) (b lt r lt v) i.e., you do not have the highest
  bid, which is less than your valuation. So, you
  shouldve bid (b v) you wouldve won the item
  and turned a profit.

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Shading Down

• Else if
• III) (v lt r) i.e., you do not have the highest
  bid, which is above your valuation. If you had
  bid (b v), you wouldve still lost.

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2nd Summary

• In cases I and III, bidding (b lt v) has the same
  result as bidding (b v).
• In case II, bidding (b lt v) is worse than bidding
  (b v).
• Again, there is no reason to bid (b lt v) instead
  of (b v) since 1/3 of the time, the result is
  worse, and 2/3 of the time, the result is equal.

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Overview

• So, we have shown that bidding your true
  valuation is better than both bidding under your
  valuation and bidding over your valuation.
• Therefore, it is clear that, in a second-price
  auction, the best strategy for each bidder is to
  bid their true valuation.
• So Vickreys Truth Serum works

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Jack v. Jill

• Lets consider a specific example. Suppose Jack
  and Jill are bidding on a painting. Assume the
  following
• a) Jill values the painting at 100.
• b) Jill considers it equal possible that Jack
  could value the painting at 100 or at 80.
• c) In the event that Jack and Jill make the
  same bid, the winner is decided by a coin toss.
• d) Jack and Jill can only make bids of 100 and
  80.
• So, this example is more limited than the last.

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Jills Point of View

• Jill know that she values the painting at 100.
  She considers the following possibilities
• a) Jack values the painting at 80 with
  probability ½. In this case, she wins the
  painting with probability 1.
• b) Jack values the painting at 100 with
  probability ½ also. In this case, she wins the
  painting with probability of ½, according to a
  coin toss.
• So she calculates her odds at winning at
• ( 1 )( ½ ) ( ½ )( ½ ) ( ¾ ).

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Jills Payoff

• So, Jills expected gain is
• ( ¾ )(100) ( ¼ )(0) 75.
• However, this equation ignores the fact that Jill
  must pay the seller. Since this is a second-price
  auction, she pays 80 with probability ½ and she
  pays 100 with probability ( ½ )( ½ )
  ( ¼ ).
• So, Jills net gain is
• ( ¾ )(100) (80)( ½ ) (100)( ¼ ) 10.

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Can Jill Increase Her Payoff?

• Can Jill increase her payoff by bidding 80?
• To answer this, Jill must consider two payoff
  matrices
• a) if Jack values the painting at 100
• b) if Jack values the painting at 80.

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If Jacks Value is 100

• If Jack values the painting at 100, Jill
  considers this matrix
• 100 80
• 100 (0, 0) (20, 0)
• 80 (0, 20) (10, 0)
• So, a strategy of bidding 100 is dominant for
  Jill.

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If Jacks Value is 80

• If Jack values the painting at 80, Jill
  considers this matrix
• 100 80
• 100 (0, 10) (20, 0)
• 80 (0, 0) (10, 0)
• So, again, a strategy of bidding 100 is dominant
  for Jill.

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Overview

• So, again we can see from this specific, yet more
  limited example that truthful bidding is the
  dominant strategy is the second-price auction.
• Again, Vickreys Truth Serum proves to be
  effective.