Chapter 19 Equivalence

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Chapter 19Equivalence

1. Types of Auctions
2. Strategic Equivalence
3. Revenue Equivalence
4. Optimal Bidding

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Types of Auctions
To begin this chapter we describe the main kinds
of auctions, and pose some questions about
auctions of interest to business men and women.
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Why study auctions?

1. Studying auctions is the simplest way of
   approaching the question of price formation.
2. Auctions serve the dual purpose of eliciting
   preferences and allocating resources between
   competing uses.
3. A less fundamental but more practical reason for
   studying auctions is that the value of goods
   exchanged each year by auction is huge.

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Auction mechanisms

• There are 5 standard types of auctions for
  auctioning a single item which are widely used
  and analyzed
• First-price sealed-bid
• Second-price sealed-bid
• English
• Japanese
• Dutch
• as well as several other types we will
  investigate.

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Sealed bid auctions

• Each bidder in a sealed bid auction submits a
  price or bid to the auctioneer simultaneously.
• The highest bidder receives the auctioned item.
• Sealed bid auctions only differ in how much
  bidders pay.
• We investigate three variations, first price,
  second price, and all pay.

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First price sealed bid auction

• In a first price sealed bid auction, the highest
  bidder pays the amount she bid in exchange for
  the object up for auction
• Suppose there are N bidders. Let bn denote the
  bid by the nth player and let vn denote how much
  she values the auctioned item. Also rank the bid
  from the highest to the lowest as b(1) through
  b(N). In a first price auction, un, the net
  payoff to the nth player is defined as
• The auctioneer receives

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Second price sealed bid auction

• Each bidder in a second price sealed bid auction
  submits a price to the auctioneer simultaneously.
  The bidder submitting the highest price pays the
  second highest price submitted. The other bidders
  neither pay nor receive anything.
• Following the same notation as in the first price
  sealed bid auction the net payoff to bidder n is
• The auctioneer receives

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All-pay sealed bid auctions

• In an all-pay sealed bid auction, each bidder
  pays what she bids, and the highest bidder wins
  the auction.
• The net payoff to the nth bidder is defined as
• The auctioneer receives

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Examples of all-pay auctions

• More generally an all-pay auction is a paradigm
  for modeling competitions of various kinds, not a
  common institution for literally conducting
  auctions.
• For example supply contracts are like all-pay
  auctions. Bidders expend considerable resources
  preparing a proposal, but only one bidder is
  awarded the contract.
• Similarly research teams in the same field use
  resources competing with each other, but the
  first team to make a discovery benefits
  disproportionately in the rewards from their
  discovery through patenting, first mover
  advantages, and so on.

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Comparing the revenue from sealed bid auctions

• Notice that
• So if all bidders adopted the same bidding
  strategy for the three auctions, then the second
  price sealed bid auction would yield least
  revenue of the three, and the all pay auction
  would yield the most.
• But would a potential buyer bid more if the
  winner pays less than their own bid? And would
  she be bid as much if she had to pay her bid
  regardless of whether hers is the winning bid or
  not?
• On reflection it is unclear which of the three
  auctions yields more revenue to the auctioneer!

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Descending auctions (Dutch auctions)

• The auctioneer begins by offering the item at a
  very high price which he confidently believes
  exceeds the willingness to pay of any bidder.
• Then he continuously lowers it until one bidder
  announces that she is willing to pay the current
  price.
• At that point the auction ends, the bidder buying
  the item at the lowest price offered.

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Payoffs in descending auctions

• To formally describe the payoffs from this game,
  let at denote the auctioneer's ask price in
  period t.
• Denote by tn the time at which the nth bidder
  accepts the bid if no other bidder has submitted
  an order by then, meaning the auction has not
  ended yet.
• Analogous to our ranking of bids in sealed bid
  auctions, let t(k) denote the kth earliest, which
  implies t(¹) t(²) t(N).
• The player's payoffs can be then defined as

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Ascending auctions

• In ascending auctions, the auctioneer raises the
  price as long as more than one person is willing
  to pay the current price. The winner pays the
  lowest price at which every other bidder has
  dropped out of the auction.
• Let rt denote the auctioneer's request price in
  period t, and now let tn denote the time at which
  the nth bidder will drop out of the auction. Also
  let t(k) denote the kth earliest time, which
  implies t(¹) t(²) t(N).
• The net payoff to the nth bidder is then

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English auction

• We will study two types of ascending auctions,
  English auctions and Japanese auctions. The
  feature differentiating these two auctions is how
  much the bidders observe as the auction proceeds.
  
• In an English auction bidders compete against
  each other by successively raising the price at
  which they are willing to pay for the auctioned
  object.
• The bidding stops when nobody is willing to raise
  the price any further, and the item is sold to
  the person who has bid the highest price, at that
  price.

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How much do bidders observe in English auctions?

• During the auction a bidder might be able to
  observe a sample of bidders who make bids, and
  thus update his beliefs about the value of the
  item as the auction progresses.
• The most restrictive assumption is that the
  bidders do not observe the identity of the other
  people making bids, and that to win the auction,
  a bidder must continuously indicate his
  willingness to pay successively higher prices.
• This simplification implies that as the auction
  progresses, a bidder willing to pay for the
  auctioned item at the current quote knows only
  that at least one other bidder has also signaled.

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Japanese auction

• Everyone willing to pay the current price for the
  auctioned indicates this to the auctioneer.
  Those who are not willing to pay the current
  price rt must leave the auction and cannot
  reenter.
• The auctioneer raises the price until the second
  last bidder drops out of contention, and the
  winner is assigned the item at that price.
• In contrast to an English auction, every bidder
  sees which which bidders have dropped out of the
  auction as the auctioneer raises the bid price.

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2. Strategic Equivalence
By definition the strategic form solutions to
strategically equivalent auctions are the same.
This section provides provides several examples
of strategically equivalent auctions.
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Strategic equivalence

• The introduction showed there are many ways of
  auctioning an item to interested buyers. However
  many auctions are closely related to each other.
• Recall that a strategy is a complete
  description of instructions to be played
  throughout the game, and that the strategic form
  of a game is the set of alternative strategies to
  each player and their corresponding expected
  payoffs from following them. Two games are
  strategically equivalent if they share the same
  strategic form.
• In strategically equivalent auctions, the set of
  bidding strategies that each potential bidders
  receive, and the mapping to the bidders payoffs,
  are the same.

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Descending auctions are strategically equivalent
to first-price auctions

• During the course of a descending auction no
  information is received by bidders.
• Each bidder sets his reservation price before
  the auction, and submits a market order to buy if
  and when the limit auctioneer's limit order to
  sell falls to that point.
• Dutch auctions and first price sealed bid
  auctions share strategic form, and hence yield
  the same realized payoffs if the initial
  valuation draws are the same.
• Rule 1 Pick the same reservation price in Dutch
  auction that you would submit in a first price
  auction

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Second-price versus ascending auctions

• When there are only 2 bidders, the two ascending
  auction mechanisms (English and Japanese) are
  strategically equivalent to the second price
  sealed bid auction (because no information is
  received during the auction).
• All three auctions are strategically equivalent
  are (almost) strategically equivalent if all the
  players have independently distributed valuations
  (because the information conveyed by the other
  bidders has no effect on a bidders valuation).
• In common value auctions the 3 mechanisms are
  not strategically equivalent if there are more
  than 2 players.
• Rule 2 If there are only two bidders, or if
  valuations are independently distributed, choose
  the same reservation price in English, Japanese
  and second price auctions.

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Summary
Rule 1 Pick the same reservation price in a
Dutch auction that you would submit in a first
price sealed bid auction. Rule 2 In private
value auctions, or if there are only two bidders,
choose the same reservation price for an English
or a Japanese auction that you would submit in a
second price sealed bid auction.
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3. Revenue Equivalence
Revenue equivalent auctions generate the same
expected revenue. Thus strategic equivalence
implies revenue equivalence, but not vice versa.
This section explores sufficient conditions for
auctions to be revenue equivalent.
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Relaxing strategic equivalence

• In strategically equivalent auctions, the
  strategic form solution strategies of the
  bidders, and the payoffs to all the players are
  identical.
• This is a very strong form of equivalence. Can
  we show that such players might be indifferent to
  certain auctions which lack strategic
  equivalence?

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Revenue equivalence defined

• The concept of revenue equivalence provides a
  useful tool for exploring this question.
• Two auction mechanisms are revenue equivalent
  if, given a set of players their valuations, and
  their information sets, the expected surplus to
  each bidder and the expected revenue to the
  auctioneer is the same.
• Revenue equivalence is a less stringent
  condition than strategic equivalence.
• Thus two strategic equivalent auctions are
  invariably revenue equivalent, but not all
  revenue equivalent auctions are strategic
  equivalent .

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Why study revenue equivalence ?

• If the auctioneer and the bidders are risk
  neutral, studying revenue equivalence yields
  conditions under which the players are
  indifferent between auctions that are not
  strategically equivalent.
• Exploiting the principle of revenue equivalence
  can sometimes give bidders a straightforward way
  of deriving their solution bid strategies.

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Preferences and Expected Payoffs
Let U(vn) denote the expected value of the nth
bidder with valuation vn bidding according to
his equilibrium strategy when everyone else does
too. P(vn) denote the probability the nth
bidder will win the auction when all players bid
according to their equilibrium strategy. C(vn)
denote the expected costs (including any fees to
enter the auction, and payments in the case of
submitting a winning bid).
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An Additivity Assumption

• We suppose preferences are additive, symmetric
  and private, meaning
• U(v) P(v) v - C(v)
• So the expected value of participating in the
  auction is additive in the expected benefits of
  winning the auction and the expected costs
  incurred.

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A revealed preference argument

• Suppose the valuation of n is vn and the
  valuation of j is vj.
• The surplus from n bidding as if his valuation
  is vj is U(vj), the value from participating if
  his valuation is vj, plus the difference in how
  he values the expected winnings compared to a
  bidder with valuation vj, or (vn vj)P(vn).
• In equilibrium the value of n following his
  solution strategy is at least as profitable as
  deviating from it by pretending his valuation is
  vj. Therefore
• U(vn) gt U(vj) (vn vj)P(vj)

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Revealed preference continued

• For convenience, we rewrite the last slide on
  the previous page as
• U(vn) - U(vj) gt (vn vj)P(vj)
• Now viewing the problem from the jth bidders
  perspective we see that by symmetry
• U(vj) gt U(vn) (vj vn)P(vn)
• which can be expressed as
• (vn vj)P(vn) gt U(vn) - U(vj)

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A fundamental equality

• Putting the two inequalities together, we
  obtain
• (vn vj) P(vn)gt U(vn) - U(vj) gt (vn vj) P(vj)
• Writing
• vn vj dv
• yields
• which, upon integration, yields

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Revenue equivalence

• This equality shows that in private value
  auctions, the expected surplus to each bidder
  does not depend on the auction mechanism itself
  providing two conditions are satisfied
• 1. In equilibrium the auction rules award the bid
  to the bidder with highest valuation.
• 2. The expected value to the lowest possible
  valuation is the same (for example zero).
• Note that if all the bidders obtain the same
  expected surplus, the auctioneer must obtain the
  same expected revenue.

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A theorem

• Assume each bidder
• - is a risk-neutral demander for the auctioned
  object
• - draws a valuation independently from a common,
  strictly increasing probability distribution
  function.
• Consider auction mechanisms where
• - the buyer with the highest valuation always
  wins
• - the bidder with the lowest feasible signal
  expects zero surplus.
• Then the same expected revenue is generated by
  the auctions, and each bidder makes the same
  expected payment as a function of her valuation.

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4. Optimal Bidding
We apply the revenue equivalence theorem to solve
for the optimal bidding rules for several types
of private value auctions.
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Steps for deriving expected revenue

• The expected revenue from any auction satisfying
  the conditions of the theorem, is the expected
  value of the second highest bidder.
• To obtain this quantity, we proceed in two
  steps
• 1. derive the probability distribution of the
  second highest valuation,
• 2. obtain its density and integrate to find
  the mean.

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Second price sealed bids

• In a sealed bid auction, the strategy of each
  player n is to submit a bid, which we label by
  bn.
• In a second price sealed bid auction, when a
  bidder knows his own valuation, there is a very
  general result available about how he should bid,
  which does not depend at all on what he knows
  about the valuations of the other players, or
  what they know about their own valuations.
• It is a weakly dominant strategy to bid his
  valuation vn.
• An immediate corollary of this general result is
  that if every bidder knows his own valuation, the
  unique solution to the game is for each bidder n
  to submit his or her true valuation. We establish
  this claim by showing that bidding vn weakly
  dominates bidding above or below it.

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Bidding in a second-price auction

• Bidding your own valuation is a weakly dominant
  strategy in second price sealed bid auctions.
• The logic supporting this result, weak
  dominance, extends beyond second price auctions
  with perfect foresight to any auction where a
  bidder knows her own valuation, that is
  regardless of the information available to the
  other bidders, and regardless of how they bid.
• This important result also applies to ascending
  auctions.
• Rule 3 In a second price sealed bid auction,
  bid your valuation if you know it.

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Proving the third rule

• Suppose you bid above your valuation, win the
  auction, and the second highest bid also exceeds
  your valuation. In this case you make a loss. If
  you had bid your valuation then you would not
  have won the auction in this case. In every other
  case your winnings are identical. Therefore
  bidding your valuation weakly dominates bidding
  above it.
• Suppose you bid below your valuation, and the
  winning bidder places a bid between your bid and
  your valuation. If you had bid your valuation,
  you would have won the auction and profited. In
  every other case your winnings are identical.
  Therefore bidding your valuation weakly dominates
  bidding below it.
• Combining the two parts of the proof, we
  conclude bidding your valuation is a weakly
  dominant strategy.

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Probability distribution of the second highest
valuation

• Since any auction satisfying the conditions for
  the theorem can be used to calculate the expected
  revenue, we select the second price auction.
• The probability that the second highest valuation
  is less than x is the sum of the the
  probabilities that
• 1. all the valuations are less than x, or
  F(x)N
• 2. N-1 valuations are less than x and the
  other one is greater than x. There are N
  ways of doing this so the probability is
  NF(x)N-11 - F(x)
• The probability distribution for the second
  highest valuation is therefore NF(x)N-1 - (N
  - 1) F(x)N

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Expected revenue from Private Value Auctions

• The probability density function for the second
  highest valuation is therefore
• N(N 1)F(x)N-2 1 - F(x)F(x)
• Therefore the expected revenue to the auctioneer,
  or the expected value of the second highest
  valuation is

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Using the revenue equivalence theorem to derive
optimal bidding functions

• We can also derive the solution bidding
  strategies for auctions that are revenue
  equivalent to the second price sealed bid
  auction.
• Consider, for example a first price sealed bid
  auctions with independent and identically
  distributed valuations.
• The revenue equivalence theorem implies that
  each bidder will bid the expected value of the
  next highest bidder conditional upon his
  valuation being the highest.

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Bidding in a first price sealed bid auction

• The truncated probability distribution for the
  next highest valuation when vn is the highest
  valuation is
• In a symmetric equilibrium to first price
  sealed bid auction, we can show that a bidder
  with valuation vn bids

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Comparison of bidding strategies

• The bidding strategies in the first and second
  price auctions markedly differ.
• In a second price auction bidders should submit
  their valuation regardless of the number of
  players bidding on the object.
• In the first price auction bidders should shave
  their valuations, by an amount depending on the
  number of bidders.

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The derivation

• The probability of the remaining valuations
  being less than w when the highest valuation is
  v(1) is
• Therefore the probability density for the
  second highest valuation when vn v(1)
  is
• This implies the expected value of the second
  highest valuation, conditional on vn v(1) is
• Integrating by parts we obtain the bidding
  function

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An example the uniform distribution

• Suppose valuations are uniformly distributed
  within a closed interval, with probability
  distribution
• Then

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Bidding function for the uniform distribution

• Thus in the case of the uniform distribution the
  equilibrium bid of the player with valuation v is
  to bid a weighted average of the lowest possible
  valuation and his own, where the weights are
  respectively 1/N and (N-1)/N

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All pay sealed bid auction with private values

• The revenue equivalence theorem implies that the
  amount bidders expect to pay in an all-pay
  auction as in all other auctions satisfying the
  conditions of the theorem.
• In contrast to a first or second price sealed
  bid auctions where only the winner bidder pays
  his bid or the second highest bid in an all pay
  auction losers also pays their bids.
• The amount paid by the nth bidder is certain,
  and not paid with the probability of winning the
  auction, that is F(vn)N-1.
• By the revenue equivalence theorem the amount
  each bidder expects to pay in the first two
  auctions, upon seeing their valuation, equals the
  amount the bidder actually does pay in all pay
  auction.

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Bidding in all pay auction

• The previous slide implies that in an all pay
  auction a bidder with vn bids the product of
  F(vn)N-1 and the amount he would bid in a first
  price auction.
• This is

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The uniform distribution revisited

• If valuations are uniformly distributed within
  in a closed interval, with probability
  distribution
• then