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Chapter 19 Equivalence

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Title: LECTURE 6 Author: Vesna Prasnikar Last modified by: Robert Miller Created Date: 12/1/2000 8:52:25 PM Document presentation format: On-screen Show – PowerPoint PPT presentation

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Title: Chapter 19 Equivalence


1
Chapter 19Equivalence
  1. Types of Auctions
  2. Strategic Equivalence
  3. Revenue Equivalence
  4. Optimal Bidding

2
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.
3
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.

4
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.

5
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.

6
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

7
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

8
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

9
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.

10
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!

11
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.

12
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

13
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

14
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.

15
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.

16
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.

17
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.
18
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.

19
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

20
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.

21
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.
22
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.
23
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?

24
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 .

25
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.

26
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).
27
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.

28
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)

29
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)

30
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

31
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.

32
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.

33
4. Optimal Bidding
We apply the revenue equivalence theorem to solve
for the optimal bidding rules for several types
of private value auctions.
34
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.

35
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.

36
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.

37
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.

38
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

39
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

40
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.

41
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

42
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.

43
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

44
An example the uniform distribution
  • Suppose valuations are uniformly distributed
    within a closed interval, with probability
    distribution
  • Then


45
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

46
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.

47
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

48
The uniform distribution revisited
  • If valuations are uniformly distributed within
    in a closed interval, with probability
    distribution
  • then

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