Title: Chapter 19 Equivalence
1 Chapter 19Equivalence
- Types of Auctions
- Strategic Equivalence
- Revenue Equivalence
- Optimal Bidding
2Types 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.
3Why study auctions?
- Studying auctions is the simplest way of
approaching the question of price formation. - Auctions serve the dual purpose of eliciting
preferences and allocating resources between
competing uses. - A less fundamental but more practical reason for
studying auctions is that the value of goods
exchanged each year by auction is huge.
4Auction 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.
5Sealed 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.
6First 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
7Second 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
8All-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
9Examples 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.
10Comparing 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!
11Descending 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.
12Payoffs 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
13Ascending 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
14English 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.
15How 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.
16Japanese 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.
172. 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.
18Strategic 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.
19Descending 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
20Second-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.
21Summary
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.
223. 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.
23Relaxing 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?
24Revenue 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 .
25Why 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.
26Preferences 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).
27An 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.
28A 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)
29Revealed 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)
30A 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
31Revenue 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.
32A 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.
334. Optimal Bidding
We apply the revenue equivalence theorem to solve
for the optimal bidding rules for several types
of private value auctions.
34Steps 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.
35Second 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.
36Bidding 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.
37Proving 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.
38Probability 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
39Expected 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
40Using 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.
41Bidding 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
42Comparison 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.
43The 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
44An example the uniform distribution
- Suppose valuations are uniformly distributed
within a closed interval, with probability
distribution - Then
45Bidding 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
46All 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.
47Bidding 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
48The uniform distribution revisited
- If valuations are uniformly distributed within
in a closed interval, with probability
distribution - then