Transcript 19_Equivalence

Chapter 19
Equivalence
1. Types of Auctions
2. Strategic Equivalence
3. Revenue Equivalence
4. Optimal Bidding
1. 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.
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.
Auction mechanisms
There are 5 standard types of auctions for
auctioning a single item which are widely used and
analyzed:
1.
2.
3.
4.
5.
First-price sealed-bid
Second-price sealed-bid
English
Japanese
Dutch
as well as several other types we will investigate.
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.
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:
un 
v n b 1if b n b 1
0 if b n b 1
The auctioneer receives:
u 0 b 1
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:
un 
v n b 2if b n b 1
0 if b n b 1
The auctioneer receives:
u 0 b 2
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:
un 
v n b 1if b n b 1
b n if b n b 1
The auctioneer receives:
u0 
N
b
n1 n
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.
Comparing the revenue from sealed
bid auctions
Notice that:

N
b
n1 n
b 1 b 2
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!
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.
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:
un 
v n a t 1 if t n t 1
0 if t n t 1
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:
un 
v n r t N1 if t n t N 
0 if t n t N 
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.
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.
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.
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.
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 bidder’s payoffs, are the same.
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
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 bidder’s 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.
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.
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.
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?
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 .
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.
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).
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.
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) > U(vj) + (vn – vj)P(vj)
Revealed preference continued
For convenience, we rewrite the last slide on the
previous page as:
U(vn) - U(vj) > (vn – vj)P(vj)
Now viewing the problem from the jth bidder’s
perspective we see that by symmetry:
U(vj) > U(vn) + (vj – vn)P(vn)
which can be expressed as:
(vn– vj)P(vn) > U(vn) - U(vj)
A fundamental equality
Putting the two inequalities together, we obtain:
(vn – vj) P(vn)> U(vn) - U(vj) > (vn – vj) P(vj)
Writing:
yields
vn = vj + dv
dU v   Pv dv
which, upon integration, yields
U vn   U v    Pv dv
vn
v v
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.
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.
4. Optimal Bidding
We apply the revenue equivalence
theorem to solve for the optimal
bidding rules for several types of
private value auctions.
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.
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.
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.
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.
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-1[1 - F(x)]
The probability distribution for the second highest
valuation is therefore: NF(x)N-1 - (N - 1) F(x)N
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:
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.
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:
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.
The derivation
The probability of the remaining valuations being less
than w when the highest valuation is v(1) is:
1N
N1
Pr
v 2 w|v 1F
v 1 F
w
Therefore the probability density for the second
highest valuation when vn = v(1) is:
N 1
F
vn 1N F
wN2 F 
w

This implies the expected value of the second
highest valuation, conditional on vn = v(1) is:
vn
N 1
F
vn   vF
vN2 F 
w
dv

1N
v
Integrating by parts we obtain the bidding function
b
vF
vn 
1N
N1
vF
v
vn
vv n
vv
vn F
vn 1N  F
vN1 dv
v
vn
F
vn   F
vN1 dv
1N
v
An example: the uniform distribution
Suppose valuations are uniformly distributed
within a closed interval, with probability distribution:
F
v
vv
v v
Then:
vn
b
vn vn F
vn   F
v dv
1N
N1
v
vn
vn 
vn v   
v v  dv
1N
v
 v /N v
N 1
/N
N1
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:
bvv /N vN 1/N
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.
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:
The uniform distribution revisited
If valuations are uniformly distributed within in
a closed interval, with probability distribution:
F
v
vv
v v
then:
b all pay vn Pr n wins auction b first price vn 

vv
v v
N1
v
N
N1
N v