Lecture 2 on Auctions
Revenue Equivalence
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. We describe the main kinds of auctions,
define strategic and revenue equivalence,
analyze optimal bidding behavior, and compare
the outcomes from using different types.
Read Chapters 19 and 20 of Strategic Play.
Relaxing strategic equivalence
In strategically equivalent auctions, the strategic
form solution strategies of the bidders, and the
payoffs to all them, are identical.
This is a very strong form of equivalence.
Can we show that such bidders 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.
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(vj).
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.
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.
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:
N  1F vn  v
1 N
wF w
vn
N 2
F ' wdw
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 in equilibrium, a player with valuation v bids a
weighted average of the lowest possible valuation and
his own, where the weights are 1/N and (N-1)/N:
vn
b
vn vn F
vn   F
vN1 dv
1N
v
vn
vn 
vn v   
v v N1 dv
1N
v
 v /N v
N 1
/N