Auction Theory By Krishna | The Revenue Equivalence Principle

Chapter 3 — The Revenue Equivalence Principle
Gota Morishita
gota.morishita@gmail.com
June 12, 2020
Gota Morishita Krishna ”Auction Theory” Chapter 3 June 12, 2020 1 / 23

OVERVIEW
1 3.0 Overview
2 3.1 MAIN RESULT
3 3.2 SOME APPLICATION OF THE REVENUE EQUIVALENCE
PRINCIPLE

Previously...
We saw that
the expected selling price is invariant in a first- and second-price
auction.
the fact holds regardless of the distribution of values F.
a risk-neutral seller is indifferent between the two formats.
Formally,
Proposition (2.3)
With independently and identically distributed private values, the expected
revenue in a first-price auction is the same as the expected revenue in a
second-price auction.

Presently...
We will show
that the invariance extends beyond ﬁrst- and second-price auctions to
a whole class of ”standard” auction,
using the invariance property, the equilibrium strategies are drawn in
some ”unusual” auction forms.

Auction Form
The auction forms we consider here have the following features:
buyers are asked to submit bids,
who wins and how much the winner pays depend on the submitted bids
alone,
the winner is the person who bids the highest amount.
We say that an auction is standard if the last condition is satisﬁed.
e.g. standard auction: a third-price auction, all-pay auction
e.g. non-standard auction: a lottery

Notation
a standard auction form: A,
a symmetric equilibrium of the auction: βA,
the equilibrium expected payment by a bidder with value x: mA(x)

Revenue Equivalence Theorem
Proposition (3.1)
Suppose that values are independently and identically distributed and all
bidders are risk neutral. Then any symmetric and increasing equilibrium of
any standard auction, such that the expected payment of a bidder with
value zero is zero, yields the same expected revenue to the seller.

Revenue Equivalence Theorem : Proof
Consider a particular bidder, say 1, and suppose that other bidders are
following the equilibrium strategy β such that mA(0) = 0.
Now, assume that bidder 1 bids β(z) instead of the equilibrium bid
β(x)1.
The winning probability is G(z) since β(z) > β(Y1) implies z > Y1.
The expected payoﬀ of bidder 1 is
ΠA
(z, x) = G(z)x − mA
(x).
1
There is such z because β increases.

Differentiating the payoff with respect to z, we have the first order
condtion:
∂
∂z
ΠA
(z, x) = g(z)x −
d
dx
mA
(z) = 0.
At an equilibrium it is optimal to report z = x, so we obtain the
following identity equation:
d
dy
mA
(y) = g(y)y.

Hence,
mA
(x) = mA
(0) +
x
0
yg(y)dy
=
x
0
yg(y)dy
= G(x)
x
0
yg(y)
G(x)
dy
= G(x)E[Y1|Y1 < x],
where p.d.f of Y1|Y1 < x is yg(y)
G(x) .

Revebye Equivalence Theorem : Example 3.1
If values are uniformly distributed on [0, 1], that is, X ∼ U(0, 1),
G(x) = xN−1.
Proposition 3.1 implies that
mA
(x) =
N − 1
N
xN
.
and the ex ante expected payment is
E[mA
(X)] =
N − 1
N(N + 1)
.
Therefore, the expected revenue is
E[RA
] = N × E[mA
(X)] =
N − 1
N + 1
.

Applications
Other than the previous boring example2, we show how Proposition 3.1
can be
used to derive equilibrium bidding strategies in alternative unusual
auction forms,
extended and applied to the case where bidders are unsure about how
many rival bidders there are.
2
This is my opinion.

Unusual Auctions
Using the revenue equivalence theorem, we draw the equilibrium bidding
strategies in the following two auction forms:
a third-price auction
The winner pays the third highest bid.
an all-pay auction
Everyone pays what they bid whether they win or lose.

All-pay Auction : equilibrium bid
An all-pay auction can be used to model lobbying activity.
Suppose that there is a symmetric, increasing equilibrium βAP such
that the expected payment of bidder with value 0 is 0.
Whether bidders win or lose, they pay what they bid. Therefore,
βAP
(x) = mA
(x)
=
x
0
yg(y)dy
This is necessary condition. To verify that it is actually an equilibrium
bidding strategy, the same procedure is conducted (see p17 in the
previous slide).

Third-Price Auction : equilibrium bidding strategy
Suppose that there are at least three bidders.
Again, suppose that there is a symmetric, increasing equilibrium βIII
such that the expected payment of a bidder with value 0 is 0.
Proposition 3.1 gives us
mIII
(x) =
x
0
yg(y)dy. (1)
Next, we write mIII(x) using βIII.
To do so, we draw the expression of the density of Y2, conditional on
Y1 < x.

The density of Y2, conditional on Y1 < x:
f
(N−1)
2 (y|Y1 < x) =
1
F
(N−1)
1 (x)
(N − 1) F(x) − F(y)
y=Y2<Y1<x
f
(N−2)
1 (y)
Y
(N−2)
1 =y
The expected payment in a third-price auction can then be written as:
mIII
(x) = F
(N−1)
1 (x)E[βIII
(Y2)|Y1 < x]
=
x
0
βIII
(y)(N − 1) F(x) − F(y) f
(N−2)
1 (y)dy
(2)

Combining 1 and 2,
x
0
βIII
(y)(N − 1) F(x) − F(y) f
(N−2)
1 (y)dy =
x
0
yg(y)dy
Solving the integral equation, we obtain
βIII
(x) = x +
F(x)
(N − 2)f (x)
.
We require F/f to be increasing to ensure that βIII is increasing.

Proposition (3.2)
Suppose that there are at least three bidders and F is log-concave.
Symmetric equilibrium strategies in a third-price auction are given by
βIII
(x) = x +
F(x)
(N − 2)f (x)
.
Comparing equilibrium bids in ﬁrst-, second-, and third-price auctions,
βI
(x) < βII
(x) = x < βIII
(x)

Uncertain Number of Bidders
So far, we assume that the following are common knowledge.
the number of bidders
the distribution from which they draw their values
We show it does not hurt to add the uncertainty about the number of
bidders

Notation
the set of potential bidders : N = {1, 2, . . . , N}.
the set of actual bidders : A ⊆ N.
the probability that any participating bidder assigns to the event that
he is facing n other bidders : pn.
We assume that everyone has the same belief about the likelihood of
meeting diﬀerent members of rivals.
The assumption is crucial because the symmetricity of auctions holds,
which results in a straightforward extension of Proposition 3.1.

Uncertain Number of Bidders
The equilibrium bidding strategy is assumed to be β.
Consider the expected payoﬀ of a bidder with value x who bids β(z)
instead of β(x).
The probability that he faces n other bidders is pn.
At an equilibrium, he wins if Y
(n)
1 < z. and the prob. is G(n) = F(z)n.
Therefore, the overall probability that he will win when he bids β(z) is
G(z) =
N−1
n=0
pnG(n)
(z).
The diﬀerence between the uncertainty and certainty cases is the
probability of winning auctions.
So, the results we have obtained so far are the same except for G.

Uncertainty Case — Second-Price Auction
In a second-price auction, even though the number of rival buyers is
uncertain, honesty pays, that is, bidding their own value is a
dominant strategy.
The expected payment in the uncertainty case under a second-price
auction is
mII
(x) =
N−1
n=0
pnG(n)
(z)E Y
(n)
1 |Y
(n)
1 < x .
Using this and the revenue equivalence theorem, we draw the
equilibrium strategy in a ﬁrst-price auction.

Uncertainty Case — First-Price Auction
Assume that β is a symmetric and increasing equilibrium.
The expected payment of an actual bidder with value x is
mI
(x) = G(x)β(x).
Applying the extended revenue equivalence theorem, mII(x) = mI (x).
Hence, we have
β(x) =
N−1
n=0
pnG(n)(x)
G(x)
E Y
(n)
1 |Y
(n)
1 < x
=
N−1
n=0
pnG(n)(x)
G(x)
β(n)
(x),
where β(n) is the equilibrium strategy in a ﬁrst-price auction in which there
are exactly n + 1 bidders for sure.

Auction Theory By Krishna | The Revenue Equivalence Principle

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