### Lecture 11: Auction Theory

Advanced Microeconomics II

Yosuke YASUDA

Osaka University, Department of Economics yasuda@econ.osaka-u.ac.jp

January 13, 2015

### Auctions

Imagine that there is a (potential) seller who has a painting that is worth nothing to her personally. She hopes to make some money by selling the art through an auction.

Suppose there are two potential buyers, called bidders 1 and 2. Let x1 and x2 denote the valuations of the two bidders. If bidder i wins the painting and has to pay b for it, then her payoff is xi− b.

where x_{1} and x_{2}are chosen independently by nature, and
each of which is uniformly distributed between 0 and 1.
The bidders observe their own valuations before engaging in the
auction. The seller and the rival do not observe a bidder’s
valuation; they only know the distribution.

In what follows, we study two prominent sealed-bid auctions: a first-price auction and a second-price auction.

The former is commonly used while the latter has great

theoretical importance and started to be applied in reality. 2 / 15

### First-Price Auction (1)

Bidders simultaneously and independently submit bids b^{1} and b^{2}.
The painting is awarded to the highest bidder i^{∗} with max b_{i} ,
who must pay her own bid, b_{i}∗.

To derive a Bayesian Nash equilibrium, we assume the bidding
strategy in equilibrium is i) symmetric, and ii) linear function of x_{i}.
That is, in equilibrium, player i chooses

β(x_{i}) = c + θx_{i}. (1)

Now suppose that player 2 follows the above equilibrium strategy, and we shall check whether player 1 has an incentive to choose the same linear strategy (1). Player 1’s optimization problem, given she received a valuation x1, is

maxb_{1} ^{(x}^{1}^{− b}^{1}^{) Pr{b}^{1} ^{> β(x}^{2}^{)}.} ^{(2)}

### First-Price Auction (2)

Since x^{2} is uniformly distributed on [0, 1] by assumption, we obtain
Pr{b1 > β(x2)} = Pr{b1 > c+ θx2}

= Pr^{ b}^{1}^{− c}
θ ^{> x}^{2}

= ^{b}^{1}^{− c}
θ ^{.}

The first equality comes from the linear bidding strategy (1), the third equality is from the uniform distribution. Substituting it into (2), the expected payoff becomes a quadratic function of b1.

maxb_{1} ^{(x}^{1}^{− b}^{1}^{)}

b1− c θ

Taking the first order condition, we obtain du1

db1

= ^{1}

θ^{[−2b}^{1}^{+ x}^{1}+ c] = 0 ⇒ b1= ^{c}
2 ^{+}

x1

2 ^{.} ^{(3)}

Comparing (3) with (1), we can conclude that c = 0 and θ = ^{1}_{2}
constitute a Bayesian Nash equilibrium.

4 / 15

### Second-Price Auction

Bidders simultaneously and independently submit bids b^{1} and b^{2}.
The painting is awarded to the highest bidder i^{∗} with max bi ,
*at a price equal to the second-highest bid, max*j6=i^{∗}bj.

Unlike the first-price auction, there is a weakly dominant strategy for each player in this game.

Theorem 1

*In a second-price auction, it is weakly dominant strategy to bid*
*according to β(x*i) = xi ^{for all i.}

Since the combination of weakly dominant strategies always becomes a Nash equilibrium, bi = xi for all i is a BNE.

✞

✝

☎

*Rm Note that there are other asymmetric equilibria.*✆

For example, β1(x1) = 1 and β2(x2) = 0 for any x1 and x2

constitute a Bayesian Nash equilibrium.

### Expectation (1)

Definition 2

Given a random variable X taking on values in [0, ω], its

cumulative distribution function (CDF) F : [0, ω] → [0, 1] is: F(x) = Pr[X ≤ x]

the probability that X takes on a value not exceeding x.

F is non-decreasing F(0) = 0 and F (ω) = 1.

We assume that F is increasing and continuously differentiable.

6 / 15

### Expectation (2)

Definition 3

If X is distributed according to F , then its expectation is E[X] =

Z ω 0

xf(x)dx

= Z ω

0

xdF(x)

and if γ : [0, ω] → R is some arbitrary function, then the expectation of γ(X) is analogously defined as

E[γ(X)] = Z ω

0

γ(x)f (x)dx

= Z ω

0

γ(x)dF (x)

.

### Expectation (3)

Definition 4

The conditional expectation of X given that X < x is
E[X | X < x] = ^{1}

F(x) Z x

0

tf(t)dt,

which can be rewritten as follows (by integrating by parts): F(x)E[X | X < x] =

Z x 0

tf(t)dt

= xF (x) −
Z _{x}

0

F(t)dt. The conditional expectation of γ(X) is defined as

E[γ(X) | X < x] = ^{1}
F(x)

Z x 0

γ(t)f (t)dt.

8 / 15

### Order Statistics

Let X^{1}, X2, . . . , X_{n}be n independent draws from a distribution F
with associated probability density function (PDF) f (= F^{′}).

Let Y1, Y2, . . . , Y_{n} be a rearrangement of these so that
Y1≥ Y2 ≥ · · · ≥ Y_{n}.

Y_{k} *is called kth(-highest) order statistic.*

Let F_{k} denote the distribution of Y_{k} (with its pdf f_{k}).
The distribution of the highest order statistic is

F1(y) = F (y)^{n}

f1(y) = nF (y)^{n−1}f(y).

The distribution of the second-highest order statistic is
F2(y) = F (y)^{n}+ nF (y)^{n−1}(1 − F (y))

= nF (y)^{n−1}− (n − 1)F (y)^{n}.
f2(y) = n(n − 1)(1 − F (y))F (y)^{n−2}f(y).

### Expected Revenue: First-Price

In a first-price auction, the payment is max{^{1}_{2}X1,^{1}_{2}X2} .
Recall that β(xi) = ^{1}_{2}xi is a BNE.

max{^{1}_{2}X1,^{1}_{2}X2} = ^{1}_{2}max{X1, X2} = ^{1}_{2}Y1.
The expectation of Y1 becomes

E[Y1] = Z 1

0

yf1(y)dy

=
Z ^{1}

0

2y^{2}dy=^{ 2}
3^{y}

3

^{1}

0

= ^{2}
3^{.}

The expected revenue of the first-price auction is ^{1}
3 ^{(=}

1
2^{×}

2
3^{).}

10 / 15

### Expected Revenue: Second-Price

In a second-price auction, the payment is min{X^{1}, X2_{} .}

Recall that β(x_{i}) = x_{i}, i.e., trugh-telling is a BNE.
min{X1, X2} = Y2.

The expectation of Y2 becomes E[Y2] =

Z 1 0

yf2(y)dy

= Z 1

0

y× 2(1 − y)dy = Z 1

0

2(y − y^{2})dy

= 2 ^{ 1}
2^{y}

2

^{1}

0

−^{ 1}
3^{y}

3

^{1}

0

!

= ^{1}
3^{.}

The expected revenue of the second-price auction is ^{1}

3^{, which is}
*identical* to the expected revenue of the first-price auction!.

### Revenue Equivalence Theorem

The two sealed-bid auctions, first-price and second-price auctions, have different equilibrium bidding strategies but yield the same expected revenue.

Interestingly, this is not by chance; the revenue equivalence result, often called as “revenue equivalence theorem (RET)”, is known to hold in much more general situations.

Theorem 5

*RET holds whenever the following conditions are satisfied:*
*Private Value Each bidder knows her value of the object.*
*Independent Bidders receives their values independently.*
*Symmetric The distribution is identical among bidders.*
*Risk Neutral Each bidder is risk neutral.*

12 / 15

### First-Price: General Model (1)

Consider a first-price auction with n bidders in which all the conditions in the previous theorem are satisfied.

Assume that bidders play a symmetric equilibrium, β(x). Given some bidding strategy b, a bidder’s expected payoff becomes

(x − b) Pr{b > Y1^{n−1}} = (x − b) × G(β^{−1}(b))

where Y1^{n−1} is the highest order statistic among n − 1 random
draws of the values and G is the associated distribution.
Maximizing w.r.t. b yields the first order condition:

g(β^{−1}(b))

β^{′}(β^{−1}(b))(x − b) − G(β^{−1}(b)) = 0 (4)
where g = G^{′} is the density of Y1^{n−1}.

### First-Price: General Model (2)

Since (4) holds in equilibrium, i.e., b = β(x), g(x)

β^{′}(x)(x − b) − G(x) = 0 ⇐⇒ G(x)β^{′}(x) + g(x)β(x) = xg(x),
which yields the differential equation

d

dx(G(x)β(x)) = xg(x).

Taking integral between 0 and x, we obtain

Z x 0

d

dy(G(y)β(y))dy =

G(x)β(x) − G(0)β(0) = Z x

0

yg(y)dy

⇒ β(x) = ^{1}
G(x)

Z x 0

yg(y)dy = E[Y1^{n−1}| Y1^{n−1} < x].

✞

✝

☎

Rm The equilibrium strategy is to bid the amount equal to the✆ conditional expectation of second-highest value given that my value x is the highest.

14 / 15

### First-Price: General Model (3)

*The expected payment (to the seller) of each bidder given x is*
G(x) × E[Y1^{n−1}| Y1^{n−1} < x],

which is identical to that of the second-price auction.

✞

✝

☎

Rm The expected revenue is just the aggregation of the expected✆
*payment of all bidders, it can be derived by*

n× Z ω

0

G(x) × E[Y1^{n−1}| Y1^{n−1} < x]f (x)dx

= n Z ω

0

G(x) × ^{1}
G(x)

Z x 0

yg(y)dy

f(x)dx

= n Z ω

0

Z x 0

yg(y)dy

f(x)dx = n Z ω

0

Z ω y

f(x)dx

yg(y)dy

= n Z ω

0

y(1 − F (y))g(y)dy = Z ω

0

yf2(y)dy

⇒ E[Y2^{n}] since f2(y) = n(1 − F (y))f1^{n−1}(y).