Lecture 11: Auction Theory
Advanced Microeconomics II
Yosuke YASUDA
National Graduate Institute for Policy Studies
January 16, 2014
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 x2are 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.
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First-Price Auction (1)
Bidders simultaneously and independently submit bids b1 and b2.
◮ The painting is awarded to the highest bidder i∗ with max bi ,
◮ who must pay her own bid, bi∗.
To derive a Bayesian Nash equilibrium, we assume the bidding strategy in equilibrium is i) symmetric, and ii) linear function of xi. That is, in equilibrium, player i chooses
β(xi) = c + θxi. (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
maxb1 (x1− b1) Pr{b1 > β(x2)}. (2)
First-Price Auction (2)
Since x2 is uniformly distributed on [0, 1] by assumption, we obtain Pr{b1 > β(x2)} = Pr{b1 > c+ θx2}
= Pr b1− c θ > x2
= b1− 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.
maxb1 (x 1− b1)
b1− c
θ
Taking the first order condition, we obtain du1
db1
= 1
θ[−2b1+ x1+ c] = 0 ⇒ b1= c 2 +
x1
2 . (3)
Comparing (3) with (1), we can conclude that c = 0 and θ = 12 constitute a Bayesian Nash equilibrium.
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Second-Price Auction
Bidders simultaneously and independently submit bids b1 and b2.
◮ The painting is awarded to the highest bidder i∗ with max bi ,
◮ at a price equal to the second-highest bid, maxj6=i∗bj. Unlike the first-price auction, there is a weakly dominant strategy for each player in this game.
Thm In a second-price auction, it is weakly dominant strategy to bid according to β(xi) = 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)
Def 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. Def 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)
.
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Expectation (2)
Def 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.
Order Statistics
Let X1, X2, . . . , Xnbe n independent draws from a distribution F with associated probability density function (PDF) f (= F′).
◮ Let Y1, Y2, . . . , Yn be a rearrangement of these so that Y1≥ Y2 ≥ · · · ≥ Yn.
◮ Yk is called kth(-highest) order statistic.
◮ Let Fk denote the distribution of Yk (with its pdf fk). The distribution of the highest order statistic is
F1(y) = F (y)n
f1(y) = nF (y)n−1f(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−2f(y).
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Expected Revenue: First-Price
In a first-price auction, the payment is max{12X1,12X2} .
◮ Recall that β(xi) = 12xi is a BNE.
◮ max{1 2X1,
1 2X2} =
1
2max{X1, X2} = 1 2Y1.
The expectation of Y1 becomes E[Y1] =
Z 1
0
yf1(y)dy
= Z 1
0
2y2dy= 2 3y
3
1
0
= 2 3.
The expected revenue of the first-price auction is 1 3 (=
1 2×
2 3).
Expected Revenue: Second-Price
In a second-price auction, the payment is min{X1, X2} .
◮ Recall that β(xi) = xi, 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 − y2)dy
= 2 1 2y
2
1
0
− 1 3y
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!.
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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.
Thm 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.
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 > Y1n−1} = (x − b) × G(β−1(b))
where Y1n−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 Y1n−1.
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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[Y1n−1| Y1n−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.
First-Price: General Model (3)
The expected payment (to the seller) of each bidder given x is G(x) × E[Y1n−1| Y1n−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[Y1n−1| Y1n−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[Y2n] since f2(y) = n(1 − F (y))f1n−1(y).
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