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Source: Linkage principle

The linkage principle is a finding of auction theory. It states that auction houses have an incentive to pre-commit to revealing all available information about each lot, positive or negative. The linkage principle is seen in the art market with the tradition of auctioneers hiring art experts to examine each lot and pre-commit to provide a truthful estimate of its value.
The discovery of the linkage principle was most useful in determining optimal strategy for countries in the process of auctioning off drilling rights (as well as other natural resources, such as logging rights in Canada). An independent assessment of the land in question is now a standard feature of most auctions, even if the seller country may believe that the assessment is likely to lower the value of the land rather than confirm or raise a pre-existing valuation.
Failure to reveal information leads to the winning bidder incurring the discovery costs himself and lowering his maximum bid due to the expenses incurred in acquiring information. If he is not able to get an independent assessment, then his bids will take into account the possibility of downside risk. Both scenarios can be shown to lower the expected revenue of the seller. The expected sale price is raised by lowering these discovery costs of the winning bidder, and instead providing information to all bidders for free.

Use in FCC auction

linkage

principle

linkage

principle

Formal derivation

, and let

g
(
⋅
∣
x
)

{\displaystyle g(\cdot \mid x)}

be the associated density. We let

v
(
x
,
y
)
=
E

[

V

1

∣

X

1

=
x
,

Y

1

=
y

]

{\displaystyle v(x,y)=E\left[V_{1}\mid X_{1}=x,Y_{1}=y\right]}

be the expectation of the value to a bidder when the signal he receives is x and the highest signal among the other bidders, Y1 is y. We assume that v is nondecreasing in y and strictly increasing in x and that v(0, 0) = 0.
For each standard auction format A, suppose that the auction has a symmetric and increasing equilibrium βA, which is a mapping from a bidder's observed signal to its bid. Let

W

A

(
z
,
x
)

{\displaystyle W^{A}(z,x)}

denote the expected payment by a bidder if he is the winning bidder when he receives a signal x but bids as if their signal were z, i.e., he bids βA(z). Let

W

1

A

(
x
,
z
)

{\displaystyle W_{1}^{A}(x,z)}

denote the derivative of WA with respect to its first argument and

W

2

A

(
x
,
z
)

{\displaystyle W_{2}^{A}(x,z)}

the derivative with respect to its second argument, evaluated at (x, z).
For specific examples, in a first-price sealed-bid auction, labeled I, where the high bidder wins and pays the amount of their bid, we have

W

I

(
z
,
x
)
=

β

I

(
z
)
,

{\displaystyle W^{I}(z,x)=\beta ^{I}(z),}

and in a second-price auction, labeled II, where the high bidder wins and pays the amount of the second-highest bid, we have

W

I
I

(
z
,
x
)
=
E

[

β

I
I

(

Y

1

)
∣

X

1

=
x
,

Y

1

<
z

]

.

{\displaystyle W^{II}(z,x)=E\left[\beta ^{II}(Y_{1})\mid X_{1}=x,Y_{1}

Now we may state:

Linkage Principle. (Krishna, 2002, Proposition 7.1) Let A and B be two standard auctions, each having a symmetric and increasing equilibrium such that
(i) for all x,

W

2

A

(
x
,
x
)
≥

W

2

B

(
x
,
x
)
;

{\displaystyle W_{2}^{A}(x,x)\geq W_{2}^{B}(x,x);}

(ii) WA(0,0) = 0 = WB(0,0).
Then the expected revenue in A is at least as large as the expected revenue in B.
Proof: The expected payoff of a bidder with signal x who bids βA(z) is

∫

−
∞

z

v
(
x
,
y
)
g
(
y
∣
x
)
d
y
−
G
(
z
∣
x
)

W

A

(
z
,
x
)

{\displaystyle \int _{-\infty }^{z}v(x,y)g(y\mid x)dy-G(z\mid x)W^{A}(z,x)}

.
In equilibrium, it is optimal to choose z = x and the resulting first-order conditions imply that

v
(
x
,
x
)
g
(
x
∣
x
)
−
g
(
x
∣
x
)

W

A

(
x
,
x
)
+
G
(
x
∣
x
)

W

1

A

(
x
,
x
)
=
0
,

{\displaystyle v(x,x)g(x\mid x)-g(x\mid x)W^{A}(x,x)+G(x\mid x)W_{1}^{A}(x,x)=0,}

which we can rewrite as

W

1

A

(
x
,
x
)
=

g
(
x
∣
x
)

G
(
x
∣
x
)

v
(
x
,
x
)
−

g
(
x
∣
x
)

G
(
x
∣
x
)

W

A

(
x
,
x
)
.

{\displaystyle W_{1}^{A}(x,x)={\frac {g(x\mid x)}{G(x\mid x)}}v(x,x)-{\frac {g(x\mid x)}{G(x\mid x)}}W^{A}(x,x).}

Letting

Δ
(
x
)
=

W

A

(
x
,
x
)
−

W

B

(
x
,
x
)
,

{\displaystyle \Delta (x)=W^{A}(x,x)-W^{B}(x,x),}

we conclude that

Δ

′

(
x
)
=
−

g
(
x
∣
x
)

G
(
x
∣
x
)

Δ
(
x
)
+

(

W

2

A

(
x
,
x
)
−

W

2

B

(
x
,
x
)

)

{\displaystyle \Delta ^{\prime }(x)=-{\frac {g(x\mid x)}{G(x\mid x)}}\Delta (x)+\left(W_{2}^{A}(x,x)-W_{2}^{B}(x,x)\right)}

.
By hypothesis (i), the second term is positive, and by hypothesis (ii), which implies Δ(0) = 0, it follows that Δ(x) and Δ′(x) cannot be of different signs, implying that for all x, Δ(x) ≥ 0. Q.E.D.
To use this proposition to rank, for example, the second-price and first-price auctions, we need to assume that the bidders signals are affiliated (see Milgrom and Weber, 1982, Appendix on Affiliation, pp. 1118–1121), which implies that

G
(
z
∣
⋅
)

{\displaystyle G(z\mid \cdot )}

is decreasing and that

W

2

I
I

(
x
,
x
)
≥
0

{\displaystyle W_{2}^{II}(x,x)\geq 0}

. Note that

W

2

I

(
x
,
x
)
=
0

{\displaystyle W_{2}^{I}(x,x)=0}

. Thus, under the assumption of affiliation,

W

2

I
I

(
x
,
x
)
≥

W

2

I

(
x
,
x
)

{\displaystyle W_{2}^{II}(x,x)\geq W_{2}^{I}(x,x)}

. In addition, WII(0,0) = 0 = WI(0,0), so the Linkage Principle implies that expected revenue from a second-price auction is at least as great as that from a first-price auction.
To use this proposition to show that expected revenue is greater when public information is made available, consider the first-price auction. Let S be a random variable denoting the information available to the seller and suppose a symmetric equilibrium strategy

β
^

(
S
,

X

1

)

{\displaystyle {\widehat {\beta }}(S,X_{1})}

that is increasing in both variables. Then let

W
^

I

(
z
,
x
)
=
E

[

β
^

(
S
,
z
)

|

X

1

=
x

]

{\displaystyle {\widehat {W}}^{I}(z,x)=E\left[{\widehat {\beta }}(S,z){\big |}X_{1}=x\right]}

be the expected payment of a winning bidder when he receives signal x but bids as if it were z. Assuming S and X1 are affiliated, so that

W
^

2

I

(
z
,
x
)
≥
0
,

{\displaystyle {\widehat {W}}_{2}^{I}(z,x)\geq 0,}

then

W
^

2

I

(
x
,
x
)
≥

W

2

I

(
x
,
x
)

{\displaystyle {\widehat {W}}_{2}^{I}(x,x)\geq W_{2}^{I}(x,x)}

and the linkage principle implies that expected revenue is at least as great when information is revealed as when it is not.
To see that an ascending-bid auction has greater expected revenue than a second-price auction, note that in an ascending-bid auction, the observed points at which other bidders cease to be active provide additional signals that are also affiliated with X1 and so the logic for information revelation increases expected revenue applies.
Although it has been shown that the linkage principle need not hold in more complex auction environments (see Perry and Reny (1999) on the failure of the linkage principle in multi-unit auctions), as argued by Loertscher, Marx, and Wilkening (2013), the intuition provided by the linkage principle for the potential benefits of open over closed auction formats, and the benefits of information revelation generally, will likely continue to influence practical auction design far into the future.

Use in FCC auction

Formal derivation

References

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