Information Technology Reference
In-Depth Information
Consequently, the auctioneer's expected revenue from the auction itself (i.e.,
excluding the payment
C
to the information provider), when the auctioneer uses
R
auc
=(
p
a
,p
1
, ..., p
k
) and the bidders use
R
bidder
, denoted
ER
(
R
auc
,R
bidder
),
is given by:
ER
(
R
auc
,R
bidder
)=
p
a
l
Pr
(
X
=
x
)
p
i
·
ER
auc
(
X
i
)
i
=1
x∈X
i
(3)
l
p
a
)+
p
a
p
i
))
+ ((1
−
Pr
(
X
=
x
)(1
−
·
ER
auc
(
∅
)
i
=1
x∈X
i
where
ER
auc
(
X
i
) is calculated according to (2) (also in the case where
X
i
=
).
Consequently the auctioneer's expected benefit, denoted
EB
(
R
auc
,R
bidder
), is
given by
EB
(
R
auc
,R
bidder
)=
ER
(
R
auc
,R
bidder
)
∅
p
a
C
.
A stable solution in terms of the mixed Bayesian Nash Equilibrium in this
case is necessarily of the form
R
auc
=
R
bidder
=
R
=(
p,p
1
, ..., p
l
)(be-
cause otherwise, if
R
auc
=
R
−
∗
=
R
bidder
then bidders necessarily have an
incentive to deviate to
R
bidder
=
R
), such that: (a) for any 0
<p
i
<
,R
bidder
)=
ER
auc
((1
,p
1
, ..., p
l
)
,R
bidder
)); (b) for any
p
i
=0(or
p
=0):
ER
auc
(
1(or0
<p<
1):
ER
auc
(
∅
,R
)=
ER
auc
(
X
i
)(or
ER
auc
(
∅
,R
bidder
)
∅
≥
,R
bidder
)
ER
auc
((1
,p
1
, ..., p
l
)
,R
bidder
); and (c) for
ER
auc
(
X
i
)(or
ER
auc
(
∅
≥
,R
bidder
)
,R
bidder
)
any
p
i
=1(or
p
=1):
ER
auc
(
≤
ER
auc
((1
,p
1
, ..., p
l
)
,R
bidder
). The proof for this derivation is similar to the proof
given in [34] (see page 39), with the exception that instead of referring to individ-
ual values of
X
we refer to subsets of values
X
i
. Therefore we need to evaluate
all the possible solutions of the form (
p,p
1
, ..., p
l
) that may hold (where each
probability is either assigned 1, 0 or a value in-between). Each mixed solution of
these 2
·
3
k
combinations (because there is only one solution where
p
= 0 is ap-
plicable) should be first solved for the appropriate probabilities according to the
above stability conditions. Since the auctioneer is the first mover in this model
(deciding on whether or not to purchase information), the equilibrium used is
the stable solution for which the auctioneer's expected profit is maximized.
We note that if the information is provided for free (
C
= 0) then information
is necessarily obtained and the resulting equilibrium is equivalent to the one
given in [11] for the pure Bayesian Nash Equilibrium case and in [29] for the
mixed Bayesian Nash Equilibrium case. Similarly, if
∅
≤
ER
auc
(
X
i
)(or
ER
auc
(
∅
i
is enforced
(i.e., the information provider provides the exact value of
X
) then the resulting
equilibrium is the same as the one given in [34].
|
X
i
|
=1
∀
4 Influencing the Information Provider's Capabilities to
Distinguish between Values
As discussed in the introduction, in various settings the auctioneer can influ-
ence the information provider's ability to distinguish between different values
the common value obtains. In this section we consider the case where the auc-
tioneer has full control over the structure of
D
, i.e., the division of
X
∗
into
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