Information Technology Reference
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set of possible subsets of
X
∗
, each containing values between which the infor-
mation provider cannot distinguish, such that
∪
X
i
∈D
X
i
=
X
∗
and
X
i
∩
X
j
=
∅
,
∀
i,j
.
If the information is purchased, the auctioneer, based on the subset obtained,
can decide either to disclose the information to the bidders or keep it to her-
self (hence disclosing
). If she discloses the information, then presumably the
information received from the information provider is disclosed as is (i.e., truth-
fully and symmetrically to all bidders), e.g., if the auctioneer is regulated or
has to consider her reputation. Finally, it is assumed that all players (auction-
eer, bidders and the information provider) are self-interested, risk-neutral and
fully rational agents, and are acquainted with the general setting parameters:
the number of bidders in the auction,
n
, the cost of purchasing the information,
C
, the possible subsets that may be obtained by the information provider,
D
,
the discrete random variables
X
and
T
, their possible values and their discrete
probability distributions.
The above model generalizes the one found in [11,29] in the sense that it
requires that the auctioneer decide whether or not to purchase the external
information rather than assume that she initially possesses it. Similarly, it gen-
eralizes the work in [34] in the sense that it allows the information provider to
provide a subset of values rather than the specific true value.
∅
3 Analysis
Our analysis uses the concept of mixed Bayesian Nash Equilibrium. Since the
auctioneer needs to decide both whether to purchase the information and if so
whether to disclose the information received, we can characterize her strategy
using
R
auc
=(
p
a
,p
1
, ..., p
l
)where
p
a
is the probability she will purchase the in-
formation from the information provider and
p
i
(1
≤ i ≤ l
) is the probability she
will disclose to the bidders the subset received if that subset is
X
i
. The dominat-
ing bid of a bidder of type
t
, when subset
X
is received (including the case where
X
=
, i.e., no information is disclosed), denoted
B
(
t, X
), is the expected pri-
vate value calculated by weighing each private value
V
t
(
x
) according to the post-
priori probability of
x
being the true common value given the information
X
,
denoted
Pr
(
X
=
x
∅
X
) [11], i.e.:
B
(
t, X
)=
x∈X
∗
V
t
(
x
)
X
). If the
|
·
Pr
(
X
=
x
|
Pr
(
X
=
x
)
y∈X
Pr
(
X
=
y
)
auctioneer discloses a subset
X
ↂ
X
∗
X
)=
=
∅
then
Pr
(
X
=
x
|
X
and
Pr
(
X
=
x
X
) = 0 otherwise. If no information is disclosed
for any
x
∈
|
(
X
=
X
=
) needs to be calculated based on the bid-
ders' belief of whether information was indeed purchased and if so, whether
that value is intentionally not disclosed by the auctioneer. Assume the bid-
ders believe that the auctioneer has purchased the information from the in-
formation provider
1
with a probability of
p
and that if indeed purchased then
if the information received was the subset
X
i
then it will be disclosed to the
∅
)then
Pr
(
X
=
x
|
∅
1
Being rational, all bidders hold the same belief in equilibrium.
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