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bidders with a probability of
p
i
. In this case the probability of any value
x
∈
X
i
being the true common value is given by:
Pr
(
X
=
x
)(
p
(1
−
p
i
)+(1
−
p
))
X
=
p
)+
p
X
j
p
j
)
y∈X
j
Pr
(
X
=
x
|
∅
)=
(1)
(1
−
(1
−
Pr
(
X
=
y
)
The term in the numerator is the probability that
x
is indeed the true value
however the subset it is in is not disclosed. If indeed
x
is the true value (i.e., with
a probability of
Pr
(
X
=
x
)) then it is not disclosed either when the information
is not purchased (i.e., with a probability of (1
−
p
)) or when purchased but not
disclosed (i.e., with a probability of
p
(1
p
i
)). The term in the denominator is the
probability information will not be disclosed. This happens when the information
is not purchased (i.e., with a probability (1
−
p
)) or when the information is
purchased however the auctioneer does not disclose the subset received (i.e.,
with a probability of
p
(1
−
p
j
)
y∈X
j
Pr
(
X
=
y
)). Further on in the paper
we refer to the strategy where information is not disclosed as an empty set.
The bidders' strategy, denoted
R
bidder
, can thus be compactly represented as
R
bidder
=(
p
b
,p
1
, ..., p
k
), where
p
b
is the probability they assign to information
purchased and
p
i
is the probability they assign to the event that the information
is indeed disclosed if purchased and becomes
X
i
.
In order to formalize the expected second-best bid if the auctioneer discloses
the subset
X
we apply the calculation method given in [34] but replace the
exact value
x
with a subset
X
. We first define two probability functions. The
first is the probability that given that the subset disclosed by the auctioneer
is
X
, the bid placed by a random bidder equals
w
, denoted
g
(
w,X
), given
−
by:
g
(
w,X
)=
B
(
t,X
)=
w
Pr
(
T
=
t
). The second is the probability that the
bid placed by a random bidder equals
w
or below, denoted
G
(
w,X
), given by:
G
(
w,X
)=
B
(
t,X
)
≤w
Pr
(
T
=
t
).
The auctioneer's expected profit when disclosing the subset
X
, denoted
ER
auc
(
X
), equals the expected second-best bid:
n
n
n−
1
−
1
ER
auc
(
X
)=
w
(
k
w∈{B
(
t,X
)
|t∈T}
k
=1
(2)
G
(
w,X
))(
g
(
w,X
))
k
(
G
(
w,X
)
g
(
w,X
))
n−k−
1
(1
−
−
n
k
(
g
(
w,X
))
k
(
G
(
w,X
)
n
g
(
w,X
))
n−k
)
+
−
k
=2
The calculation iterates over all of the possible second-best bid values, assign-
ing to each its probability of being the second-best bid. As we consider discrete
probability functions, it is possible to have two bidders place the same highest
bid (in which case it is also the second-best bid). For any given bid value,
w
,we
therefore consider the probability of either: (i) one bidder bidding more than
w
,
k
1) bidders bidding exactly
w
and all of the other bidders bidding
less than
w
; or (ii)
k
∈
1
, ...,
(
n
−
∈
2
, ..., n
bidders bidding exactly
w
and all of the others
bidding less than
w
.
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