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likelihood techniques since rather than simply providing a single 'most likely' parame-
ter value, we derive a distribution that describes our belief over all possible values.
To illustrate this process, we describe a general setting, in which an auctioneer im-
plements a regularly repeating auction, and in each auction a single identical item is
sold. As described earlier, we assume that there is a large pool of potential bidders, who
have private independent valuations that are drawn from a common distribution. Each
potential bidder has a small probability of actively participating in any auction, and thus
each repeated auction faces a number of bidders that is described by the Poisson dis-
tribution shown in equation 4. Note that whilst their numbers are similar, these bidders
are different individuals with different valuations and, since we are explicitly interested
in the actions of the auctioneer, we assume that their bidding behaviour is unaffected by
their own observations of previous auctions
1
. Thus, our goal is to estimate the typical
number of bidders who participate in each auction,
, and also the parameters that de-
scribe their common valuation distribution. These estimated parameter values can then
be used to calculate optimal discrete bid levels in subsequent auctions.
ν
5.1
Estimating the Mean Number of Bidders
We first consider an example in which the bidders' valuation distribution is known,
but,
, the parameter that characterises the Poisson distribution and represents the mean
number of bidders participating in each repeated auction, is unknown. Thus, if at time
t
the auctioneer implemented an auction that used the discrete bid levels
l
t
=
ν
l
t
0
...
l
t
m
}
{
and closed at bid level
l
t
w
, we wish to find the value
that best explains this outcome.
In other words, we wish to calculate the probability distribution
P
(
ν
l
t
w
,
F
(
x
)
,
l
t
).Now,
in equation 6 we have already derived the probability of the auction closing at any bid
level, in terms of the mean number of bidders, the bidders' valuation distribution and
the actual bid levels implemented. Thus, in the notation we are using here, we have
already derived
P
(
l
t
w
|
ν
ν
|
,
F
(
x
)
,
l
t
). With this expression, we can use Bayes' theorem in
order to calculate the required result:
P
ν
|
l
t
w
,
F
(
x
)
,
l
t
=
P
(
l
t
w
|
ν
,
F
(
x
)
,
l
t
)
P
(
ν
)
(9)
P
(
l
t
w
|
F
(
x
)
,
l
t
)
Now, this described the case where the auctioneer has made an observation of a single
auction. In general, if
t
such auctions have been observed, the auctioneer can use all of
this evidence to improve its estimate. Thus if the bid levels used in these auctions were
L
=
l
1
,...,
l
t
l
w
,...,
l
t
w
}
{
}
, and the observed closing prices were
l
w
=
{
,wehave:
i
=1
P
l
i
w
|
ν
,
F
(
x
)
,
l
i
P
(ν)
Z
t
P
(ν
|
l
w
,
F
(
x
)
,
L
)=
(10)
In this expression,
Z
is a normalising factor that ensures that
P
(
ν
|
l
w
,
F
(
x
)
,
L
) sums to
one over the range of possible values of
ν
.Now,
P
(
ν
|
l
w
,
F
(
x
)
,
L
) is a continuous prob-
1
This assumption is reasonable in circumstances where historical auction data is not available
to the bidders. However, we intend to investigate the full implications of this assumption in
future work.
