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In-Depth Information
Posterior Probability
P
(
v
,
α
|
l
w
,
F
(
x
)
,
L
)
Actual Number Of Bidders (
n
)
30
x 10
−3
20
6
4
10
1
5
10
15
20
Repeated Auctions
2
Bid Levels & Observed Closing Prices (
L
,
l
w
)
8
0
2
6
4
Valuation
Distribution
Parameter
(
1
100
2
80
60
40
α
)
0
Mean Number
Of Bidders
(v)
20
1
5
10
15
20
0
Repeated Auctions
(a)
(b)
Fig. 6.
Plots showing (a) the actual number of bidders that participated in the auction (unknown
to the auctioneer) and the actual bid levels and closing prices observed by the auctioneer and (b)
the joint posterior belief distributions of the auctioneer after 20 repeated auctions
ment is realised after the first few auction and after this point, the revenue exceeds that
generated with fixed bid increments.
5.2 Estimating Multiple Parameters
The algorithm that we have presented here is certainly not restricted to learning sin-
gle parameters. In figure 6 we present a second example, this time for the exponential
valuation distribution presented in section 4.2. In this case we infer both the value of
parameter that describes the distribution of the number of bidders,
, and the value of
the parameter that describes the bidders' exponential valuation distribution,
ν
α
. Thus we
must calculate the two-dimensional joint probability distribution
P
(
l
w
,
F
(
x
)
,
L
).
Again, despite the stochastic nature of the auction process, after twenty repeated auc-
tions the probability distribution shows a clear peak around the true values of
ν
,
α
|
ν
= 20 and
α
= 1, and thus the bid levels converge toward the true optimal bid levels. Space does
not allow us present a full analysis of the convergence, however, in general, increasing
the number of parameters that are learnt reduces the convergence rate.
We can extend this method to estimate more parameters, by simply calculating larger
joint probability distributions in more dimensions. However, in so doing, the cost of per-
forming this exact calculation increases geometrically. Fortunately Bayesian inference
is a well developed field with sophisticated methods that allow us to approximate these
distributions. For example, variational methods (which we intend to explore in the fu-
ture) allow us to approximate the full
n
-dimensional joint distribution as the product of
n
independent distributions, with a corresponding computational saving [9].
6
Conclusions
In this paper we considered the optimal design of English auctions with discrete bid lev-
els and our aim was to automate their configuration to generate the maximum revenue
