Information Technology Reference
In-Depth Information
Mean Absolute Estimation Error
Mean Auction Revenue (%)
25
Learning Algorithm
100
20
99
Fixed bid increment (reserve=
x
/
2)
15
98
10
97
5
Fixed bid increment (no reserve)
0
96
1
5
10
15
20
1
5
10
15
20
Repeated Auctions
Repeated Auction
(a)
(b)
Fig. 5.
Plots showing (a) the converging estimates generated by the learning algorithm, and (b)
how this results in improvements in the auctioneer's revenue
figure 4a we show the actual number of bidders that participated (unknown to the auc-
tioneer) and the bid levels that were implemented in each repeated auction, along with
the actual bid level at which the auction closed (denoted by a filled circle on the appro-
priate bid level and observed by the auctioneer). In figure 4b, we show the probability
distribution,
P
(
ν
(shown after 2, 4 and 20 auctions). The variance in the observed auction closing prices
is driven by the stochastic nature of the number of bidders, their valuations and also
the changing auction bid levels. However, despite this variance, the auctioneers' belief
in the most likely value of
ν
|
l
w
,
F
(
x
)
,
L
), that describes the auctioneers' belief in the values of
converges rapidly to the true value. Thus the bid levels
implemented by the auctioneer also converge to the those that generate the maximum
revenue.
To demonstrate the convergence of this algorithm, after each repeated auction we
calculate the error in the estimate that it produced (i.e. the difference between the es-
timated value and the true value). We repeat the process 1000 times using the same
parameter values (i.e.
ν
ν = 20 and a uniform bidders' valuation distribution where
x
= 0
and
x
= 4) and average over the results. Figure 5a shows the mean absolute estimation
error plotted against the number of repeated auctions. The plot shows that the estimates
improve rapidly after the first few auctions and then converge to the true value.
Figure 5b shows the improvement in revenue that results from more accurately esti-
mating the mean number of bidders who are participating in the auctions, and then use
this result to optimise the discrete bid levels used in subsequent auctions. For the same
simulation runs presented in figure 5a, we show the efficiency of the auction, calculated
in terms of the percentage of the second highest bidder's valuation that the auction
was able to extract. We compare this revenue to that which would have been achieved
with an auction that used the more commonly implemented fixed bid increment, with
and without setting a reserve price. Clearly, as the estimates of the auction parameters
improve, so the revenue of the auctioneer increases. Significantly, the greatest improve-
