In the model delineated in this paper, the producer faces
a profit maximization problem with two independent variables:
the normalized subscription fee f and the price-per-item 
. For
this problem, the
amoeba employs a two dimensional simplex (i.e., a triangle) to search in
the profit landscape. In the scenario in which the consumers know
their valuations and the producer knows both g and h, the producer
can compute the profit landscape and use amoeba offline to compute
the optimal f and to several digits of accuracy within a
few dozen iterations. This takes little more than a second on a modest
processor. It then sets f and to their computed values.
If the consumers know their valuations, but
the producer does not know g and h, the producer can still
use amoeba, but it must do so in an online configuration.
In this case, each iteration
of the amoeba samples the profit landscape by experimentation. Thus
each step is slow and costly, and the producer has to expect non-optimal
profit during the exploration phase of the algorithm. Nonetheless, after
a few dozen iterations, the producer will have found near-optimal settings
of f and , and will enjoy the same steady-state profits as
a producer that was fully informed from the beginning.
The situation becomes more interesting when the producer uses
amoeba in a market in which the consumers learn
their own valuations by sampling items and updating their
estimates via Eq. 14. The producer's knowledge
of g and h is now immaterial: its profit landscape is
determined by the consumers' beliefs about their valuations.
As has been seen, the distribution of estimated 
can diverge substantially from 
, resulting in a profit
landscape that shifts dramatically over time. Can the amoeba algorithm
cope with a dynamic landscape?
Fig. 6a illustrates
what happens when the amoeba algorithm is started with very small,
low-profit values of f and . Quickly, it finds
profitable values for these parameters.
Having reached a high-profit region, the amoeba begins to contract
its simplex. However, once the price parameters
stabilize, consumer leakage begins to set in, and the amoeba is unable
to escape from an ever-deepening trough similar in nature to that of
Fig. 5c. Amoeba falls into this trap because
it never bothers to resample vertices that it has already evaluated.
In particularly, it fails to consider that the profit obtained at
the best vertex in its simplex may change over time -- it (falsely)
believes that the best vertex is an ever-sharper needle sitting in a
ever-deepening trough in the profit landscape.
Once the amoeba finds a high-profit region, it is compelled to
shrink its simplex, and it is completely blind to the ever-diminishing
profits that occur throughout that simplex.
Figure 6: (a)Profit 
(solid line; normalized to ``ideal'' value of
0.41367) and proportion of subscribed
consumers m (dashed line) vs. time (in subscription periods) and
(b) f (solid) and (dashed) vs. time
when the producer uses amoeba for online learning and the consumers
estimate their 
values with flightiness parameter 
.
The horizontal dashed lines
indicate the optimal f and values for fully informed consumers.
As before, the market consists of M=10000 consumers and
one seller offering N=10 articles per subscription period.
For a simplex in two dimensions (a triangle), we have used the standard
coefficients 
for
the amoeba algorithm. In this and all subsequent experiments involving
the amoeba algorithm, the initial simplex consists of the following
values: (0.10, 0.10), (0.10, 0.15), and (0.15, 0.10).
The amoeba algorithm can be improved considerably for this application by having it occasionally resample previously sampled points and by conditionally expanding its simplex from time to time. Specifically, we propose a modified amoeba that is more suitable for dynamic landscapes by replacing Step 5 (Shrink Simplex) in the amoeba algorithm as follows:
at 
.
Denote the new evaluate as 
. 
then expand simplex, by
computing the n new vertices for 
:
Else, (i.e., 
)
shrink simplex, and compute the n new vertices for
:
Fig. 7 illustrates that the modified amoeba
algorithm avoids long-term profit and consumer leakage.
Leakage is thwarted because
the modified amoeba briefly lowers its prices to recapture disenfranchised
consumers, and then raises them again to enjoy increased profits
from consumers who have learned again to appreciate the
value of the producer's wares. In steady state, the average profit
(computed between iterations 50 and 200 to avoid transient effects)
is 0.31683, or 0.766 of the ideal profit. This is approximately
a 12% improvement over the highest profit that can be sustained
with fixed prices, which was 
.
Fig. 8 illustrates how
the distribution of responds to lowering of prices
by the modified amoeba. In iteration 40,
the producer makes a profit 
by setting
.
In Fig. 8a, less than
65% of the consumer population subscribes -- the ones
whose estimates are
below the critical threshold for those price parameters,
.
The situation changes dramatically in iteration 41 when the
producer sets 
. While
the profit decreases temporarily ( 
), all consumers
are now encouraged to subscribe because the critical
threshold has risen to 
.
The consumers re-estimate their values, and the overall
distribution now shifts to that illustrated in
Fig. 8b. This sets the stage for
the producer to obtain profit 
in the next four time steps
by increasing f (but reducing ).
Fig. 9 depicts the
corresponding profit landscapes at iterations a) 40 and b) 41 conditioned
on the distribution of . While the
profit landscape still shifts continually, the modified amoeba manages
to set prices in such a way as to avoid the formation of long-lived
troughs.
Additionally, Fig. 9c, which shows the
profit landscape at iteration 200, indicates that the algorithm can be
be successful for long periods of time. Ironically, it is the modified
amoeba's recognition that the landscape may be dynamic that helps
stabilize that landscape sufficiently to yield long-term profits.
Figure: (a)Profit (solid line; normalized to ``ideal'' value of
0.41367) and proportion of subscribed
consumers m (dashed line) vs. time (in subscription periods) and
(b) f (solid) and (dashed) vs. time
when the producer uses the modified amoeba algorithm for online learning.
The parameter settings remain unchanged from Fig. 6.
Figure 8: Estimated distributions at (a) time 40 and (b) time 41
(in subscription periods)
when the producer uses modified amoeba for online learning.
The dashed line represents the initial distribution of .
Figure:
Profit landscape showing the
mean profit per
article per consumer as a function of f and conditioned on
the distribution of
a) at iteration 40, b) at iteration 41, and c) at iteration 200
when the producer uses modified amoeba for online learning.
The parameter settings remain unchanged from Fig. 6.