Figures 1 and 2 show our results for the neural network and amoeba, respectively, on the five price schedules when N = 10. Each line indicates the average cumulative per-period (per-article, per-customer) profit for a particular schedule. For example, the value of about 0.65 for mixed bundling at iteration 10 in Figure 1 means that over the first 10 iterations, profit averaged 0.65 (so cumulative profit is approximately 6.5). Higher lines indicate schedules which have been more profitable to date.
With neural net learning, mixed bundling performs extremely well. It has the highest profits of any schedule for about 100 periods, and thereafter is close to the leader, two-part tariff. Contrary to our expectations, the one-parameter schedules didn't perform particularly well, even during the learning phase. Their initial explorations lead to a substantial profit drop for 20 periods; after period 10, either or both of the two-parameter schemes nearly always dominate. Nonlinear pricing, which has the highest profit potential, performs very badly, getting stuck at a unit profit of about 0.6, only about 40% of the maximum possible. Note that the rate of convergence is not a good measure for comparing price schedules, since it does not reflect the quality of the final solution. For example, nonlinear pricing converges quickly to a local optimum, and thereafter obtains very poor profits.
The amoeba results are notably different. The one-parameter schemes (linear pricing and pure bundling) perform quite well for the first 30-40 iterations. Then the two-parameter schemes overtake the simpler schemes and head to a much higher average profit level. Amoeba also does much better with nonlinear pricing. As predicted, learning for the more complicated scheme is slower, and its cumulative profits are much lower than the other schemes for the first 400-1000 periods, but once the algorithm begins to converge on a solution its performance is swift and durable. The qualitative results are clear with amoeba: simpler schemes with lower profit potential nevertheless outperform more complex schedules during the early stages of learning.
Figure 1: Cumulative profit per article,
per period, per customer (N = 10, neural net)
Figure 2: Cumulative profit per article,
per period, per customer (N = 10, amoeba)
The experimental results for the neural network and amoeba when
N = 100 are shown in Figures 3 and 4. Increasing the number of goods
from 10 to 100 helped to smooth out the profit landscape and improved
the solution quality for a number of the schedules. For example, as
the number of goods increases, the two-part tariff landscape is
smoothed out. Also, it contains an easy-to-climb hill, and many
non-optimal values have relatively high profit, making exploration
less costly. Additionally, there are no large chasms, making it easy
to move between optima. Contrast this with mixed bundling, in which
valleys separate the optimal solution from the solutions for
linear pricing and pure bundling. Because of this, even though these
two-parameter schedules have the same potential optimal profit, the
two-part tariff performs better on average after learning is
complete
. Two-part tariff and mixed bundling also differ in the degree
to which their parameters are coupled. Mixed bundling offers two
independent prices. This makes non-optimal solutions more likely to be
profitable, but means that it is difficult to use one parameter to
``tune'' the other. In contrast, the fee and per-article price in
two-part tariff are more closely coupled; as one rises, the other will
fall, creating a crest in the landscape which a learner can ascend.
For the most part, increasing the number of articles that may be
transacted reinforces the results discussed above. Amoeba
again finds higher profit levels, and the qualitative results are
consistent across schedules. As seen in Figure 4, the one-parameter
schedules are much more profitable during the learning phase, but
after about fifty periods the more complex schedules
dominate. The pattern
is less systematic for neural net learning, but the results are
qualitatively similar. Note that the producer was able to find more
optimal solutions for each of the schedules; this is presumably due to
the smoothing of the profit landscape.
Figure 3: Cumulative profit per article,
per period, per customer (N = 100, neural net)
Figure 4: Cumulative profit per article,
per period, per customer (N = 100, amoeba)