The amoeba learner experiments support our predictions: simpler pricing schemes perform better during the phase of high exploration; when learning is nearly complete, exploitation dominates and the higher dimensional schemes perform better. However, the learning phase may be quite long, depending on the complexity of the schedule and the frequency of observable transactions. In particular, exploration is slow for the most general scheme, nonlinear pricing.
These results suggest that if an environment is changing significantly at moderate frequencies, a producer might be better off using a simpler scheme which has lower potential profit but is more robust to uncertainty. The results also suggest that more sophisticated agents might try to use statistical modeling to endogenously switch between simpler and more complex schemes, depending on indicators of stability and their progress in learning the current schedule. Although we have not modeled this phenomenon yet, the idea is suggestive of the variety of special promotions and trial prices that are used when new products are introduced to speed up exploration without sacrificing too much profit.
Our results also indicate the sensitivity of performance to the choice of learning method. We discuss this more fully in the next section, noting now only that in these experiments amoeba typically found better results than the neural net.
One of the more interesting and less expected results is the performance gap between schedules with the same dimensionality and the same potential profit. In an environment which contains uncertainty and learning, the smoothness of the profit landscape is an important determinant of effective complexity; it is harder to get consistently good results on a craggy landscape; cf. two-part tariffs vs. mixed bundling in Figures 5 and 6.
A number of recent papers have ignored uncertainty regarding the distribution of consumer preferences and advocate forms of mixed bundling for its exploitative performance; our results suggest that when exploration is important, two-part tariffs (and perhaps other two or more parameter schedules with smooth landscapes) may be superior.
Figure 5: Exact profit landscape with
N=10 and two-part tariff pricing
Figure 6: Exact profit landscape with
N=10 and mixed bundling pricing