The two optimization techniques studied in this section are instances of two fundamentally different approaches: Function approximation (exemplified by the neural net) and direct search for optima (exemplified by amoeba).
Function approximation techniques such as that used by the neural net comprise two steps. First, a model of the entire profit landscape is estimated from an observed series of price schedules and resultant profits. Second, this model landscape is used to predict expected profits for a large number of proposed price schedules, and the schedule for which the model predicts the largest expected profit is chosen as the (approximately) optimal one. In general, a function approximating approach would appear to have the advantage that optimization can be done in ``virtual'' time, i.e. once the model landscape has been learned, the method can explore a vast number of possible price schedules without risking the loss of real time or real money. On the other hand, a large number of data samples may be required to learn the model in the first place, so overall the method may be expensive and slow. Furthermore, function approximation is likely to fare poorly if the learned model is insufficiently accurate in the vicinity of the peak - a problem to which the technique is vulnerable because it strives for a good global fit to the entire landscape rather than a good localized fit to the peak. This study provides some evidence for both of these effects: relative to amoeba, the neural net took longer to reach a solution, and this solution was generally less optimal than that attained by amoeba, as is seen in Table 2.
Table 2: A comparison of the performance of neural net and amoeba over the different pricing schedules for N = 10
Also compare the craggy landscape shown in Figure 5 to the estimated landscape produced by the neural net in Figure 7.
Figure 7: Neural net-learned profit
landscape with N=10 and two-part tariff pricing
In Figure 7, we can see that the peaks and fissures that are a part of the two-part tariff price schedule have been replaced within the neural net by an overly smooth landscape, leading to a loss of some optimal solutions.
In contrast, direct-search optimizers such as amoeba make no attempt to establish a functional relationship between prices and profits. Amoeba single-mindedly seeks the optimal price schedule without retaining any information about the portions of the profit landscape that lie beyond the borders of its ever-shrinking simplex. Ultimately, amoeba comes to rest at or near an optimum (almost always the global optimum in these experiments), but knows nothing of its environs. In the problem studied here, this ignorance is not a liability since the landscape is static; consumers are fixed in their preferences and behavior and there are no competitors which can introduce prices and alter the landscape.
While amoeba appears to outperform the neural network for this specific problem, it would be premature to conclude that function approximation techniques are inferior to direct-search optimizing ones. There are two basic reasons why further exploration of function approximation approaches ought to be encouraged.
First, the neural network is only one of a great variety of function approximation techniques, and has not been tuned for this particular problem. Other training methods might allow faster training with less data. Off-the-shelf Quickprop is very generic, and does function approximation using sigmoidal basis functions. These might be replaced with basis functions that are more appropriate to the application at hand. Also, there is no a priori reason why the algorithm must minimize the error over the entire landscape; it could be altered to minimize the error at the landscape's peaks, at the cost of a loss of resolution of lower parts of the landscape.
A more sophisticated learning approach might apply domain knowledge and analysis to formulate a more useful and appropriate model of the landscape. It might prove advantageous to model the consumer preferences rather than the profit landscape itself, particularly if the number of parameters that define consumer preferences is smaller than the number of parameters defining the price schedule.
A second reason to encourage further exploration of function approximation
approaches is that the situation studied here is not truly
representative of the complex, dynamic environments in which we
expect producer agents to operate, and function approximating approaches may
prove to be advantageous in such environments. A more global
knowledge of the landscape is likely to be helpful in dealing with
shifting consumer preferences, and is probably essential in dealing
with competitors, who are continually trying to knock one another off
of the peak in an endless game of ``King of the Hill''.
Finally, it should be noted that function approximating and direct-search approaches might be complementary. If direct-search techniques prove to be generally faster in their convergence, but fragile in the face of competition that alters the profit landscape, then one could quietly learn a model of the entire landscape in parallel while the direct-search optimizer is being used to seek a good price schedule. Once a reasonable model of the landscape has been developed, the direct-search optimizer can start using this model rather than real-world data, resulting in a tremendous optimization speedup.