Uniformly distributed consumer interests

If the consumer interest levels are distributed uniformly, the broker profitability per consumer I(J, P) can be calculated analytically in the limit , if we make the assumption that for all j, i.e., all categories are equally prevalent. This makes all categories equally profitable, effectively replacing the question of which categories to offer with the simpler one of how many to offer. Fig. 2 shows the result in the form of profit cross-sections I(J, P) vs. P for , for a characteristic choice of extrinsic costs , and consumer value V.

  
Figure 2: I(J, P) vs. P for J=1, 2, 3, 10, when .

Focusing for the moment on the curve for J=1, we observe that the profit per customer has a single peak at . As P increases beyond , the broker's profit is reduced as fewer and fewer consumers purchase articles. On the other hand, as P decreases away from , the broker's profit per customer per article is less.

More important, however, is the fact that as the number of categories J increases, the profitability value at the peak changes drastically. The analytic solution shows that is a strictly decreasing function of J, and as . In other words, the broker can always charge a higher price if it offers fewer categories. However, the number of consumers actually buying articles, and hence profit per consumer, may increase or decrease with the number of offered categories. For J = 1, the peak profit per customer is , while for J = 10, the peak value is . Evidently, for optimal profit, the broker should not automatically offer as many categories as possible. In this case, for example, the best situation is J=2, because it has the highest peak, even though the overall curve is not universally higher than other curves for arbitrary P.

  
Figure 3: vs. J for , and V=1 for all three curves.

The dependence of the broker's profit on the number of categories is summarized in Fig. 3. It shows the peak profit per customer vs. J, for three different settings of and , with V=1. For and , the broker's profit is maximized for a small number of categories (J=1 and J=2, respectively). For , however, profit increases monotonically with J. We may call this the ``spam'' regime, in which the extrinsic costs of computation and transportation are so low that the broker can always make more profit by offering more categories.

As Fig. 3 shows, for each setting of extrinsic costs costs and , there is an optimal number of categories that maximizes broker profit. Fig. 4 shows a contour plot of as a function of and , again for V = 1. Note that, for , . In this region, it is impossible to simultaneously satisfy the consumers' and broker's conflicting desires as indictated in Eq. 4. Even for a maximally interested consumer ( ), for all positive prices P.

  
Figure 4: The optimal number of categories for the broker to offer as a function of and , for the uniform distribution of consumer interests. As before, V=1.