Single broker case

Given that effective monopolies can occur even in multi-broker systems, it is useful and instructive to establish a few simple results for systems with a single broker. First, suppose the broker offers a single category, and that the number of consumers is arbitrarily large, i.e. B=1, J=1, and . The broker tries to maximize its utility by choosing a set of preferred consumers (those c for which ), setting a price P, and setting its interest level . The consumers try to maximize their utility simply by declaring whether they wish to subscribe to the broker (in which case ).

Analysis of Eqs. 1 and 2 shows that, for a wide range of parameters, the equilibrium state is ). The subscription matrix element , which defines whether or not consumer c has subscribed to the broker, is the product of two step functions, which can be understood intuitively as follows. The first step function, , represents the veto power of the broker: it only wishes to serve consumers that are interested enough (and the price is high enough) so that the expected revenue from an article will exceed the cost of sending it. The second step function, , represents the consumer's veto power: it only wishes to subscribe to the broker if it is interested enough (and the price is low enough) so that the expected net benefit of receiving an article exceeds the cost of processing it. The monopolistic equilibrium price is constrained by the step functions and the restriction to be in the range ; its exact value depends in detail upon , , V, and the distribution of consumer interest probabilities in the population. For example, when is a uniform distribution, is the solution to a cubic equation involving , , and V.

Now suppose that the number of categories J offered by the broker is arbitrary [7]. Assuming that the category prevalences are all equal ( ) and the distribution of consumer interests within a given category is given by , one can derive analytic expressions for the monopolistic equilibrium price as a function of the number of categories. Substituting this function back into Eq. 2, one can derive the broker's optimal utility as a function of the number of categories, and then from this the optimal number of categories. Illustrative results for two very different distributions are shown in Fig. 2.

  
Figure 2: The optimal number of categories for the broker to offer as a function of and (with V=1). a) is a uniform distribution in the interval (0,1), so . b) is an all-or-nothing distribution: with probability , else .

For a wide class of distributions, three behavioral regimes are observed. When the combined cost of transport and processing is sufficiently high ( ), the optimal number of categories is 0. In this ``dead'' regime, an article costs more to send and process than it is worth, even if the consumer is guaranteed to be interested in it, and so no articles will be bought or sold. At the other extreme, when the costs are sufficiently low ( , where is the mean of the distribution ), the broker is motivated to offer all categories that exist ( ). In real information filtering applications, one expects to be quite small, since each consumer regards most information as junk. It is useful to think of as a (presumably tiny) spam regime, in which it costs so little to send information, and the financial impact on a consumer of receiving junk is so minimal, that it makes economic sense to send all articles to all consumers. In between these two regimes, the optimal number of categories is finite.