3 CDAP functions

3.2. Preferences and the determination of market strength

We define a context-dependent attribute preference by an ideal value, a tolerance index and an importance index for each type of attribute included in the model. In The application to a market for spirits reported in section 5, for example, the four contexts are functional drinking, social, reward-seeking and novelty-seeking. The attributes of the brands sold in that market which were specified by the marketing professionals as uniqueness, specialness and expensiveness. Expensiveness is not the same as price or relative price since an "expensive" drink can sometimes be acquired (relatively) cheaply in a sales promotion.

A natural representation of ideal and tolerance is as a preference distribution function. This reflects the expectation that different consumers will have different ideals but that there is, at the same time, a central tendency to their ideal attribute values. For social purposes, for example, the tendency will be to value specialness quite highly and uniqueness very little. The dispersion of ideals for this purpose will be small and, in general, individual tolerance to deviation from the ideal is seen by the marketing professionals to be very limited. For this model, preferences are represented by a transform of the normal distribution such that the ideal value of an attribute is in effect the mean and the tolerance determines the variance. Importance is represented formally in a manner which takes advantage of the recognition that our preference-distribution function is not a probability function and is not required to integrate to unity.

In Figure 1, ps on the vertical axis is the preference index determined by attribute value c for consumers in the context which entails undertaking activity s. The domain of ps is the unit interval. The value of c is represented on the horizontal axis. Clearly, for a given ideal attribute value c*, the dashed preference distribution entails more tolerance to deviations from the ideal than the solid-lined distribution. Moreover, the dashed distribution is more sensitive to attribute values below the ideal than to actual values above the ideal while the effect of deviations from the ideal is symmetrical for the solid-lined distribution

.

The preference index corresponding to the activity s context for brand b, denoted Gsb, is the product of the preference indices for actual attribute value associated with the brand. Formally,

(11)

where C is the set of defined attributes.

With this background, we turn now to the representation of importance.

It is easily seen that flatter and higher (in the sense of closer to 1) is the preference distribution, the smaller the effect if can have on the preference index of the brand for consumers undertaking the activity. If an attribute is completely unimportant, the preference distribution will be a horizontal line at the preference level equal to 1.

In Figure 2, we have the distributions differ only in their degrees of importance. Clearly, the flatter distribution is less important than the steeper distribution in that deviations from the ideal value entail preference indices closer to unity and, so, reduce the value of the context-dependent preference index for a brand by a lesser proportion

.

The preference distribution function used in the model reported here is a transform of the normal distribution since we are assuming that, at any time, individuals purchasing in a particular context are drawn at random from the population of potential purchasers and that there are random differences among them in respect of their ideal attribute values. In further developments, we will consider distributions of the importance and tolerance indices as well but, for the present, the preference distribution functional form is

(12)

where c is the value of attribute C, c* is the ideal value, mc is the index of the importance and tc is the corresponding tolerance index. In practice, we have found that we get good empirical results with these models by mapping tolerance indices into the unit interval and importance indices into the [0, 0.5]-interval.

The derivation of the brand strength measure is now straightforward.

For strength of brand b in context s is

(13)

which is the preference index for brand b expressed as a proportion of the sum of the preference indices for all brands in context s.

For any brand b, the market strength measure used in equation (8) is the sum of the context-related brand strengths as defined in equation (13) weighted by the proportion of sales accounted for by the corresponding contexts:

(14)

We also use the context-related brand strength together with the values of the reach of brand b vis a vis all other brands to determine its notional demand index in context s:

(15)

This value is not to be confused with an actual level of demand since the scale of these variable values is determined by the preference distribution parameters and not in any way by the data.

The total brand-demand index is the context-weighted sum of the values of the :

(16)

The market share is then determined by the brand-demand indices as if those indices were actual demands. That is,

(17)

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