﻿ The Exponent (Scale Factor)

# The Exponent (Scale Factor)

The Exponent is used for adjusting the steepness or flatness of the shares of preference (under Randomized First Choice or Share of Preference rules).  It is a positive multiplicative factor that is applied to part-worth utilities prior to conducting market simulations.  The most typical use of the Exponent is to adjust the conjoint predictions to approximate the response error (noise) as found in out-of-sample data, such as real world market shares or differently formatted holdout tasks completed by a different group of respondents.

Assume that you set up a simulation as defined in the previous section with three products: A, B and C.  Also assume that you are simulating the projected choice for just one individual under a Share of Preference model.  Simulation results for this individual might come back as follows:

 Product Share of Choice A 10.8% B 24.0% C 65.2% Total: 100.0%

Note that in choice simulations, the resulting shares are normalized to sum to 100%.  We interpret these results to mean that if this respondent was faced with the choice of A, B, or C, he would have a 10.8% probability of choosing A, a 24.0% probability of choosing B, and a 65.2% probability of choosing C.  Note that B is more than twice as likely to be selected as A, and C is more than twice as likely to be chosen as B.

Let's suppose, however, that the differences in share seen in this simulation are really greater than what we would observe in the real world.  Suppose that random forces come to bear in the actual market (e.g. out-of-stock conditions, satisficing, buyer confusion or apathy) and the shares (probabilities of choice) are really flatter.  We can often tune the results of market simulations using an adjustment factor called the Exponent (controlled by clicking the Exponent icon within the External Effects ribbon group on the My Scenario Settings tab).

The table below shows results for the previous simulation under different settings for the Exponent (0.01, 0.5, 1.0, 2.0, and 5.0):

 Share of Choice under Different Exponent Values 0.01 0.5 1.0 2.0 5.0 Product A 33.0% 20.2% 10.8% 2.4% 0.0% Product B 33.3% 30.1% 24.0% 11.6% 0.7% Product C 33.7% 49.7% 65.2% 86.0% 99.3% Total 100% 100% 100% 100% 100%

The exponent is applied as a multiplicative factor to all of the utility part-worths prior to computing shares.  As the exponent approaches 0 the differences in share are minimized and preference is divided equally among the various product offerings.  As the exponent becomes large, the differences in share are maximized, with nearly all the share allocated to the single best product.  (Given a large enough multiplier, the approach is identical to the First Choice model, with all of the share given to a single product.)

Leading researchers tend to find that respondents make choices in conjoint questionnaires with less error than choices made in the real world, leading to market simulators exhibiting relatively "sharper" share differences.  So, if the Exponent is adjusted, the typical direction is to adjust below 1.0, often to something in the range of 0.3 to 0.8. Exponent adjustments below about 0.2 (for conjoint part-worths estimated via logit-based methods) would seem extreme and point to possible problems in the data (either the part-worth utilities or the holdout judgments being used to tune the exponent).