The Exponent (Scale Factor)

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 (described in greater detail later). After clicking Compute!, 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 conjoint 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, 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.

The table below shows results for the previous simulation under different settings for the Exponent:

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.)

If you have solid external information (such as existing market share data) and have reason to expect that conjoint shares should resemble market shares (see earlier assumptions), you may want to tune the exponent within simulations. Or perhaps you have choice data from a holdout choice scenario included in your conjoint survey. You may decide to tune the exponent so that simulated shares resemble these holdout shares. If you do not have solid external information, you probably should not change the exponent from the default value of 1.

Once the exponent is "set" for a data set, one typically does not change it from one simulation to the next.