Setting prices in mixed logit model designs

Abstract

We investigate different procedures to set prices in designs for choice-based conjoint analysis using the mixed logit model which captures latent consumer heterogeneity. Besides discrete attributes, we include a linear price term in the deterministic utility function thereby treating price as continuous variable. We consider two different price intervals and several price sets which contain either two or three prices. We compare these alternatives to set prices by simulating choices for different constellations on the basis of the mixed logit model. Furthermore, we generate ten designs simultaneously instead of just one. Using these simulated choices, we estimate the parameters of the mixed logit model in the next step. To reduce the needed sample size and computation time caused by accounting for latent consumer heterogeneity, we apply Halton draws and set a minimum potential design for prior draws. ANOVA with root mean squared error between estimated and true price coefficient values of individual consumers as dependent variable shows that using more extreme prices as interval bounds and one intermediate price positioned to the right of the interval performs best.

Appendices

Appendix A: Minimum potential design

Let μ ₁,⋯ ,μ ₂₀ be 20 points on a r-dimensional hypersphere around the zero vector with radius 1. The e-th point and f-th point are labeled μ _e and μ _f . Then the Euclidean distance d _{e f} between these two points is computed by

$$ d_{ef}=\sqrt{\sum\limits_{i=1}^{r}(\mu_{ei}-\mu_{fi})^{2}.} $$

(19)

The minimum potential design is now specified as the combination of points μ ₁…μ ₂₀ that have the lowest potential energy E _{p o t} , i.e., the combination where the points are farthest away from each other:

$$ minimize\ E_{pot} = \sum\limits_{e=1}^{19}\sum\limits_{f=e+1}^{20}(d_{ef}^{2}+\frac{1}{d_{ef}}). $$

(20)

We solve this optimization problem by the following heuristic. We draw 100⋅2^r random r-dimensional points, which we standardize, so that these points all are on the hypersphere. This means we draw each point out of the same uniform distribution and divide them by their length. By this means, every point will have the length of 1 and be on a hypersphere with radius 1.

Then we use an interface with R (Baier and Neuwirth 2007) and the function ”cover.design” of R to pick those 20 points out of the generated ones, which have the lowest potential energy and thereby are furthest from each other (Furrer et al. 2012).

Appendix B: Estimation of the mixed logit model

We set initial coefficients b ⁰ to a random value in the interval [$\bar {\beta } - 2 \cdot \sigma , \bar {\beta } + 2 \cdot \sigma$] with $\bar {\beta }$ being the true value of the consumer coefficient and σ=0.1, which is the higher value of factor 2. The initial covariance matrix is diagonal with all diagonal elements being 5. We initialize parameters this way, as we focus on the comparison of design procedures and do not intend to test estimation algorithms. In contrast to the original version of the algorithm in Train (2008) we had take more draws per person as we need person specific coefficients and do not distinguish finite components. Its pseudo-algorithmic description is as follows:

• 1. For each person, ϱ=75r-dimensional random values are drawn from the normal distribution N(b ⁰,V ⁰). The ρ-th draw for person n is labeled as $\hat {\beta }_{n\rho }^{0}$. Note that just like γ is included $\tilde {\beta }$ the draw of the price coefficient, $\hat {\gamma }_{n\rho }$, is included in $\hat {\beta }_{n\rho }^{0}$.

• 2. Let z _{n s} denote the alternative that person n chooses in choice set s. The probability of choice path z _n =(z _n1,z _n2…z _{n S} ) conditional on $\hat {\beta }_{n\rho }^{0}$ is

  $$K_{n}(\hat{\beta}_{n\rho}^{0})=\prod\limits_{s=1}^{S}{\frac{e^{\hat{\beta}_{n\rho}^{0}x_{nz_{ns}s}}}{{\sum}_{j}{e^{\hat{\beta}_{n\rho}^{0}x_{njs}}}}},$$

  where x _{n j s} is the attribute vector including the negative price of alternative j in choice set s in the design given to person n. Now a weight is calculated for each person and each draw as

  $$h_{n\rho}^{0}=\frac{K_{n}(\hat{\beta}_{n\rho}^{0})}{1/\varrho{\sum}_{\rho} K_{n}(\hat{\beta}_{n\rho}^{0})}.$$

• 3. Coefficients and covariances are updated by

  $$b^{1}=\frac{{\sum}_{n}{\sum}_{\rho} h_{n\rho}^{0}\hat{\beta}_{n\rho}^{0}}{{\sum}_{n}{\sum}_{\rho} h_{n\rho}^{0}} $$

  and

  $$V^{1}=\frac{{\sum}_{n}{\sum}_{\rho} h_{n\rho}^{0}[(\hat{\beta}_{n\rho}^{0}-b^{1})(\hat{\beta}_{n\rho}^{0}-b^{1})']}{{\sum}_{n}{\sum}_{\rho} h_{n\rho}^{0}}, $$

  respectively.

• 4. If the absolute difference between previous and updated parameter values is less than 10⁻⁶ or 10,000 iterations have been made, the algorithm stops, otherwise it sets the old values of parameters to their updated values and goes back to step 1.