Choice Models for Product Optimization and Pricing

1. Ask customers to choose among alternative products … something consumers do every day!
2. Fit a model to infer feature and price tradeoffs from observed choices.
3. Predict preference among alternative product designs using the model.

Respondents might answer 8-12 of these types of questions (sometimes more!), which gives us lots of data to infer preferences.

Respondents' choices are modeled as conditional multinomial responses (choice among alternatives), modeling the likelihood to choose a product as a function of its features and price. (We'll see details in a moment.)

Data collection — the survey itself — is commonly done using Sawtooth Software, a survey platform used to design a questionnaire, field it online, and analyze results.

Given the data, simple modeling can be done in Sawtooth Software, while more advanced modeling can be done with packages in R.

Hands-on available!

A more extensive version of the code is at the website for Chapman and Feit (2015), R for Marketing Research and Analytics:

goo.gl/j8oUez == http://r-marketing.r-forge.r-project.org/

You'll want the “Code” tab, and the .R file for Chapter 13.

Data is simulated and comes from Chapman & Feit (2015):

cbc.df <-
  read.csv("http://goo.gl/5xQObB",
           colClasses = c(seat = "factor",
                          price = "factor"))
head(cbc.df[ , c(-4, -5)])

  resp.id ques alt cargo  eng price choice
1       1    1   1   2ft  gas    35      0
2       1    1   2   3ft  hyb    30      0
3       1    1   3   3ft  gas    30      1
4       1    2   1   2ft  gas    30      0
5       1    2   2   3ft  gas    35      1
6       1    2   3   2ft elec    35      0

Outcome variable “choice” is multinomial … choice among options.

xtabs(choice ~ seat, data=cbc.df)

seat
   6    7    8 
1164  854  982 

xtabs(choice ~ eng, data=cbc.df)

eng
elec  gas  hyb 
 608 1444  948 

• Univariate summaries only tell part of the story.
• Although counts show that customers prefer 6-seat minivans and gas engines, it is hard to say whether seats or engines have a stronger influence.
  • What if the 6-seat minivan options tended to have gas engines in the survey?
• As with many multivariate problems, the solution is a regression model (of a special type).

library(mlogit)

Tell mlogit which column is which in our choice data:

cbc.mlogit <-
  mlogit.data(data    = cbc.df, 
              choice  = "choice",
              shape   = "long",   
              varying = 3:6,
              id.var  = "resp.id",
              alt.levels = paste("pos", 1:3))

Choice as a function of the minivan features:

m1 <- mlogit(choice ~ 0 + seat + cargo + eng + price, data = cbc.mlogit)
summary(m1)

Call:
mlogit(formula = choice ~ 0 + seat + cargo + eng + price, data = cbc.mlogit, 
    method = "nr", print.level = 0)

Frequencies of alternatives:
  pos 1   pos 2   pos 3 
0.32700 0.33467 0.33833 

nr method
5 iterations, 0h:0m:0s 
g'(-H)^-1g = 7.84E-05 
successive function values within tolerance limits 

Coefficients :
          Estimate Std. Error  t-value  Pr(>|t|)    
seat7    -0.535280   0.062360  -8.5837 < 2.2e-16 ***
seat8    -0.305840   0.061129  -5.0032 5.638e-07 ***
cargo3ft  0.477449   0.050888   9.3824 < 2.2e-16 ***
enggas    1.530762   0.067456  22.6926 < 2.2e-16 ***
enghyb    0.719479   0.065529  10.9796 < 2.2e-16 ***
price35  -0.913656   0.060601 -15.0765 < 2.2e-16 ***
price40  -1.725851   0.069631 -24.7856 < 2.2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Log-Likelihood: -2581.6

We get estimates of choice by feature level:

Coefficients :
          Estimate Std. Error  t-value  Pr(>|t|)
seat7    -0.535280   0.062360  -8.5837 < 2.2e-16 ***
seat8    -0.305840   0.061129  -5.0032 5.638e-07 ***
cargo3ft  0.477449   0.050888   9.3824 < 2.2e-16 ***
enghyb   -0.811282   0.060130 -13.4921 < 2.2e-16 ***
engelec  -1.530762   0.067456 -22.6926 < 2.2e-16 ***
price35  -0.913656   0.060601 -15.0765 < 2.2e-16 ***
price40  -1.725851   0.069631 -24.7856 < 2.2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

.. and then what?

As a conceptual model:

Choice likelihood = $\frac{utilityOneProduct}{utilityAllProducts}=\frac{e^{oneProductCoefs}}{\sum_{\textrm{all products in set}}{e^{productCoefs}}}$

As a conceptual model:

Choice likelihood = $\frac{utilityOneProduct}{utilityAllProducts}=\frac{e^{oneProductCoefs}}{\sum_{\textrm{all products in set}}{e^{productCoefs}}}$

Formally: If $X_{tj}$ is a vector of attributes of alternative $j$ in choice task $t$, the probability of choosing option $j$ in choice task $t$ is

$P(y_t=j) = \frac{exp(\beta X_{tj})}{\sum_{\textrm{all }j'}{exp(\beta X_{tj'})}}$

where $\beta$ is the vector of estimated coefficients.

Coefficients may be estimated by maximum likelihood estimation (MLE) or Bayesian methods (MCMC sampling). mlogit uses MLE.

This uses a design matrix for feature combinations, plus model coefficients, to estimate share via MNL model:

predict.mnl <- function(model, data) {
  data.model <-
    model.matrix(
      update(model$formula, 0 ~ .),
      data = data)[ , -1]
  utility <- data.model %*% model$coef
  share <- exp(utility) / sum(exp(utility))
  cbind(share, data)
}

Given some defined products (rows 8, 1, 3, etc., from a design matrix):

predict.mnl(m1, new.data)

        share seat cargo  eng price
8  0.44643895    7   2ft  hyb    30
1  0.16497955    6   2ft  gas    30
3  0.12150814    8   2ft  gas    30
41 0.02771959    7   3ft  gas    40
49 0.06030713    6   2ft elec    40
26 0.17904663    7   2ft  hyb    35

Suppose you are designing minivan #8 to compete against the other five. Looks like you have a pretty good design, with estimated 44.6% share of preference.

If we estimate a linear parameter for price, we can compute the average willingness-to-pay for a particular level of an attribute by dividing the coefficient for that level by the price coefficient.

coef(m3)["cargo3ft"] / 
  (-coef(m3)["as.numeric(as.character(price))"] / 1000)

cargo3ft 
2750.601 

On average, customers are willing to pay $2750 more for 3ft of cargo space versus 2ft in these data.

We know that different customers have different preferences. I might prefer 6-seats for my 3-person family, while you prefer the 8-seats.

We do this this using a hierarchical model.

$P(y_t=j) = \frac{exp(\beta_i X_{tj})}{\sum_{\textrm{all }j'}{exp(\beta_i X_{tj'})}}$

where $\beta_i$ is the coefficient vector for each customer, $i$. We assume customers are drawn from a multivariate normal population:

$\beta_i \sim N_k(\mu_\beta, \Sigma_\beta)$

This can also be estimated by MLE or Bayesian methods.

Define which parameters are heterogeneous for respondents:

m1.rpar <- rep("n", length=length(m1$coef))
names(m1.rpar) <- names(m1$coef)
m1.rpar

   seat7    seat8 cargo3ft   enggas   enghyb  price35  price40 
     "n"      "n"      "n"      "n"      "n"      "n"      "n" 

m2.hier <-
  mlogit(choice ~ 0 + seat + eng + cargo + price,
                  data = cbc.mlogit,
                  panel=TRUE, rpar = m1.rpar,
                  correlation = TRUE)

We predict at individual level and aggregate those:

library(MASS)
predict.hier.mnl <- function(model, data, nresp=1000) {
  data.model <-
    model.matrix(update(model$formula, 0 ~ .),
                 data = data)[,-1]
  coef.Sigma <- cov.mlogit(model)
  coef.mu <- m2.hier$coef[1:dim(coef.Sigma)[1]]
  draws <- mvrnorm(n=nresp, coef.mu, coef.Sigma)
  shares <- matrix(NA, nrow=nresp, 
                   ncol=nrow(data))
  for (i in 1:nresp) {
    utility <- data.model %*% draws[i, ]
    share = exp(utility) / sum(exp(utility))
    shares[i, ] <- share
  }
  cbind(colMeans(shares), data)
}

predict.hier.mnl(m2.hier, data=new.data)

   colMeans(shares) seat cargo  eng price
8        0.45369314    7   2ft  hyb    30
1        0.18472196    6   2ft  gas    30
3        0.13321905    8   2ft  gas    30
41       0.01794268    7   3ft  gas    40
49       0.05410889    6   2ft elec    40
26       0.15631428    7   2ft  hyb    35

After individual level estimation, the answers are slightly different. We could further examine prediction by individual.

• These models can be estimated by Bayesian MCMC using the ChoiceModelR package, which builds on the bayesm package.
• In the Bayesian framework, you can also introduce characteristics of the decision makers as covariates to $\beta_i$ in the hierarchy.
  • For instance, you can allow preference for seats to be a function of whether the respondent uses their minivan for carpooling.
• The data formatting for ChoiceModelR is a little bit fussy and doesn't follow other hierarchial modeling packages like lme4.

Everything we've talked about can be used with any observed choices – not just choices from a survey.

• Selection of a subscription plan from a menu
• Purchases of televisions from a website
• Actual car purchases (good luck getting the data!)

You can also combine survey data with observational data (Feit, Beltramo and Feinberg, 2010)

• Feature preference for consumers, B2B customers, developers
• Product prioritization: preferred areas to work on
• Understanding civic engagement (Krontiris, Webb, Chapman: https://research.google.com/pubs/pub44180.html)
• Assessing response to design elements
• Product pricing
• Quantitative attitudinal profiles (Chapman, Webb, Krontiris, forthcoming)

Chris Chapman
Principal Quantitative Experience Researcher, Google
cnchapman+r@gmail.com
@cnchapman

Elea McDonnell Feit
Assistant Professor of Marketing, Drexel University
efeit@drexel.edu
@eleafeit

These slides as an RStudio notebook:
https://goo.gl/vsrenL

This presentation is based on Chapter 13 of Chapman and Feit, R for Marketing Research and Analytics © 2015 Springer.

All code in the presentation is licensed under the Apache License, Version 2.0 (the “License”); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0\ Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an “AS IS” BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License.