Utility Models

EMSE 6035: Marketing Analytics for Design Decisions

John Paul Helveston

October 14, 2026

Week 8: Utility Models

1. Utility models

2. Exploring choice data

3. Linear & discrete parameters

4. No choice

5. Team project utility models

Week 8: Utility Models

1. Utility models

2. Exploring choice data

3. Linear & discrete parameters

4. No choice

5. Team project utility models

Random utility model

The utility for alternative $j$ is

\[\tilde{u}_j = v_j + \tilde{\varepsilon}_j\]

$v_j$ = Things we observe (non-random variables)

$\tilde{\varepsilon}_j$ = Things we don’t observe (random variable)

\[\tilde{u}_j = v_j + \tilde{\varepsilon}_j\]

Practice Question 1

A random variable, $\tilde{x}$, has the PDF, $f_{\tilde{x}}(x)$. Write the equation to compute its total probability (hint: think area under the curve!). What is the answer to the equation?

A random variable, $\tilde{x}$, has a uniform distribution between the values 0 and 1. Draw the probability density function (PDF) and Cumulative Density Function (CDF) of $\tilde{x}$.

The value of a random variable, $\tilde{x}$, is determined by rolling one fair, 6-sided dice. Draw the PDF and CDF of $\tilde{x}$.

Logit model: Assume that $\tilde{\varepsilon}_j$ ~ Gumbel Distribution

\[\tilde{u}_j = v_j + \tilde{\varepsilon}_j\]

Probability of choosing alternative $j$:

\[P_j = \frac{e^{v_j}}{\sum_k{e^{v_k}}}\]

Practice Question 2

a) A consumer is making a choice between two bars of chocolate:

Milk chocolate $(m)$
Dark chocolate $(d)$

Assume that we know the observed utility of each bar to be $v_m = 3$ and $v_d = 4$. Using a logit model, compute the probabilities of choosing each bar: $P_m$ and $P_d$.

b) A third bar of chocolate is now added to the choice set. It is the exact same as the milk chocolate bar, but it has a slightly different wrapper (which has no effect on the consumer’s utility). Now, $v_{m1} = v_{m2} = 3$, and $v_d = 4$. Based on the probabilities from question a), what would we expect the probabilities of choosing each bar to be? What probabilities does the logit model produce?

“Observed utility” $(v_j)$ is a weighted sum of attribute values

\[v_j = \beta_1 x_{j}^{\mathrm{A}} + \beta_2 x_j^{\mathrm{B}} + \dots\]

Each $x_j$ is an observable attribute (price, etc.)

We know $x_{j}^{\mathrm{A}}, x_{j}^{\mathrm{B}}, \dots$,
we want to estimate $\beta_1, \beta_2, \dots$

Notation Convention

Continuous: $x_j$

\[u_j = \beta_1 x_{j}^{\mathrm{price}} + \dots\]

#>   price
#> 1     1
#> 2     2
#> 3     3

Discrete: $\delta_j$

\[u_j = \beta_1 \delta_{j}^{\mathrm{ford}} + \beta_2 \delta_{j}^{\mathrm{gm}} \dots\]

#>   brand brand_BMW brand_Ford brand_GM
#> 1  Ford         0          1        0
#> 2    GM         0          0        1
#> 3   BMW         1          0        0

Practice Question 3

Attribute	Bar 1	Bar 2	Bar 3
Price	$1.20	$1.50	$3.00
% Cacao	10%	60%	80%

Write out a model for the observed utility of each chocolate bar in the above set.

If the coefficient for the price attribute was -0.1 and the coefficient for % Cacao attribute was 0.1, what is the difference in the observed utility between bars 3 and 1?

With the addition of the brand attribute, repeat part a.

Attribute	Bar 1	Bar 2	Bar 3
Price	$1.20	$1.50	$3.00
% Cacao	10%	60%	80%
Brand	Hershey	Lindt	Ghirardelli

20:00

Your Turn

Let’s say our utility function is:

\[v_j = \beta_1 x_j^{\mathrm{price}} + \beta_2 x_j^{\mathrm{cacao}} + \beta_3 \delta_j^{\mathrm{hershey}} + \beta_4 \delta_j^{\mathrm{lindt}}\]

And we estimate the following coefficients:

Parameter	Coefficient
$\beta_1$	-0.1
$\beta_2$	0.1
$\beta_3$	-2.0
$\beta_4$	-0.1

What are the expected probabilities of choosing each of these bars using a logit model?

Attribute	Bar 1	Bar 2	Bar 3
Price	$1.20	$1.50	$3.00
% Cacao	10%	60%	80%
Brand	Hershey	Lindt	Ghirardelli

What price would Bar 2 have to be to get a 50% market share?

Week 8: Utility Models

1. Utility models

2. Exploring choice data

3. Linear & discrete parameters

4. No choice

5. Team project utility models

Download the logitr-cars repo from GitHub

Exploring choice data

1. Open the logitr-cars folder in Positron

2. Open code/2.1-explore-data.R

Week 8: Utility Models

1. Utility models

2. Exploring choice data

3. Linear & discrete parameters

4. No choice

5. Team project utility models

Dummy-coded variables

Dummy coding: 1 = “Yes”, 0 = “No”

Data frame with one variable: price

data <- data.frame(
  price = c(10, 20, 30)
)

data

#>   price
#> 1    10
#> 2    20
#> 3    30

Add dummy columns for each price “level”

library(fastDummies)

dummy_cols(data, "price")

#>   price price_10 price_20 price_30
#> 1    10        1        0        0
#> 2    20        0        1        0
#> 3    30        0        0        1

Model price as continuous

$v_j = \beta_1 x^\mathrm{price}$

model <- logitr(
    data = data,
    choice = "choice",
    obsID = "obsID",
    pars = "price"
)

Coef.	Interpretation
β1	how utility changes with increasing price

Model price as discrete

$v_j = \beta_1 \delta^\mathrm{price = 20} + \beta_2 \delta^\mathrm{price = 30}$

model <- logitr(
    data = data,
    choice = "choice",
    obsID = "obsID",
    pars = c("price_20", "price_30")
)

Reference level: price=10

Coef.	Interpretation
β1	utility for price=20 relative to price=10
β2	utility for price=30 relative to price=10

Dummy coding categorical variables with `cbc_encode()`

Standard coding (human-readable)

design |>
  select(altID, powertrain)

#>   altID powertrain
#> 1     1   Gasoline
#> 2     2     Hybrid
#> 3     3   Electric

Dummy coding (what logitr() needs)

design |>
  cbc_encode(coding = "dummy") |>
  select(altID, starts_with("powertrain"))

#>   altID powertrainHybrid powertrainElectric
#> 1     1                0                  0
#> 2     2                1                  0
#> 3     3                0                  1

Rule:
Use dummy_cols() for any data frame (simulated or from a file)
Use cbc_encode() to preview {cbcTools} designs, e.g. with a no choice option

Choosing the reference level

Default: first level is the reference

cbc_encode(
  design,
  coding = "dummy"
)

$v_j = \beta_1 \delta^\mathrm{hybrid} + \beta_2 \delta^\mathrm{electric}$

Coef.	Interpretation
β1	utility of Hybrid relative to Gasoline
β2	utility of Electric relative to Gasoline

Pick a different reference level:

cbc_encode(
  design,
  coding = "dummy",
  ref_levels = list(powertrain = "Electric")
)

$v_j = \beta_1 \delta^\mathrm{gasoline} + \beta_2 \delta^\mathrm{hybrid}$

Coef.	Interpretation
β1	utility of Gasoline relative to Electric
β2	utility of Hybrid relative to Electric

Estimating utility models

1. Open the logitr-cars folder in Positron

2. Open code/3.1-model-mnl.R

20:00

Your Turn

Run the code chunk to read in the data.csv file in the “data” folder, which contains choice observations from chocolate bars with the following attributes:

Attribute	Description
`price`	Price in $
`percent_cacao`	% Cacao (how “dark” the chocolate is)
`crispy_rice`	0 or 1 for if the bar contains crispy rice
`brand`	“Hershey”, “Lindt”, or “Ghirardelli”

Write code to estimate the following utility model:

\[u_j = \beta_1 x_j^{\mathrm{price}} + \beta_2 x_j^{\mathrm{\%cacao}} + \beta_3 \delta_j^{\mathrm{crispy}} + \beta_4 \delta_j^{\mathrm{hershey}} + \beta_5 \delta_j^{\mathrm{lindt}} + \varepsilon_j\]

Write code to plot the change in utility for the price attribute.

Break

05:00

Week 8: Utility Models

1. Utility models

2. Exploring choice data

3. Linear & discrete parameters

4. No choice

5. Team project utility models

Estimating utility models with a No Choice option

1. Open the logitr-cars folder in Positron

2. Open code/4.1-model-no_choice.R

Dummy coding with a No Choice option

The “no choice” rows have NA for every attribute

data <- data %>%
  dummy_cols("powertrain")

powertrain_NA = 1 on the no-choice rows. Do not include it in the model!

Replace the NAs with 0 for the named columns only

data <- data %>%
  mutate(
    across(
      c(
        price,
        fuelEconomy,
        accelTime,
        powertrain_Gasoline,
        powertrain_Hybrid,
        powertrain_Electric
      ),
      ~ replace_na(.x, 0)
    )
  )

Week 8: Utility Models

1. Utility models

2. Exploring choice data

3. Linear & discrete parameters

4. No choice

5. Team project utility models

Simulating choice data

Random choices

data <- cbc_choices(
  design = design
)

Choices according to assumed model

$v_j = -0.7 x_j^{\mathrm{price}} + 0.1 x_j^{\mathrm{fuelEconomy}} - 0.2 x_j^{\mathrm{accelTime}} + 0.5 \delta_j^{\mathrm{hybrid}} - 4 \delta_j^{\mathrm{electric}}$

priors <- cbc_priors(
  profiles = profiles,
  price = -0.7,
  fuelEconomy = 0.1,
  accelTime = -0.2,
  powertrain = c(
    Hybrid = 0.5,
    Electric = -4.0
  )
)
data <- cbc_choices(
  design = design,
  priors = priors
)

Not sure what coefficients to use?

cbc_suggest_priors() proposes a starting point based on your attribute levels

cbc_suggest_priors(profiles)

priors <- cbc_priors(
  profiles = profiles,
  price = -0.065,
  fuelEconomy = 0.065,
  accelTime = -0.325,
  powertrain = c(
    Hybrid = 0.2,
    Electric = -0.4
  )
)

These are only a starting point:

Continuous attributes get small, same-sized effects
Categorical levels are relative to the reference level
Signs are all positive, so flip them where you expect a negative effect (e.g. price)
Adjust using your own domain knowledge

Estimate a choice model

\[v_j = \beta_1 x_j^{\mathrm{price}} + \beta_2 x_j^{\mathrm{fuelEconomy}} + \beta_3 x_j^{\mathrm{accelTime}} + \beta_4 \delta_j^{\mathrm{hybrid}} + \beta_5 \delta_j^{\mathrm{electric}}\]

# Simulated choices use standard coding, so dummy code first
library(fastDummies)
data <- dummy_cols(data, "powertrain")

model <- logitr(
  data = data,
  outcome = "choice",
  obsID = "obsID",
  pars = c(
    "price",
    "fuelEconomy",
    "accelTime",
    "powertrain_Hybrid",
    "powertrain_Electric"
  )
)

20:00

Your Turn

As a team:

Go back to your code from last week where you created your choice questions.
Write out a utility model for your project.
Write code to simulate data according to your utility model - pick some parameter values (see simulate-choices.R).
Write code to estimate a model using your simulated data.

Week 8: Utility Models

1. Utility models

2. Exploring choice data

3. Linear & discrete parameters

4. No choice

5. Team project utility models

Week 8: Utility Models

1. Utility models

2. Exploring choice data

3. Linear & discrete parameters

4. No choice

5. Team project utility models

\[\tilde{u}_j = v_j + \tilde{\varepsilon}_j\]

Practice Question 1

Logit model: Assume that \(\tilde{\varepsilon}_j\) ~ Gumbel Distribution

Practice Question 2

“Observed utility” \((v_j)\) is a weighted sum of attribute values

\[v_j = \beta_1 x_{j}^{\mathrm{A}} + \beta_2 x_j^{\mathrm{B}} + \dots\]

Each \(x_j\) is an observable attribute (price, etc.)

Notation Convention

Practice Question 3

Your Turn

Week 8: Utility Models

1. Utility models

2. Exploring choice data

3. Linear & discrete parameters

4. No choice

5. Team project utility models

Download the logitr-cars repo from GitHub

Exploring choice data

Week 8: Utility Models

1. Utility models

2. Exploring choice data

3. Linear & discrete parameters

4. No choice

5. Team project utility models

Dummy-coded variables

Dummy coding categorical variables with cbc_encode()

Choosing the reference level

Estimating utility models

Your Turn

Break

Week 8: Utility Models

1. Utility models

2. Exploring choice data

3. Linear & discrete parameters

4. No choice

5. Team project utility models

Estimating utility models with a No Choice option

Dummy coding with a No Choice option

Week 8: Utility Models

1. Utility models

2. Exploring choice data

3. Linear & discrete parameters

4. No choice

5. Team project utility models

Simulating choice data

Not sure what coefficients to use?

Estimate a choice model

Your Turn

As a team:

Dummy coding categorical variables with `cbc_encode()`