Heterogeneous Demand Modeling

MGT 100 Week 5

Professor of Marketing and Analytics

University of California, San Diego

SVP-Analytics

GBK Collective

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Segmentation Case Study: Quidel

Leading B2B manufacturer of home pregnancy tests
Tests were quick and reliable
Wanted to enter the B2C HPT market
Market research found 2 segments of equal size;
what were they?

Hint: Both segments were about the same size, and both had very similar distributions of customer demographics

	Segment 1	Segment 2
Segment Name
Product Name
Positioning
Image on Package
Package Size
Location in Store
Product Line Extensions
Price

What does this heterogeneity distribution imply for relevant strategic choices?

A phone shopper first snaps a photo and examines the result. How might the photo quality influence their perception of the phone’s less evaluable features?

Full article | 2022 Pixel Ad

Digital Default Frequencies

Classic study: 95% of MS Word users maintained original default options
Classic study: 81-90% of users don’t use Ctrl+F
Classic study: Randomizing top 2 search results only changed click rates from 42%/8% to 34%/12%
2021 data: Safari has 90% share on iPhone, Chrome (74%) and Samsung Internet (15%) have 89% share on Android
- Chrome always preinstalled on Android, Samsung Internet preinstalled on 58% of Android devices
Why? Defaults are good enough; changing is hard, uncertain; users assume designer knows best; popularity implies utility; possible fear of norm deviation

Figures indicate the importance of understanding digital user needs, not assuming that usage indicates satisfaction, and using experiments to indicate optimal default choices

Het. Demand Models

Heterogeneity by customer attributes
Heterogeneity by segment
Individual-level demand parameters

We’ll code 1 & 2
3 is often best but needs advanced techniques → graduate study

MNL Demand

Recall our MNL market share function $s_{jt}=\frac{e^{x_{jt}\beta-\alpha p_{jt}}}{\sum_{k=1}^{J}{e^{x_{kt}\beta-\alpha p_{kt}}}}$
- Recall that the model can predict how much any change in any product’s x or p would affect all products’ market shares
- What is $\alpha$? What is $\beta$?
What are the model’s main limitations?
- Assumes all customers have the same attribute preferences
- Assumes all customers have same price sensitivity
- IIA: Predictions become unreliable when choice sets change
- Requires exogenous price variation to estimate $\alpha$ (all demand models)
- Assumes iid $\epsilon$ distribution: Convenient but unrealistic

Modeling heterogeneity can alleviate the first 3 and enable better predictions, but not price endogeneity.
However, predictive performance will worsen if we overfit the model. Have you heard of cross-validation?

Mapping Heterogeneity Extensions

Analyst can impose heterogeneity assumptions or seek to learn heterogeneity structure from data (very demanding). Tradeoff depends on empirical reasonableness of modeling assumptions vs. data thickness/sparsity

Heterogeneity by Customer Attributes

Let $w_{it}\sim F(w_{it})$ be observed customer attributes, needs or behaviors that drive demand
$w_{it}$ is often a vector of customer attributes including an intercept
Assume $\beta=\delta w_{it}$ and $\alpha=w_{it} \gamma$ :: $\delta$ & $\gamma$ conformable matrices
- Suppose $x_{jt}$ is 1x5, and $w_{it}$ is 2x1, then $\delta$ would be 5x2
Then $u_{ijt}=x_{jt}\delta w_{it}- w_{it} \gamma p_{jt} +\epsilon_{ijt}$ and

\[s_{jt}=\int \frac{e^{x_{jt}\delta w_{it}- w_{it}\gamma p_{jt} }}{\sum_{k=1}^{J}e^{x_{kt}\delta w_{it}- w_{it}\gamma p_{kt} }} dF(w_{it}) \approx \frac{1}{N_t}\sum_i \frac{e^{x_{jt}\delta w_{it}- w_{it}\gamma p_{jt}}}{\sum_{k=1}^{J}e^{x_{kt}\delta w_{it}- w_{it}\gamma p_{kt}}}\]

We usually approximate this integral with a Riemann sum

We often restrict elements in $\delta$ and $\beta$ to zero, if we think the interaction is unimportant, to avoid overparameterizing the model. What goes into $w_{it}$? What if $dim(x)$ and/or $dim(w)$ is large?

Heterogeneity by Segment

Assume each customer $i=1,...,N_t$ is in exactly 1 of $l=1,...,L$ segments with sizes $\sum_{l=1}^{L}N_{lt}=N_t$
- We will use 3 kmeans segments based on 6 usage variables
Assume preferences are uniform within segments & vary between segments
- Consistent with the definition of segments
Replace $u_{ijt}=x_{jt}\beta-\alpha p_{jt}+\epsilon_{ijt}$ with $u_{ijt}=x_{jt}\beta_l-\alpha_l p_{jt}+\epsilon_{ijt}$
That implies $s_{ljt}=\frac{e^{x_{jt}\beta_l-\alpha_l p_{jt}}}{\sum_{k=1}^{J}e^{x_{kt}\beta_l-\alpha_l p_{kt}}}$ and $s_{jt}=\sum_{l=1}^{L}N_{lt} s_{ljt}$

Alternatively, it is possible to estimate segment memberships, rather than supplying them via kmeans. Pros: We don’t have to pre-specify segment memberships. Cons: Noisy, so we need a lot of data to do this well.

Individual Heterogeneity

Assume $(\alpha_i,\beta_i)\sim F(\Theta)$
- Includes the Hierarchical Bayesian Logit
Then $s_{jt}=\int\frac{e^{x_{jt}\alpha_i-\beta_i p_{jt}}}{\sum_{k=1}^{J}e^{x_{jt}\alpha_i-\beta_i p_{jt}}}dF(\Theta)$
Typically, we assume $F(\Theta)$ is multivariate normal, for convenience, and estimate $\Theta$
- We usually have to approximate the integral, often use Bayesian techniques (MSBA/PhD)
Or, we can estimate every $\alpha_i$ and $\beta_i$, that is very very data intensive
- In theory, we can estimate all $(\alpha_i,\beta_i)$ pairs without $\sim F(\Theta)$ assumption, but requires numerous observations & sufficient variation for each $i$. Most data intensive

How to Choose?

“All models are wrong. Some models are useful” (Box)
“Truth is too complicated to allow anything but approximations.” (von Neumann)
“The map is not the territory” (Box)
“Scientists generally agree that no theory is 100% correct. Thus, the real test of knowledge is not truth, but utility” (Hariri)
No model is ever “correct,” No assumption is ever “true” (why not?)
Model selection: A Judgment Problem
- How do you choose among plausible specifications?
- Involves both model selection — which f() in y=f(x) — and predictor selection

Use modeling purposes and constraints as model selection criteria. Purposes can include prediction, explanation and decision-making. Constraints can include privacy and ethics. What criteria would you prioritize?

Should you model weight=f(height) or height=f(weight)? Notice the many-to-one correspondence and the discrete nature of height measurements.

Model Specification

Bias-variance tradeoff
- Adding predictors always increases model fit
- Yet parsimony often improves predictions
Many criteria drive model selection
- Modeling objectives
- Theoretical properties
- Model flexibility
- Precedents & prior beliefs
- In-sample fit
- Prediction quality
- Computational properties

Imagine you prepared for a quiz by perfectly memorizing the textbook but without understanding the material. Could you accurately apply the concepts in unseen settings?

Choosing a model that maximize a single criterion, such as R-square, can lead to bad decisions.

Here we want to visualize the process of overfitting
Let’s imagine you’re estimating demand at a B2B firm with few customers
Suppose you had 2 data points. You standardize your variables and fit a line. R2 = 1 !!
Then you get another data point. You standardize again and fit a quadratic. R2 = 1 !!
Then you get a 4th data point. Standardize again & fit a cubic. R2 = 1 undefeated
… you see the problem here? … consider # of degrees of freedom & # of parameters
Finally, suppose you had lots of data as in lower-right. Then you might discover the relationship is actually quite weak, and you have been dramatically overestimating the strength of the relationship up until now
Is the true relationship actually changing with your sample size? Or are you just getting more data points?
Source
Another example: Suppose you studied for an exam by memorizing every word of the textbook. Then the professor asked you questions that asked you to apply your understanding of the text. If the only thing you did was return the text contents, you would get the questions wrong. That would be overfitting.

Regularization: Lasso, Ridge & Elastic Net

Adding predictors always improves in-sample fit but risks overfitting. Regularization methods penalize model complexity on the theory that simpler models predict better:

Lasso (L1 penalty): $\min_\beta \sum(y_i - x_i\beta)^2 + \lambda\sum|\beta_k|$
- Can shrink coefficients to exactly zero → automatic variable selection
- Useful when only a few predictors matter
Ridge (L2 penalty): $\min_\beta \sum(y_i - x_i\beta)^2 + \lambda\sum\beta_k^2$
- Shrinks all coefficients toward zero, but seldom to exactly zero
- Useful when many predictors each contribute small effects
Elastic Net (L1 + L2): $\min_\beta \sum(y_i - x_i\beta)^2 + \lambda_1\sum|\beta_k| + \lambda_2\sum\beta_k^2$
- Combines Lasso and Ridge

The tuning parameter $\lambda$ controls penalty strength: higher $\lambda$ → more shrinkage, fewer effective parameters. Cross-validation is typically used to choose $\lambda$.

If you have 100 product attributes but suspect only 10 matter, Lasso identifies which ones while Ridge keeps all 100 with smaller weights. Penalties are assumed as proxies for overfitting, but overfitting is not directly optimized.

How to Evaluate Overfitting?

Retrodiction = “RETROspective preDICTION”
- Quantifies model’s ability to generalize to non-training data
- We can compare different models and different specifications on retrodictive accuracy
We can even train a model to maximize retrodiction quality (“Cross-validation”)
- Most helpful when the model’s purpose is prediction

Be careful not to confuse retrodiction with prediction. Why not?

Cross-Validation Algorithm

General approach to evaluate retrodiction performance and overfitting risk among a set of competing models $m=1,...,M$. Algorithm:

Randomly divide the data into $K$ distinct folds
Hold out fold $k$, use remaining $K-1$ folds to estimate model $m$, then predict outcomes in fold $k$; store prediction errors
Repeat 2 for each $k$
Repeat 2&3 for every model $m$
Retain model $m$ with minimal prediction errors, usually MAPE or MSPE

You estimate every model K times (K=4 in the graphic); each uses a different (K-1)/K proportion of the data
We evaluate each model’s retrodiction quality K times, then average them
When K=N, we call that “leave-one-out” cross-validation
Cross-validation is just one criterion; final model selection also depends on theory, objectives, other criteria

Non-Random Holdout Evaluations

When are models most robust to major changes in the data-generating process?
Non-random holdouts are strong tests, but can only be retrospective

This paper evaluated retrodictive performance of various models of auto production and pricing estimated using pre-2008 data after the 2008 gas price shock. It supported the generalizability of microfounded models. In what scenarios are tests like this most valuable?

Source

Het Demand: Misuse Risks

Customer attributes should reflect differences in customer needs
Customer data should be high quality (GIGO, Errors-in-variables biases)
Use needs to consider qualitative factors {effectiveness, legality, morality, privacy, conspicuousness, equity, reactance}
- Guiding principle (not a rule):
  Using data to legally, genuinely serve customers’ interests is usually OK
- Using private data against customer interest can harm some consumers, break laws, incur liability. One lawsuit can kill a start-up
- Major US laws: COPPA, GLBA, HIPAA, patchwork of state laws

Some Evidence

How does demand model performance depend on heterogeneity specification and training data?

Research Question

Suppose we
1. Train demand model $M$ to predict mayonnaise sales …
2. … using information set $X$ …
3. … & choose targeted discounts for each consumer to maximize firm profits
  - Essentially 3rd-degree price discrimination
Separately, using different data, we nonparametrically estimate how each individual responds to price discounts
- This gives us ground-truth to assess each household’s response to price discount
- But, the nonparametric estimate can’t give counterfactual predictions; we need M for that
How do targeted coupon profits depend on $M$ and $X$?
- We use model M and data X to predict profits of offering targeted price discounts to particular households
- We use ground-truth to calculate household response, then calculate profits across all households
- We’ll also compare to no-discount and always-discount strategies

Optimal Price Targeting (MkSc 2022)

Lil Bit of Theory

For any price discount < contribution margin, giving a targeted discount to…
- … our own brand-loyal customer directly reduces profit
- … a marginal customer may increase profit
- … another brand’s loyal customer does not change profit
So the demand model’s challenge is to distinguish marginal customers from loyal customers
- This research disregards the ‘post-promotion dip’ for simplicity

Information Sets $X$

Base Demographics:
Income, HHsize, Retired, Unemployed, SingleMom
Extra Demographics: Age, HighSchool, College, WhiteCollar, #Kids, Married, #Dogs, #Cats, Renter, #TVs
Purchase History: BrandPurchaseShares, BrandPurchaseCounts, DiscountShare, FeatureShare, DisplayShare, #BrandsPurchased, TotalSpending

Which information set do you expect to be most predictive, and why?

Demand Models $M$

Bayesian Logit models (3)
- Based on utility maximization in which consumers compare utility and price of each available product
- Includes Hierarchical and Pooled versions
Multinomial Logit Regressions (2)
- Estimated via Lasso and Elastic Net to reduce overfitting
Neural Network (2)
- Including single-layer and deep NN
KNN: Nearest-Neighbor Algorithm (1)
Random Forests (2)
- Including standard RF for bagging and XGBoost for boosting

All models trained using 5-fold cross validation.

Optimal Price Targeting (MkSc 2022) | estimation code

How Do We Answer the Question?

Economic criteria:
What profit does each $M$-$X$ combination imply?
- Depends on counterfactual predictions: What if we had selected different customers to receive coupons?
- Quantifies prediction quality in profit terms
Statistical criteria:
How well does each $M$-$X$ fit its training data?
- Generally, what the models are generally trained to maximize

Economic and statistical criteria can be very different
- Doing well on one does not imply doing well on the other
- Which one do we care more about in customer analytics?

Why might these criteria disagree? “Pessimists are often right. Optimists are often wealthy.”

Used the degenerate strategies as benchmarks. What do you see?

Notice first that giving everyone a coupon gives higher profits than giving nobody a coupon, suggesting maybe the firm was pricing too high previously. We can’t typically assume optimal pricing in the marketplace, because demand changes, and most firms don’t estimate demand
Bayesian Hierarchical Logit performs well even without purchase history data, because it constructs a likelihood for every consumer based on observed purchases
Among base demographics only, none of the ML models outperform the blanket coupon
With expanded demographics, ML models come closer to blanket-coupon profits
ML models perform much better when they have better purchase history data to learn from
Bayesian hierarchical logit is not defined for purchase history column because the model’s likelihood essentially builds in-sample purchase history into the posterior through the unobserved heterogeneity distribution; that’s why it does well even when it only has base demos
pooled logit does not allow for unobserved heterogeneity

How did statistical criteria compare to economic criteria?

Past purchases on discount are, by far, the strongest predictor of models’ targeting decisions.

Takeaways

To predict behavior, use past behavior
Economic theory can help demand models to perform well with limited behavioral data
ML model performance depends critically on data quality & abundance. Counterfactual predictions do not always outperform economic models
Statistical performance $\ne$ economic performance

Conjoint Analysis

Generates stated-preference data to estimate heterogeneous demand model, to enable counterfactual predictions and optimal product designs and prices
Probably the most popular quant marketing framework

Choosing Product Attributes

Until now, we studied existing product attributes
- What about choosing new product attribute levels?
- Or what about introducing new products?
Enter conjoint analysis: Attributes are Considered jointly
Survey and model to estimate attribute utilities
- Autos, phones, hardware, durables
- Travel, hospitality, entertainment
- Professional services, transportation
- Consumer package goods
Combines well with cost data to select optimal attributes

Conjoint extends demand modeling from “predict demand for existing products” to “predict demand for products that don’t exist yet.” When would a firm need this capability?

In cluttered categories, conjoint survey designs sometimes seek to replicate the experience of choosing from a typical product shelf. Should Red Bull be in its own refrigerator within the choice task?

Sample Choice Task: Trucks

Conjoint respondents often have trouble deciding when choice tasks become too complex, so we usually limit stated-preference surveys to just a few product attributes. Then we use the stated-choice data to estimate heterogeneous $\beta$ parameters for each level of each attribute.

Conjoint Analysis Algorithm

Identify product attributes and levels/values : These constitute points in your attribute space
- Screen size: 5.5”, 6”, 6.5”, 7”
- Memory: 16 GB, 32 GB, 64 GB, 128 GB, 256 GB
- Price: $199, $399, $599, $799, $999
Recruit consumer participants to make choices
- Recruit a representative sample of your target market
- Offer 8-15 choices among 3-5 hypothetical attribute bundles
Sample from product space, record consumer choices
Specify model, e.g. $u_{jl}=x_{j}\beta_l-\alpha_l p_j+\epsilon_j$ and $s_j=\frac{\sum_{l}x_{j}\beta_l-\alpha_l p_j}{\sum_l\sum_{k}x_{k}\beta_l-\alpha_l p_k}$
Estimate choice model
Combine estimated model with projected cost data to choose product attributes and maximize profits

Let’s fill out a class conjoint about food at the ballpark; use code 920-89

Case Study: UberPOOL

In 2013, Uber hypothesized
- some riders would wait and walk for lower price
- some riders would trade pre-trip predictability for lower price
- shared ridership could ↓ average price and ↑ quantity
- more efficient use of drivers, cars, roads, fuel

UberPOOL white paper

Business Case Was Clear! But …

Shared rides were new for Uber
- Rider/driver matching algo could reflect various tradeoffs
- POOL reduces routing and timing predictability
Uber had little experience with price-sensitive segments
- What price tradeoffs would incentivize new behaviors?
- How much would POOL expand Uber usage vs cannibalize other services?
Coordination costs were unknown
- “I will never take POOL when I need to be somewhere at a specific time”
- Would riders wait at designated pickup points?
- How would communicating costs upfront affect rider behavior?

Uber used Conjoint Analysis to design UberPOOL, so that key product attribute decisions could be informed by customer data.

Approach

23 in-home diverse interviews in Chicago and DC
- Interviewed {prospective, new, exp.} riders to (1) map rider’s regular travel, (2) explore decision factors and criteria, (3) a ride-along for context
- Findings identified 6 attributes for testing
Online Maximum Differentiation Survey
- Selected participants based on city, Uber experience & product; N=3k, 22min

Open-ended interviews revealed key service attributes, then a forced-choice maxdiff measured which ones mattered most. How does this make the Conjoint survey more efficient?

Maxdiff Results

The Maxdiff results showed that walking and number of stops mattered less than arrival time, speed and duration. Do you think UberPOOL would be competing more with Uber or with public transit?

Conjoint Attribute Space

4 ETAs × 3 walking levels × 5 trip lengths × 3 variances × 5 discounts = 900 hypothetical services.

Conjoint Choice Task

Each respondent saw 3 alternatives per choice task, and 7 choice tasks per respondent.